Overview
This book collects five related, but independent, sets of documentation:
 The Apalache User Manual
 Apalache Tutorials
 Apalache HOWTOs
 A TLA+ Language Reference Manual
 Guidelines for Idiomatic TLA+
Overview
 Entrylevel Tutorial on the Model Checker
 Tutorial on the Type Checker Snowcat
 Apalache trail tips: how to check your specs faster
 Checking Pluscal specifications
 Symbolic Model Checking
 Specifying temporal properties and understanding counterexamples
Entrylevel Tutorial on the Model Checker
Difficulty: Blue trail – Easy
In this tutorial, we show how to turn an implementation of binary search into a TLA+ specification. This implementation is known to have an outofbounds error, which once existed in Java, see Nearly All Binary Searches and Mergesorts are Broken by Joshua Bloch (2006). Our goal is to write a specification after this implementation, not to write a specification of an abstract binary search algorithm. You can find such a specification and a proof in Proving Safety Properties and Binary search with a TLAPS proof by Leslie Lamport (2019).
This tutorial is written under the assumption that the reader does not have any
knowledge of TLA+ and Apalache. Since we are not diving into protocol and
algorithm specifications too quickly, this is a nice example to start with. We
demonstrate how to use Apalache to find errors that are caused by integer
overflow and the outofbounds error, which is caused by this overflow. We
also show that the same overflow error prevents the algorithm from terminating
in the number of steps that is expected from the binary search. Normally it is
expected that the binary search terminates in log2(n)
steps, where n
is the
length of the search interval.
Sometimes, we refer to the model checker TLC in this text. TLC is another model checker for TLA+ and was introduced in the late 90s. If you are new to TLA+ and want to learn more about TLC, check the TLC project and the TLA+ Video Course by Leslie Lamport. If you are an experienced TLC user, you will find this tutorial helpful too, as it demonstrates the strong points of Apalache.
Related documents
 Tutorial on Snowcat shows how to write type annotations for Apalache.
 TLA+ Cheatsheet in HTML summarizes the common TLA+ constructs. If you prefer a printable version in pdf, check the Summary of TLA+.
Setup
We assume that you have Apalache installed. If not, check the manual page on Apalache installation. The minimal required version is 0.22.0.
We provide all source files referenced in this tutorial as a ZIP archive download. We still recommend that you follow along typing the TLA+ examples yourself.
Running example: Binary search
We are not going to explain the idea of binary search in this tutorial. If you need more context on this, check the Wikipedia page on the Binary search algorithm. Let's jump straight into the Java code that is given in Nearly All Binary Searches and Mergesorts are Broken:
1: public static int binarySearch(int[] a, int key) {
2: int low = 0;
3: int high = a.length  1;
4:
5: while (low <= high) {
6: int mid = (low + high) / 2;
7: int midVal = a[mid];
8:
9: if (midVal < key)
10: low = mid + 1
11: else if (midVal > key)
12: high = mid  1;
13: else
14: return mid; // key found
15: }
16: return (low + 1); // key not found.
17: }
As was found by Joshua Bloch, the addition in line 6 may throw
an out of bounds exception at line 7, due to an integer overflow. This is because low
and high
are signed integers, with a maximum value of 2^31  1
.
However, the sum of two values, each smaller than 2^311
, may be greater than 2^31 1
. If this is the case, low + high
can wrap into a negative number.
This bug was discussed in the TLA+ User Group in 2015. Let's see how TLA+ and Apalache can help us here. A bit of warning: The final TLA+ specification will happen to be longer than the 17 lines above. Don't get disappointed too fast. There are several reasons for that:

TLA+ is not tuned towards one particular class of algorithms, e.g., sequential algorithms.

Related to the previous point, TLA+ and Apalache are not tuned to C or Java programs. A software model checker such as CBMC, Stainless, or Coral would probably accept a shorter program, and it would check it faster. However, if you have a sledgehammer like TLA+, you don't have to learn other languages.

We explicitly state the expected properties of the algorithm to be checked by Apalache. In imperative languages, these properties are usually omitted or written as plaintext comments.

We have to introduce a bit of boilerplate, to make Apalache work.
Step 0: Introducing a template module
Source files for this step: BinSearch0.tla.
TLA+ is built around the concept of a state machine. The specified system
starts in a state that is picked from the set of its initial states. This
set of states is described with a predicate over states in TLA+. This predicate
is usually called Init
. Further, the state machine makes a transition from
the current state to a successor state. These transitions are described with a
predicate over pairs of states (current, successor)
in TLA+. This predicate
is usually called Next
.
We start with the simplest possible specification of a singlestate machine. If we visualize it as a state diagram, it looks like follows:
Let's open a new file called BinSearch0.tla
and type a very minimal module
definition:
 MODULE BinSearch0 
EXTENDS Integers, Sequences, Apalache
Init == TRUE
Next == TRUE
===============================================================================
This module does not yet specify any part of the binary search implementation. However, it contains a few important things:

It imports constants and operators from three standard modules:
Integers
,Sequences
, andApalache
. 
It declares the predicate
Init
. This predicate describes the initial states of our state machine. Since we have not declared any variables, it defines the single possible state. 
It declares the predicate
Next
. This predicate describes the transitions of our state machine. Again, there are no variables andNext == TRUE
, so this transition defines the entire set of states as reachable in a single step.
Now it is a good time to check that Apalache works. Run the following command:
$ apalachemc check BinSearch0.tla
The tool output is a bit verbose. Below, you can see the important lines of the output:
...
PASS #13: BoundedChecker
Step 0: picking a transition out of 1 transition(s)
Step 1: picking a transition out of 1 transition(s)
...
Step 10: picking a transition out of 1 transition(s)
The outcome is: NoError
Checker reports no error up to computation length 10
...
We can see that Apalache runs without finding an error, as expected.
If you are curious, replace TRUE
with FALSE
in either Init
or Next
,
run Apalache again and observe what happens.
It is usually a good idea to start with a spec like BinSearch0.tla
, to ensure
that the tools are working.
Step 1: Introducing specification parameters
Source files for this step: BinSearch1.tla.
Diffs: BinSearch1.tla.patch.
The Java code of binarySearch
accepts two parameters: an array of integers
called a
, and an integer called key
. Similar to these parameters, we introduce
two specification parameters (called CONSTANTS
in TLA+):
 the input sequence
INPUT_SEQ
, and  the element to search for
INPUT_KEY
.
 MODULE BinSearch1 
EXTENDS Integers, Sequences, Apalache
CONSTANTS
\* The input sequence.
\*
\* @type: Seq(Int);
INPUT_SEQ,
\* The key to search for.
\*
\* @type: Int;
INPUT_KEY,
Importantly, the constants INPUT_SEQ
and INPUT_KEY
are prefixed with type
annotations in the comments:
INPUT_SEQ
has the typeSeq(Int)
, that is, it is a sequence of integers (sequences in TLA+ are indexed), andINPUT_KEY
has the typeInt
, that is, it is an integer.
Recall that we wanted to specify signed and unsigned Java integers, which are
32 bit long. TLA+ is not tuned towards any computer architecture. Its integers
are mathematical integers: always signed and arbitrarily large (unbounded).
To model fixed bitwidth integers, we introduce another constant INT_WIDTH
of
type Int
:
\* Bitwidth of machine integers.
\*
\* @type: Int;
INT_WIDTH
The benefit of defining the bit width as a parameter is that we can try
our specification for various bit widths of integers: 4bit, 8bit, 16bit, 32bit,
etc. Similar to many programming languages, we introduce several constant
definitions (a^b
is a
taken to the power of b
):
\* the largest value of an unsigned integer
MAX_UINT == 2^INT_WIDTH
\* the largest value of a signed integer
MAX_INT == 2^(INT_WIDTH  1)  1
\* the smallest value of a signed integer
MIN_INT == 2^(INT_WIDTH  1)
Note that these definitions do not constrain integers in any way. They are simply convenient names for the constants that we will need in the specification.
To make sure that the new specification does not contain syntax errors or type errors, execute:
$ apalachemc check BinSearch1.tla
Step 2: Specifying the base case
Source files for this step: BinSearch2.tla.
Diffs: BinSearch2.tla.patch.
We start with the simplest possible case that occurs in binarySearch
. Namely,
we consider the case where low > high
, that is, binarySearch
never enters
the loop.
Introduce variables. To do that, we have to finally introduce some
variables. Obviously, we have to introduce variables low
and high
. This is
how we do it:
VARIABLES
\* The low end of the search interval (inclusive).
\* @type: Int;
low,
\* The high end of the search interval (inclusive).
\* @type: Int;
high,
The variables low
and high
are called state variables. They define a state of our state machine. That is, they are never introduced and
never removed. Remember, TLA+ is not tuned towards any particular computer
architecture and thus it does not even have the notion of an execution stack.
You can think of low
and high
as being global variables. Yes, global
variables are generally frowned upon in programming languages. However, when dealing with a
specification, they are much easier to reason about than the execution stack.
We will demonstrate how to introduce local definitions later in this tutorial.
We introduce two additional variables, the purpose of which might be less obvious:
\* Did the algorithm terminate.
\* @type: Bool;
isTerminated,
\* The result when terminated.
\* @type: Int;
returnValue
The variable isTerminated
indicates whether our search has terminated. Why do
we even have to introduce it? Because, some computer systems are not designed
with termination in mind. For instance, such distributed systems as the
Internet and Bitcoin are designed to periodically serve incoming requests
instead of producing a single output for a single input.
If we want to specify the Internet or Bitcoin, do we understand what it means for them to terminate?
The variable returnValue
will contain the result of the binary search, when
the search terminates. Recall, there is no execution stack. Hence, we introduce
the variable returnValue
right away. The downside is that we have to do
bookkeeping for this variable.
Initialize variables. Having introduced the variables, we have to initialize them. That is, we want to specify lines 23 of the Java code:
1: public static int binarySearch(int[] a, int key) {
2: int low = 0;
3: int high = a.length  1;
...
17: }
To this end, we change the body of the predicate Init
to the following:
Init ==
low = 0 /\ high = Len(INPUT_SEQ)  1 /\ isTerminated = FALSE /\ returnValue = 0
You probably have guessed, what the above line means. Maybe you are a bit
puzzled about the mountainlike operator /\
. It is called conjunction,
which is usually written as &&
or and
in programming languages. The
important effect of the above expression is that every variable in the
lefthand side of =
is required to have the value specified in the righthand
side of =
^{1}.
As it is hard to fit many expressions in one line, TLA+ offers special syntax
for writing a big conjunction. Here is the standard way of writing Init
(indentation is important):
\* Initialization step (lines 23)
Init ==
/\ low = 0
/\ high = Len(INPUT_SEQ)  1
/\ isTerminated = FALSE
/\ returnValue = 0
The above lines do not deserve a lot of explanation. As you have probably guessed,
Len(INPUT_SEQ)
computes the length of the input sequence.
It is important to know that TLA+ does not impose any
particular order of evaluation for /\
. However, both Apalache and TLC
evaluate some expressions of the form x = e
in the initialization predicate
as assignments. Hence, it is often a good idea to think about /\
as being
evaluated from left to right.
Update variables. Having done all the preparatory work, we are now ready to
specify the behavior in lines 5 and 16 of binarySearch
.
1: public static int binarySearch(int[] a, int key) {
2: int low = 0;
3: int high = a.length  1;
4:
5: while (low <= high) {
...
15: }
16: return (low + 1); // key not found.
17: }
To this end, we redefine Next
as follows:
\* Computation step (lines 516)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614: not implemented yet
UNCHANGED <<low, high, isTerminated, returnValue>>
ELSE \* line 16
/\ isTerminated' = TRUE
/\ returnValue' = (low + 1)
/\ UNCHANGED <<low, high>>
Most likely, you have no problem reading this definition, except for the part
that includes isTerminated'
, returnValue'
, and UNCHANGED
. Recall that a
transition predicate, like Next
, specifies the relation between two states of
the state machine; the current state, the variables of which are referenced by
unprimed names, and the successorstate, the variables of which are referenced
by primed names.
The expression isTerminated' = TRUE
means that only states where
isTerminated
equals TRUE
can be successors of the current state. In
general, isTerminated'
could also depend on the value of isTerminated
, but
here, it does not. Likewise, returnValue' = (low + 1)
means that
returnValue
has the value (low + 1)
in the next state. The expression
UNCHANGED <<low, high>>
is a convenient shortcut for writing low' = low /\ high' = high
. Readers unfamiliar with specification languages might question
the purpose of UNCHANGED
, since in most programming languages variables only
change when they are explicitly changed. However, a transition predicate, like
Next
, establishes a relation between pairs of states. If we were to omit
UNCHANGED
, this would mean that we consider states in which low
and high
have completely arbitrary values as valid successors. This is clearly not
how Java code should behave. To encode Java semantics, we must therefore
explicitly state that low
and high
do not change in this step.
It is important to understand that an expression like returnValue' = (low + 1)
does not immediately update the variable on the lefthand side. Hence,
returnValue
still refers to the value in the state before evaluation of
Next
, whereas returnValue'
refers to the value in the state that is
computed after evaluation of Next
. You can think of the effect of x' = e
being delayed until the whole predicate Next
is evaluated.
Step 2b: Basic checks for the base case
Source files for this step: BinSearch2.tla and MC2_8.tla.
As we discussed, it is a good habit to periodically run the model checker, as you are writing the specification. Even if it doesn't check much, you would be able to catch the moment when the model checker slows down. This may give you a useful hint about changing a few things before you have written too much code.
Let us check BinSearch2.tla
:
$ apalachemc check BinSearch2.tla
If it is your first TLA+ specification, you may be surprised by this error:
...
PASS #13: BoundedChecker
This error may show up when CONSTANTS are not initialized.
Check the manual: https://apalache.informal.systems/docs/apalache/parameters.html
Input error (see the manual): SubstRule: Variable INPUT_SEQ is not assigned a value
...
Apalache complains that we have declared several constants (INPUT_SEQ
,
INPUT_KEY
, and INT_WIDTH
), but we have never defined them.
Adding a model file. The standard approach in this case is either to fix all constants, or to introduce another module that fixes the parameters and instantiates the general specification. Although Apalache supports TLC Configuration Files, for the purpose of this tutorial, we will stick to toolagnostic TLA+ syntax.
To this end, we add a new file MC2_8.tla
with the following contents:
 MODULE MC2_8 
\* an instance of BinSearch2 with all parameters fixed
\* fix 8 bits
INT_WIDTH == 8
\* the input sequence to try
\* @type: Seq(Int);
INPUT_SEQ == << >>
\* the element to search for
INPUT_KEY == 10
\* introduce the variables to be used in BinSearch2
VARIABLES
\* @type: Int;
low,
\* @type: Int;
high,
\* @type: Bool;
isTerminated,
\* @type: Int;
returnValue
\* use an instance for the fixed constants
INSTANCE BinSearch2
===============================================================================
As you can see, we fix the values of all parameters. We are instantiating
the module BinSearch2
with these fixed parameters. Since instantiation
requires all constants and variables to be defined, we copy the variables
definitions from BinSearch2.tla
.
Since we are fixing the parameters with concrete values, MC2_8.tla
looks very
much like a unit test. It's a good start for debugging a few things, but since
our program is entirely sequential, our specification is as good as a unit
test. Later in this tutorial we will show how to leverage Apalache to check
properties for all possible inputs (up to some bound).
Let us check MC2_8.tla
:
$ apalachemc check MC2_8.tla
...
Checker reports no error up to computation length 10
This time Apalache has not complained. This is a good time to stop and think about whether the model checker has told us anything interesting. Kind of. It told us that it has not found any contradictions. But it did not tell us anything interesting about our expectations. Because we have not set our expectations yet!
Step 3: Specifying an invariant and checking it for the base case
Source files for this step: BinSearch3.tla and MC3_8.tla.
Diffs: BinSearch3.tla.patch and MC3_8.tla.patch.
What do we expect from binary search? We can check the Java documentation, e.g., Arrays.java in OpenJDK:
...the return value will be >= 0 if and only if the key is found.
This property is actually quite easy to write in TLA+. First, we
introduce the property that we call ReturnValueIsCorrect
:
\* The property of particular interest is this one:
\*
\* "Note that this guarantees that the return value will be >= 0 if
\* and only if the key is found."
ReturnValueIsCorrect ==
LET MatchingIndices ==
{ i \in DOMAIN INPUT_SEQ: INPUT_SEQ[i] = INPUT_KEY }
IN
IF MatchingIndices /= {}
THEN \* Indices in TLA+ start with 1, whereas the Java returnValue starts with 0
returnValue + 1 \in MatchingIndices
ELSE returnValue < 0
Let us decompose this property into smaller pieces. First, we define the set
MatchingIndices
:
ReturnValueIsCorrect ==
LET MatchingIndices ==
{ i \in DOMAIN INPUT_SEQ: INPUT_SEQ[i] = INPUT_KEY }
With this TLA+ expression we define a local constant called MatchingIndices
that is equal to the set of indices i
in INPUT_SEQ
so that the sequence
elements at these indices are equal to INPUT_KEY
. If this syntax
is hard to parse for you, here is how we could write a similar definition in a
functional programming language (Scala):
val MatchingIndices =
INPUT_SEQ.indices.toSet.filter { i => INPUT_SEQ(i) == INPUT_KEY }
Since the
sequence indices in TLA+ start with 1, we require that returnValue + 1
belongs to MatchingIndices
when MatchingIndices
is nonempty. If
MatchingIndices
is empty, we require returnValue
to be negative.
We can check that the property ReturnValueIsCorrect
is an invariant, that
is, it holds in every state that is reachable from the states specified by Init
via a sequence of transitions specified by Next
:
$ apalachemc check inv=ReturnValueIsCorrect MC3_8.tla
This property is violated in the initial state. To see why, check the file
counterexample1.tla
.
Actually, we only expect this property to hold after the computation terminates,
that is, when isTerminated
equals to TRUE
. Hence, we add the following
invariant:
\* What we expect from the search when it is finished.
Postcondition ==
isTerminated => ReturnValueIsCorrect
Digression: Boolean connectives. In the above code, the operator =>
is
the Classical implication. In general, A => B
is equivalent to IF A THEN B ELSE TRUE
. The implication A => B
is also equivalent to the TLA+
expression ~A \/ B
, which one can read as "not A holds, or B holds". The
operator \/
is called disjunction. As a reminder, here is the standard
truth table for the Boolean connectives, which are no different from the
Boolean logic in TLA+:
A  B  ~A  A \/ B  A /\ B  A => B 

FALSE  FALSE  TRUE  FALSE  FALSE  TRUE 
FALSE  TRUE  TRUE  TRUE  FALSE  TRUE 
TRUE  FALSE  FALSE  TRUE  FALSE  FALSE 
TRUE  TRUE  FALSE  TRUE  TRUE  TRUE 
Checking Postcondition.
Let us check Postcondition
on MC3_8.tla
:
$ apalachemc check inv=Postcondition MC3_8.tla
This property holds true. However, it's a small win, as MC3_8.tla
fixes all
parameters. Hence, we have checked the property just for one data point. In
Step 5, we will check Postcondition
for all sequences admitted by INT_WIDTH
.
Step 4: Specifying the loop (with a caveat)
Source files for this step: BinSearch4.tla and MC4_8.tla.
Diffs: BinSearch4.tla.patch and MC4_8.tla.patch.
We specify the loop of binarySearch
in TLA+ as follows:
\* Computation step (lines 516)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614
LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = mid 1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
ELSE \* line 16
/\ isTerminated' = TRUE
/\ returnValue' = (low + 1)
/\ UNCHANGED <<low, high>>
ELSE \* isTerminated
UNCHANGED <<low, high, returnValue, isTerminated>>
Let's start with these two definitions:
LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
As you have probably guessed, we define two (local) values mid
and midVal
.
The value mid
is the average of low
and high
. The operator \div
is
simply integer division, which is usually written as /
or //
in programming
languages. The value midVal
is the value at the location mid + 1
. Since
the TLA+ sequence INPUT_SEQ
has indices in the range 1..Len(INPUT_SEQ)
,
whereas we are computing zerobased indices, we are adjusting the index by one,
that is, we write INPUT_SEQ[mid + 1]
instead of INPUT_SEQ[mid]
.
Warning: The definitions of mid
and midVal
do not properly reflect the
Java code of binarySearch
. We will fix them later. It is a good exercise
to stop here and think about the source of this imprecision.
The following lines look like ASCII graphics, but by now you should know enough to read them:
LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = mid 1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
These lines are the indented form of \/
for three cases:
 when
midVal < INPUT_KEY
, or  when
midVal > INPUT_KEY
, or  when
midVal = INPUT_KEY
.
We could write these expressions with IFTHENELSE
or even with the TLA+
operator CASE
(see Summary of TLA+). However, we find the disjunctive
form to be the least cluttered, though unusual.
Now we can check the postcondition again:
$ apalachemc check inv=Postcondition MC4_8.tla
The check goes through, but did it do much? Recall, that we fixed INPUT_SEQ
to be the empty sequence << >>
in MC4_8.tla
. Hence, we never enter the loop
we have just specified.
Actually, Apalache gives us a hint that it never tries some of the cases:
...
PASS #13: BoundedChecker
State 0: Checking 1 state invariants
Step 0: picking a transition out of 1 transition(s)
Step 1: Transition #0 is disabled
Step 1: Transition #1 is disabled
Step 1: Transition #2 is disabled
Step 1: Transition #3 is disabled
State 1: Checking 1 state invariants
Step 1: picking a transition out of 1 transition(s)
Step 2: Transition #0 is disabled
Step 2: Transition #1 is disabled
Step 2: Transition #2 is disabled
Step 2: Transition #4 is disabled
Step 2: picking a transition out of 1 transition(s)
...
Digression: symbolic transitions. Internally, Apalache decomposes the
predicates Init
and Next
into independent pieces like Init == Init$0 \/ Init$1
and Next == Next$0 \/ Next$1 \/ Next$2 \/ Next$3
. If you want to see
how it is done, run Apalache with the options writeintermediate
and rundir
:
$ apalachemc check inv=Postcondition writeintermediate=1 rundir=out MC4_8.tla
Check the file out/intermediate/XX_OutTransitionFinderPass.tla
, which contains the
preprocessed specification that has Init
and Next
decomposed. You can find
a detailed explanation in the section on Assignments in Apalache.
Step 5: Checking Postcondition for plenty of inputs
Source files for this step: BinSearch5.tla and MC5_8.tla.
Diffs: BinSearch5.tla.patch and MC5_8.tla.patch.
In this step, we are going to check the invariant Postcondition
for all
possible input sequences and all input keys (for a fixed bitwidth).
Create the file MC5_8.tla
with the following contents:
 MODULE MC5_8 
\* an instance of BinSearch5 with all parameters fixed
EXTENDS Apalache
\* fix 8 bits
INT_WIDTH == 8
\* We do not fix INT_SEQ and INPUT_KEY.
\* Instead, we reason about all sequences with ConstInit.
CONSTANTS
\* The input sequence.
\*
\* @type: Seq(Int);
INPUT_SEQ,
\* The key to search for.
\*
\* @type: Int;
INPUT_KEY
\* introduce the variables to be used in BinSearch5
VARIABLES
\* @type: Int;
low,
\* @type: Int;
high,
\* @type: Bool;
isTerminated,
\* @type: Int;
returnValue
\* use an instance for the fixed constants
INSTANCE BinSearch5
==================
Note that we introduced INPUT_SEQ
and INPUT_KEY
as parameters again. We
cannot check MC5_8.tla
just like that. If we try to check MC5_8.tla
,
Apalache would complain again about a value missing for INPUT_SEQ
.
To check the invariant for all sequences, we will use two advanced features of Apalache: ConstInit predicate and Value generators.
ConstInit. This idiom allows us to initialize CONSTANTS
with a TLA+
formula. Let us introduce the following operator definition in MC5_8.tla
:
ConstInit ==
/\ INPUT_KEY \in Int
\* Seq(Int) is a set of all sequences that have integers as elements
/\ INPUT_SEQ \in Seq(Int)
This is straightforward definition. However, it does not work in Apalache:
$ apalachemc check cinit=ConstInit inv=Postcondition MC5_8.tla
...
MC5_8.tla:39:2239:29: unsupported expression: Seq(_) produces an infinite set of unbounded sequences.
Checker has found an error
...
The issue with our definition of ConstInit
is that it requires the model
checker to reason about the infinite set of sequences, namely, Seq(Int)
.
Interestingly, the model checking does not complain about the expression
INPUT_KEY \in Int
. The reason is that this expression requires the model
checker to reason about one integer, though it ranges over the infinite set of
integers.
Value generators. Fortunately, this problem can be easily circumvented by using Apalache Value generators^{2}.
Let us rewrite ConstInit
in MC5_8.tla
as follows:
ConstInit ==
/\ INPUT_KEY = Gen(1)
/\ INPUT_SEQ = Gen(MAX_INT)
In this new version, we use the Apalache operator Gen
to:
 produce an unrestricted integer to be used as a value of
INPUT_KEY
and  produce a sequence of integers to be used as a value of
INPUT_SEQ
. This sequence is unrestricted, except its length is bounded withMAX_INT
, which is exactly what we need in our case study.
The operator Gen
introduces a data structure of proper type whose size is
bounded with the argument of Gen
. For instance, the type of INPUT_SEQ
is
the sequence of integers, and thus Gen(MAX_INT)
produces an unrestricted
sequence of up to MAX_INT
elements. This sequence is bound to the name
INPUT_SEQ
. For details, see Value generators. This lets Apalache check
all instances of the data structure, without enumerating the instances!
By doing so, we are able to check the specification for all the inputs, when we
fix the bit width. To quickly get feedback from Apalache, we fix INT_WIDTH
to 8 in the model MC5_8.tla
.
If you know propertybased testing, e.g., QuickCheck, Apalache generators are inspired by this idea. In contrast to propertybased testing, an Apalache generator is not randomly producing values. Rather, Apalache simply introduces an unconstrained data structure (e.g., a set, a function, or a sequence) of the proper type. Hence, Apalache is reasoning about all possible instances of this data structure, instead of reasoning about a small set of randomly chosen instances.
Let us check Postcondition
again:
$ apalachemc check cinit=ConstInit inv=Postcondition MC5_8.tla
...
State 2: state invariant 0 violated. Check the counterexample in:
/[a long path]/counterexample1.tla
...
Let us inspect the counterexample:
 MODULE counterexample 
EXTENDS MC5_8
(* Constant initialization state *)
ConstInit == INPUT_KEY = 1 /\ INPUT_SEQ = <<0, 1>>
(* Initial state *)
State0 ==
INPUT_KEY = 1
/\ INPUT_SEQ = <<0, 1>>
/\ high = 1
/\ isTerminated = FALSE
/\ low = 0
/\ returnValue = 0
(* Transition 1 to State1 *)
State1 ==
INPUT_KEY = 1
/\ INPUT_SEQ = <<0, 1>>
/\ high = 1
/\ isTerminated = FALSE
/\ low = 0
/\ returnValue = 0
(* Transition 3 to State2 *)
State2 ==
INPUT_KEY = 1
/\ INPUT_SEQ = <<0, 1>>
/\ high = 1
/\ isTerminated = TRUE
/\ low = 0
/\ returnValue = 1
...
Is it a real issue? It is, but it is not the issue of
the search, rather our invariant Postcondition
is imprecise.
Step 5b: Fixing the postcondition
Source files for this step: BinSearch5.tla and MC5_8.tla.
If we check our source of truth, that is, the Java documentation in Arrays.java in OpenJDK, we will see the following sentences:
The range must be sorted (as by the {@link #sort(int[], int, int)} method)
prior to making this call. If it is not sorted, the results are undefined.
If the range contains multiple elements with the specified value, there is
no guarantee which one will be found.
It is quite easy to add this constraint ^{3}. This is where TLA+ starts to shine:
InputIsSorted ==
\* The most straightforward way to specify sortedness
\* is to use two quantifiers,
\* but that would produce O(Len(INPUT_SEQ)^2) constraints.
\* Here, we write it a bit smarter.
\A i \in DOMAIN INPUT_SEQ:
i + 1 \in DOMAIN INPUT_SEQ =>
INPUT_SEQ[i] <= INPUT_SEQ[i + 1]
...
\* What we expect from the search when it is finished.
PostconditionSorted ==
isTerminated => (~InputIsSorted \/ ReturnValueIsCorrect)
If we check PostconditionSorted
, we do not get any error after 10 steps:
$ apalachemc check cinit=ConstInit inv=PostconditionSorted MC5_8.tla
...
The outcome is: NoError
Checker reports no error up to computation length 10
It takes some time to explore all executions of length up to 10 steps, for all
input sequences of length up to 2^7  1
arbitrary integers. If we think about
it, the model checker managed to crunch infinitely many numbers in several
hours. Not bad.
Exercise. If you are impatient, you can check PostconditionSorted
for the
configuration that has integer width of 4 bits. It takes only a few seconds to
explore all executions.
Instead of checking whether INPUT_SEQ
is sorted in the
postcondition, we could restrict the constant INPUT_SEQ
to be sorted in
every execution. That would effectively move this constraint into the
precondition of the search. Had we done that, we would not be able to observe
the behavior of the search on the unsorted inputs. An important property is
whether the search is terminating on all inputs.
Step 6: Checking termination and progress
Source files for this step: BinSearch6.tla and MC6_8.tla.
Diffs: BinSearch6.tla.patch and MC6_8.tla.patch.
Actually, we do not need 10 steps to check termination for the case INT_WIDTH = 8
. If you recall the complexity of the binary search, it needs
ceil(log2(Len(INPUT_SEQ)))
steps to terminate.
To check this property, we add the number of steps as a variable in
BinSearch6.tla
and in MC6_8.tla
:
VARIABLES
...
\* The number of executed steps.
\* @type: Int;
nSteps
Also, we update Init
and Next
in BinSearch6.tla
as follows:
Init ==
...
/\ nSteps = 0
Next ==
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614
/\ nSteps' = nSteps + 1
/\ LET mid == (low + high) \div 2 IN
...
ELSE \* line 16
/\ isTerminated' = TRUE
/\ returnValue' = (low + 1)
/\ UNCHANGED <<low, high, nSteps>>
ELSE \* isTerminated
UNCHANGED <<low, high, returnValue, isTerminated, nSteps>>
Having nSteps
, we can write the Termination
property:
\* We know the exact number of steps to show termination.
Termination ==
(nSteps >= INT_WIDTH) => isTerminated
Let us check Termination
:
$ apalachemc check cinit=ConstInit inv=Termination MC6_8.tla
...
Checker reports no error up to computation length 10
It took me 0 days 0 hours 0 min 19 sec
Even if did not know precisely complexity of the binary search, we could write a simpler property, which demonstrates progress of the search:
Progress ==
~isTerminated' => (low' > low \/ high' < high)
It takes about 10 seconds to check Progress
as well.
Step 7: Fixedwidth integers
Source files for this step: BinSearch7.tla and MC7_8.tla.
Diffs: BinSearch7.tla.patch and MC7_8.tla.patch.
Do you recall that our specification of the loop had a caveat? Let us have a look at this piece of the specification again:
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614
/\ nSteps' = nSteps + 1
/\ LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = mid 1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
You can see that all arithmetic operations are performed over TLA+ integers, that is, unbounded integers. We have to implement fixedwidth integers ourselves. Fortunately, we do not have to implement the whole set of integer operators, but only the addition over signed integers, which has a potential to overflow. To this end, we have to recall how signed integers are represented in modern computers, see Two's complement. Fortunately, we do not have to worry about an efficient implementation of integer addition. We simply use addition over unbounded integers to implement addition over fixedwidth integers:
\* Addition over fixwidth integers.
IAdd(i, j) ==
\* add two integers with unbounded arithmetic
LET res == i + j IN
IF MIN_INT <= res /\ res <= MAX_INT
THEN res
ELSE \* wrap the result over 2^INT_WIDTH (probably redundant)
LET wrapped == res % MAX_UINT IN
IF wrapped <= MAX_INT
THEN wrapped \* a positive integer, return as is
ELSE \* complement the value to represent it with an unbounded integer
(MAX_UINT  wrapped)
Having defined IAdd
, we replace addition over unbounded integers with IAdd
:
Next ==
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614
/\ nSteps' = nSteps + 1
/\ LET mid == IAdd(low, high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = IAdd(mid, 1)
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = IAdd(mid,  1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
ELSE \* line 16
...
This finally gives us a specification that faithfully represents the Java code
of binarySearch
. Now we can check all expected properties once again:
$ apalachemc check cinit=ConstInit inv=PostconditionSorted MC7_8.tla
...
State 2: state invariant 0 violated.
...
Total time: 2.786 sec
$ apalachemc check cinit=ConstInit inv=Progress MC7_8.tla
...
State 1: action invariant 0 violated.
...
Total time: 2.935 sec
$ apalachemc check cinit=ConstInit inv=Termination MC7_8.tla
...
State 8: state invariant 0 violated.
...
Total time: 39.540 sec
As we can see, all of our invariants are violated. They all demonstrate the issue that is caused by the integer overflow!
Step 8: Checking the boundaries
Source files for this step: BinSearch8.tla and MC8_8.tla.
Diffs: BinSearch8.tla.patch and MC8_8.tla.patch.
As we have seen in Step 7, the cause of all errors in PostconditionSorted
,
Termination
, and Progress
is that the addition low + high
overflows and
thus the expression INPUT_SEQ[mid + 1]
accesses INPUT_SEQ
outside of its
domain.
Why did Apalache not complain about access outside of the domain? Its behavior is actually consistent with Specifying Systems (p. 302):
A function f has a domain DOMAIN f, and the value of f[v] is specified only if v is an element of DOMAIN f.
Hence, Apalache returns some value of a proper type, if v
is outside of
DOMAIN f
. As we have seen in Step 7, such a value would usually show up in a
counterexample. In the future, Apalache will probably have an automatic check
for outofdomain access. For the moment, we can write such a check as a state
invariant. By propagating the conditions from INPUT_SEQ[mid + 1]
up in Next
,
we construct the following invariant:
\* Make sure that INPUT_SEQ is accessed within its bounds
InBounds ==
LET mid == IAdd(low, high) \div 2 IN
\* collect the conditions of IFTHENELSE
~isTerminated =>
((low <= high) =>
(mid + 1) \in DOMAIN INPUT_SEQ)
Apalache finds a violation of this invariant in a few seconds:
$ apalachemc check cinit=ConstInit inv=InBounds MC8_8.tla
...
State 1: state invariant 0 violated.
...
Total time: 3.411 sec
If we check counterexample1.tla
, it contains the following values for low
and high
:
State0 ==
/\ high = 116
/\ low = 0
...
State1 ==
/\ high = 116
/\ low = 59
...
In state 1, we have low + high = 116 + 59 > 2^7
. Since we have INT_WIDTH = 8
, we have IAdd(116, 59) = 81
.
Step 9: Fixing the access bug
Source files for this step: BinSearch9.tla and MC9_8.tla.
Diffs: BinSearch9.tla.patch and MC9_8.tla.patch.
Let us apply the fix that was proposed by Joshua Bloch in Nearly All Binary
Searches and Mergesorts are Broken. Namely, we update this line of
BinSearch8.tla
:
/\ LET mid == IAdd(low, high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
The fix is as follows:
/\ LET mid == IAdd(low, IAdd(high, low) \div 2) IN
LET midVal == INPUT_SEQ[mid + 1] IN
We also update InBounds
as follows:
\* Make sure that INPUT_SEQ is accessed within its bounds
InBounds ==
LET mid == IAdd(low, IAdd(high, low) \div 2) IN
...
Now we check the four invariants: InBounds
, PostconditionSorted
,
Termination
, and Progress
.
$ apalachemc check cinit=ConstInit inv=InBounds MC9_8.tla
...
The outcome is: NoError
...
Total time: 76.352 sec
$ apalachemc check cinit=ConstInit inv=Progress MC9_8.tla
...
The outcome is: NoError
...
Total time: 63.578 sec
$ apalachemc check cinit=ConstInit inv=Termination MC9_8.tla
...
The outcome is: NoError
...
Total time: 72.682 sec
$ apalachemc check cinit=ConstInit inv=PostconditionSorted MC9_8.tla
...
The outcome is: NoError
...
Total time: 2154.646 sec
Exercise: It takes quite a bit of time to check PostconditionSorted
.
Change INT_WIDTH
to 6 and check all these invariants once again. Observe that
it takes Apalache significantly less time.
Exercise: Change INT_WIDTH
to 16 and check all these invariants once again.
Observe that it takes Apalache significantly more time.
Step 10: Beautifying the specification
Source files for this step: BinSearch10.tla and MC10_8.tla.
Diffs: BinSearch10.tla.patch and MC10_8.tla.patch.
We have reached our goals: TLA+ and Apalache helped us in finding the access bug and showing that its fix works. Now it is time to look back at the specification and make it easier to read.
Let us have a look at our definition of Next
:
\* Computation step (lines 516)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN \* lines 614
/\ nSteps' = nSteps + 1
/\ LET mid == IAdd(low, IAdd(high, low) \div 2) IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = IAdd(mid, 1)
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = IAdd(mid,  1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
ELSE \* line 16
/\ isTerminated' = TRUE
/\ returnValue' = (low + 1)
/\ UNCHANGED <<low, high, nSteps>>
ELSE \* isTerminated
UNCHANGED <<low, high, returnValue, isTerminated, nSteps>>
Next
contains a massive expression. We can decompose it nicely in smaller
pieces:
\* loop iteration
LoopIteration ==
/\ ~isTerminated
/\ low <= high \* lines 614
/\ nSteps' = nSteps + 1
/\ LET mid == IAdd(low, IAdd(high, low) \div 2) IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 910
/\ low' = IAdd(mid, 1)
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 1112
/\ high' = IAdd(mid,  1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 1314
/\ returnValue' = mid
/\ isTerminated' = TRUE
/\ UNCHANGED <<low, high>>
\* loop termination
LoopExit ==
/\ ~isTerminated \* line 16
/\ low > high
/\ isTerminated' = TRUE
/\ returnValue' = (low + 1)
/\ UNCHANGED <<low, high, nSteps>>
\* instead of terminating the computation, we keep variables unchanged
StutterOnTermination ==
/\ isTerminated
/\ UNCHANGED <<low, high, returnValue, isTerminated, nSteps>>
\* Computation step (lines 516)
Next ==
\/ LoopIteration
\/ LoopExit
\/ StutterOnTermination
The definitions LoopIteration
, LoopExit
, and StutterOnTermination
are
called actions in TLA+. It is usually a good idea to decompose a large Next
formula into actions. Normally, an action contains assignments to all primed
variables.
Discussion
The final specifications can be found in BinSearch10.tla and MC10_8.tla.
In this tutorial we have shown how to:
 Specify the behavior of a sequential algorithm (binary search).
 Specify invariants that check safety and termination.
 Take into account the specifics of a computer architecture (fixed bit width).
 Automatically find examples of simultaneous invariant violation.
 Efficiently check the expected properties against our specification.
We have written our specification for parameterized bit width. This lets us
check the invariants relatively quickly and get fast feedback from the model
checker. We chose a bit width of 8, a nontrivial value for which
Apalache terminates within reasonable time. Importantly, the specification
for the bit width of 32 stays the same; we only have to change INT_WIDTH
. Of
course, Apalache reaches its limits when we set INT_WIDTH
to 16 or 32. In
these cases, it has to reason about all sequences of length up to 32,767
elements or 2 Billion elements, respectively!
Apalache gives us a good idea whether the properties of our binary search specification hold true. However, it does not give us an ultimate proof of correctness for Java integers. If you need such a proof, you should probably use TLAPS. Check the paper on Proving Safety Properties by Leslie Lamport.
This tutorial is rather long. This is because we wanted to show the evolution of a TLA+ specification, as we were writing it and checking it with Apalache. There are many different styles of writing TLA+ specifications. One of our goals was to demonstrate the incremental approach to specification writing. In fact, this approach is not very different from incremental development of programs in the spirit of Testdriven development. It took us 23 hours to iteratively develop a specification that is similar to the one demonstrated in this tutorial.
This tutorial touches upon the basics of TLA+ and Apalache. For instance, we did not discuss nondeterminism, as our specification is entirely deterministic. We will demonstrate advanced features in future tutorials.
If you are experiencing a problem with Apalache, feel free to open an issue or drop us a message on Zulip chat.
Tutorial on the Snowcat❄🐱 Type Checker
Difficulty: Blue trail – Easy
Revision: August 24, 2022
In this tutorial, we introduce the Snowcat ❄ 🐱 type checker. We give concrete steps on running the type checker and annotating a specification with types.
This tutorial uses Type System 1.2, which guarantees safe record access. To see how to upgrade to Type System 1.2, check Migrating to Type System 1.2.
Related documents
 ADR002 that introduces Type Systems 1 and 1.2, used by Snowcat.
 A more technical HOWTO on writing type annotations.
 ADR004 that introduces Javalike annotations in TLA+ comments.
Setup
We assume that you have Apalache installed. If not, check the manual page on Apalache installation. The minimal required version is 0.29.0.
Running example: Lamport's mutex
As a running example, we are using the specification of Lamport's Mutex
(written by Stephan Merz). We recommend to reproduce the steps in this
tutorial. So, go ahead and download the specification file
LamportMutex.tla. We will add type annotations to this file. Let's rename
LamportMutex.tla
to LamportMutexTyped.tla
.
Step 1: Running Snowcat
Before we start writing any type annotations, let's run the type checker and see, whether it complains:
$ apalachemc typecheck LamportMutexTyped.tla
The tool output is a bit verbose. Below, you can see the important lines of the output:
...
PASS #1: TypeCheckerSnowcat
> Running Snowcat .::.
Typing input error: Expected a type annotation for VARIABLE clock
...
Step 2: Annotating variables clock, req, ack, crit
In Step 1, Snowcat complained about the name clock
. The reason
for that is very simple: clock
is declared as a variable, but Snowcat
does not find a type annotation for it.
The comment next to the declaration of clock
does not tell us precisely
what clock
should be:
clock, \* local clock of each process
Let's dig a bit further and check TypeOK
, which is usually a good source of
type hints in untyped specifications:
TypeOK ==
(* clock[p] is the local clock of process p *)
/\ clock \in [Proc > Clock]
(* req[p][q] stores the clock associated with request from q received by p, 0 if none *)
/\ req \in [Proc > [Proc > Nat]]
(* ack[p] stores the processes that have ack'ed p's request *)
/\ ack \in [Proc > SUBSET Proc]
(* network[p][q]: queue of messages from p to q  pairwise FIFO communication *)
/\ network \in [Proc > [Proc > Seq(Message)]]
(* set of processes in critical section: should be empty or singleton *)
/\ crit \in SUBSET Proc
This is better, but we have to figure out the types of Proc
and Clock
.
Let's have a look at their definitions.
Proc == 1 .. N
Clock == Nat \ {0}
From the definitions of Proc
and Clock
, it is clear that they are both
subsets of integers. We could annotate these two definitions with the type
Set(Int)
, but this is not necessary, since Snowcat will infer these types
itself.
Together with TypeOK
, this gives us enough information to annotate all but
one variable (we will annotate the variable network
later):
VARIABLES
\* @type: Int > Int;
clock, \* local clock of each process
\* @type: Int > (Int > Int);
req, \* requests received from processes (clock transmitted with request)
\* @type: Int > Set(Int);
ack, \* acknowledgements received from processes
\* @type: Set(Int);
crit, \* set of processes in critical section
In our type annotations:

clock
belongs to the set[Proc > Clock]
. Hence, it is a function of integers to integers, that is,Int > Int
. 
req
belongs to the set[Proc > [Proc > Nat]]
. Hence, it is a function of integers to functions of integers to integers, that is,Int > (Int > Int)
. 
ack
belongs to the set[Proc > SUBSET Proc]
. Hence, it is a function of integers to sets of integers, that is,Int > Set(Int)
. 
crit
is a subset ofProc
, so it is a set of integers, that is,Set(Int)
.
Note: We place the annotation for clock
between the keyword VARIABLES
and clock
, not before the keyword VARIABLES
. Similarly, we added a type
annotation immediately above every other variable name.
We used the oneline TLA+ comment for clock
:
\* @type: Int > Int;
Alternatively, we could use the multiline comment:
(*
Clock is a function of integers to integers.
@type: Int > Int;
*)
Note: Importantly, every type annotation must end with a semicolon ;
.
Let's run the type checker again:
$ apalachemc typecheck LamportMutexTyped.tla
...
Typing input error: Expected a type annotation for VARIABLE network
Not surprisingly, the type checker tells us that we still have to annotate the
variable network
.
Step 3: Annotating the variable network
Let's have a look at the operator TypeOK
again:
TypeOK ==
(* clock[p] is the local clock of process p *)
/\ clock \in [Proc > Clock]
(* req[p][q] stores the clock associated with request from q received by p, 0 if none *)
/\ req \in [Proc > [Proc > Nat]]
(* ack[p] stores the processes that have ack'ed p's request *)
/\ ack \in [Proc > SUBSET Proc]
(* network[p][q]: queue of messages from p to q  pairwise FIFO communication *)
/\ network \in [Proc > [Proc > Seq(Message)]]
(* set of processes in critical section: should be empty or singleton *)
/\ crit \in SUBSET Proc
From this we can see that network
is a function of integers to a function of
integers to a sequence of messages. So its type should look like:
Int > (Int > Seq(/* message type */))
But what is the message type? To find out, we have to continue our archaeology
trip and check the definition of Message
and related operators:
ReqMessage(c) == [type > "req", clock > c]
AckMessage == [type > "ack", clock > 0]
RelMessage == [type > "rel", clock > 0]
Message == {AckMessage, RelMessage} \union {ReqMessage(c) : c \in Clock}
From these four definitions, we can see that Messages
is a set of records
that have two fields: the field type
that should be a string, and the field
clock
that should be an integer. In Apalache, we write such a record type as:
{ type: Str, clock: Int }
Hence, the type of network
should be:
Int > (Int > Seq({ type: Str, clock: Int }))
We could write it as above, but that type is a bit hard to read. Hence, we
split it into two parts: the type alias message
that defines the type of
messages, and the type of network
that refers to the type alias message
.
This can be done in the following way:
\* @typeAlias: message = {
\* type: Str,
\* clock: Int
\* };
\* @type: Int > (Int > Seq($message));
network \* messages sent but not yet received
Note: We are lucky that ReqMessage
, AckMessage
, and RelMessage
are
producing records of the same shape. In some specifications, the shapes of
records differ, while these records should be added to the same set. This is a
bit problematic for the type checker, as it expects set elements to have the
same type. In this case, we have three options:

Slightly rewrite the specification to homogenize records,

Partition the set of messages into several sets, see Idiom 15, or

Use variants. This is a more advanced topic, see the HOWTO on writing type annotations.
Now we should run the type checker again:
$ apalachemc typecheck LamportMutexTyped.tla
...
Typing input error: Expected a type annotation for CONSTANT maxClock
The type checker is still not happy: We have not annotated CONSTANTS
.
Step 4: Annotating constants
Now we have to figure out the types of the constants: N
and maxClock
. This
is fairly easy, as these constants are accompanied by the two assumptions:
ASSUME NType == N \in Nat
ASSUME maxClockType == maxClock \in Nat
From these assumptions, we can conclude that both N
and maxClock
are
integers. We add the type annotations:
CONSTANT
\* @type: Int;
N,
\* @type: Int;
maxClock
Let's run the type checker once again:
$ apalachemc typecheck LamportMutexTyped.tla
...
> All expressions are typed
Type checker [OK]
Snowcat is happy, and so we are too!
Discussion
To see the complete code, check LamportMutexTyped.tla. We have added seven type annotations for 184 lines of code. Not bad.
It was relatively easy to figure out the types of constants and variables in our example, though it required some exploration of the specification.
As a rule, you always have to annotate constants and variables with types.
Hence, we did not have to run the type checker seven times to see the error messages. The type annotations are useful on its own, since we do not have to traverse the spec to figure out the types of constants and states. Our more engineeringoriented peers find these annotations to be quite important.
Sometimes, the type checker cannot find a unique type of an expression. This usually happens when you declare an operator of a parameter that can be: a function, a tuple, a record, or a sequence (or a subset of these four types that has at least two elements). For instance, here is a definition from GameOfLifeTyped.tla:
Pos ==
{ <<x, y>>: x, y \in 1..N }
Although it is absolutely clear that x
and y
have the type Int
,
the type of <<x, y>>
is ambiguous. This expression can either be
a tuple <<Int, Int>>
, or a sequence Seq(Int)
. In this case, we have to
help the type checker by annotating the operator definition:
\* @type: () => Set(<<Int, Int>>);
Pos ==
{<<x, y>>: x, y \in 1..N}
Since it is common to have operators that take no arguments, Snowcat supports the following syntax sugar for operators without arguments:
\* @type: Set(<<Int, Int>>);
Pos ==
{<<x, y>>: x, y \in 1..N}
Further reading
For more advanced type annotations, check the following examples:
We have not discussed type aliases, which are a more advanced feature of the type checker. To learn about type aliases, see HOWTO on writing type annotations.
If you are experiencing a problem with Snowcat, feel free to open an issue or drop us a message on Zulip chat.
Tutorial on checking PlusCal specifications with Apalache
Difficulty: Blue trail – Easy
In this short tutorial, we show how to annotate a PlusCal specification of the Bakery algorithm, to check it with Apalache. In particular, we check mutual exclusion by bounded model checking (which considers only bounded executions). Moreover, we automatically prove mutual exclusion for unbounded executions by induction.
We only focus on the part related to Apalache. If you want to understand the Bakery algorithm and its specification, check the comments in the original PlusCal specification.
Setup
We assume that you have Apalache installed. If not, check the manual page on Apalache installation. The minimal required version is 0.22.0.
We provide all source files referenced in this tutorial as a ZIP archive download. We still recommend that you follow along typing the TLA+ examples yourself.
Running example: Bakery
We start with the PlusCal specification of the Bakery algorithm. This specification has been checked with the model checker TLC. Moreover, Leslie Lamport has proved safety of this algorithm with the TLAPS.
Step 0: Remove the TLAPS proof
Since we are not interested in the TLAPS proof, we copy Bakery.tla to BakeryWoTlaps.tla and modify it as follows:

Remove
TLAPS
from the list of extended modulesEXTENDS Naturals

Remove the theorem and its proof:
THEOREM Spec => []MutualExclusion ...
Step 1: Add a module with type annotations
Let us check the types of BakeryWoTlaps.tla with Apalache:
$ apalachemc typecheck BakeryWoTlaps.tla
...
Typing input error: Expected a type annotation for VARIABLE max
The type checker complains about missing type annotations. See the Tutorial on Snowcat for details. When we try to add type annotations to the variables, we run into an issue. Indeed, the variables are declared with the PlusCal syntax:
algorithm Bakery
{ variables num = [i \in Procs > 0], flag = [i \in Procs > FALSE];
fair process (p \in Procs)
variables unchecked = {}, max = 0, nxt = 1 ;
The most straightforward approach would be to add type annotations directly in the PlusCal code. As reported in Issue 1412, this does not work as expected, as the PlusCal translator erases the comments.
A simple solution is to add type annotations directly to the declarations in
the generated TLA+ code. However, this solution is fragile. If we change the
PlusCal code, our annotations will get overridden. We propose another solution
that is stable under modification of the PlusCal code. To this end, we
introduce a new module called BakeryTyped.tla
with the following contents:
 MODULE BakeryTyped 
CONSTANT
\* @type: Int;
N
VARIABLES
\* @type: Int > Int;
num,
\* @type: Int > Bool;
flag,
\* @type: Int > Str;
pc,
\* @type: Int > Set(Int);
unchecked,
\* @type: Int > Int;
max,
\* @type: Int > Int;
nxt
ConstInit4 ==
N = 4
INSTANCE BakeryWoTlaps
==============================================================================
Due to the semantics of INSTANCE
, the constants and variables declared in
BakeryTyped.tla
substitute the constants and variables of
BakeryWoTlaps.tla
. By doing so we effectively introduce type annotations.
Since we introduce a separate module, any changes in the PlusCal code do not
affect our type annotations.
Additionally, we add a constant initializer ConstInit4
, which we will use
later. See the manual section about the ConstInit predicate for a detailed
explanation.
Step 2: Annotate the operator \prec
Let us run the type checker against BakeryTyped.tla
:
$ apalachemc typecheck BakeryTyped.tla
...
[BakeryWoTlaps.tla:66:1766:20]: Cannot apply a to the argument 1 in a[1].
...
The type checker complains about types of a
and b
in the operator \prec
:
a \prec b == \/ a[1] < b[1]
\/ (a[1] = b[1]) /\ (a[2] < b[2])
The issue is that the type checker is not able to decide whether a
and b
are functions, sequences, or tuples. We help the type checker by adding type
annotations to the operator \preceq
.
\* A type annotation introduced for Apalache:
\*
\* @type: (<<Int, Int>>, <<Int, Int>>) => Bool;
a \prec b == \/ a[1] < b[1]
\/ (a[1] = b[1]) /\ (a[2] < b[2])
When we run the type checker once again, it computes all types without any problem:
$ apalachemc typecheck BakeryTyped.tla
...
Type checker [OK]
Note that our annotation of \preceq
would not get overwritten, when we update
the PlusCal code. This is because \preceq
is defined in the TLA+ section.
Step 3: Checking mutual exclusion for bounded executions
Once we have annotations, we run Apalache to check the property of mutual exclusion for four processes and executions of length up to 10 steps:
$ apalachemc apalachemc check \
cinit=ConstInit4 inv=MutualExclusion BakeryTyped.tla
...
It took me 0 days 0 hours 32 min 2 sec
Apalache reports no violation of MutualExclusion
. This is a good start.
However, since Apalache only analyzes executions that make up to 10
transitions by default, this analysis is incomplete.
Step 4: Checking mutual exclusion for unbounded executions
To analyze executions of arbitrary length with Apalache, we can check an inductive invariant. For details, see the section on Checking inductive invariants. The Bakery specification contains such an invariant written by Leslie Lamport:
(***************************************************************************)
(* Inv is the complete inductive invariant. *)
(***************************************************************************)
Inv == /\ TypeOK
/\ \A i \in Procs :
/\ (pc[i] \in {"e4", "w1", "w2", "cs"}) => (num[i] # 0)
/\ (pc[i] \in {"e2", "e3"}) => flag[i]
/\ (pc[i] = "w2") => (nxt[i] # i)
/\ pc[i] \in {(*"e2",*) "w1", "w2"} => i \notin unchecked[i]
/\ (pc[i] \in {"w1", "w2"}) =>
\A j \in (Procs \ unchecked[i]) \ {i} : Before(i, j)
/\ /\ (pc[i] = "w2")
/\ \/ (pc[nxt[i]] = "e2") /\ (i \notin unchecked[nxt[i]])
\/ pc[nxt[i]] = "e3"
=> max[nxt[i]] >= num[i]
/\ (pc[i] = "cs") => \A j \in Procs \ {i} : Before(i, j)

To prove that Inv
is an inductive invariant for N = 4
, we run Apalache
twice. First, we check that the initial states satisfy the invariant Inv
:
$ apalachemc apalachemc check cinit=ConstInit4 \
init=Init inv=Inv length=0 BakeryTyped.tla
...
The outcome is: NoError
It took me 0 days 0 hours 0 min 6 sec
Second, we check that for every state that satisfies Inv
, the following
holds true: Its successors via Next
satisfy Inv
too. This is done as
follows:
$ apalachemc apalachemc check cinit=ConstInit4 \
init=Inv inv=Inv length=1 BakeryTyped.tla
...
The outcome is: NoError
It took me 0 days 0 hours 0 min 28 sec
Now we know that Inv
is indeed an inductive invariant. Hence, we check
the property MutualExclusion
against the states that satisfy Inv
:
$ apalachemc apalachemc check cinit=ConstInit4 \
init=Inv inv=MutualExclusion length=0 BakeryTyped.tla
...
The outcome is: NoError
It took me 0 days 0 hours 0 min 9 sec
In particular, these three results allow us to conclude that MutualExclusion
holds for all states that are reachable from the initial states (satisfying
Init
) via the available transitions (satisfying Next
). Since we have fixed
the constant N
with the predicate ConstInit4
, this result holds true for N = 4
. If you want to check MutualExclusion
for other values of N
, you can
define a predicate similar to ConstInit4
. We cannot check the invariant for
all values of N
, as this would require Apalache to reason about unbounded
sets and functions, which is currently not supported.
Dealing with the define block
PlusCal specifications may contain the special defineblock. For example:
 MODULE CountersPluscal 
(*
Pluscal code inside TLA+ code.
algorithm Counters {
variable x = 0;
define {
\* This is TLA+ code inside the PlusCal code.
IsPositive(x) == x > 0
}
...
}
*)
================================
Unfortunately, the PlusCal transpiler erases comments when translating PlusCal code to TLA+. Hence, the simplest solution is to move the defineblock outside the PlusCal code. For example:
 MODULE CountersPluscal 
\* @type: Int => Bool;
IsPositive(x) == x > 0
(*
algorithm Counters {
variable x = 0;
...
}
*)
================================
Conclusion
The final specifications can be found in BakeryTyped.tla and BakeryWoTlaps.tla.
In this tutorial we have shown how to:
 Annotate a PlusCal spec with types by introducing an additional TLA+ module.
 Check safety of Bakery for bounded executions by bounded model checking (for
N=4
).  Check safety of Bakery for unbounded executions by invariant checking (for
N=4
).
If you are experiencing a problem with Apalache, feel free to open an issue or drop us a message on Zulip chat.
Further reading
 Entrylevel Tutorial on the Model Checker
 Tutorial on Snowcat shows how to write type annotations for Apalache.
 TLA+ Cheatsheet in HTML summarizes the common TLA+ constructs. If you prefer a printable version in pdf, check the Summary of TLA+.
Apalache trail tips: how to check your specs faster
Difficulty: Red trail – Medium
This tutorial collects tips and tricks that demonstrate the strong sides of Apalache.
Tip 1: Use TLA+ constructs instead of explicit iteration
The Apalache
antipatterns mention that
one should not use explicit iteration (e.g., ApaFoldSet
and
ApaFoldSeqLeft
), unless it is really needed. In
this tip, we present a concrete example that demonstrates how explicit
iteration slows down Apalache.
In our example, we model a system of processes from a set Proc
that are
equipped with individual clocks. These clocks may be completely unsynchronized.
However, they get updated uniformly, that is, all clocks have the same speed.
Let's have a look at the first part of this specification:
 MODULE FoldExcept 
(*
* This specification measures performance in the presence of an antipattern.
*)
EXTENDS Integers, Apalache
CONSTANT
\* A fixed set of processes
\*
\* @type: Set(Str);
Proc
VARIABLES
\* Process clocks
\*
\* @type: Str > Int;
clocks,
\* Drift between pairs of clocks
\*
\* @type: <<Str, Str>> > Int;
drift
As we can see, the constant Proc
is a specification parameter. For instance,
it can be equal to { "p1", "p2", "p3" }
. The variable clocks
assigns a
clock value to each process from Proc
, whereas the variable drift
collects
the clock difference for each pair of processes from Proc
. This relation
is easy to see in the predicate Init
:
Init ==
/\ clocks \in [ Proc > Nat ]
/\ drift = [ <<p, q>> \in Proc \X Proc > clocks[p]  clocks[q] ]
Further, we write down a step of our system:
\* Uniformly advance the clocks and update the drifts.
\* Constructing functions without iteration.
NextFast ==
\E delta \in Nat \ { 0 }:
/\ clocks' = [ p \in Proc > clocks[p] + delta ]
/\ drift' = [ <<p, q>> \in Proc \X Proc > clocks'[p]  clocks'[q] ]
The transition predicate NextFast
uniformly advances the clocks of all
processes by a nonnegative number delta
. Simultaneously, it updates the
clock differences in the function drift
.
It is easy to see that drift
actually does not change between the steps. We
can formulate this observation as an action
invariant:
\* Check that the clock drifts do not change
DriftInv ==
\A p, q \in Proc:
drift'[p, q] = drift[p, q]
Our version of NextFast
is quite concise and it uses the good parts of
TLA+. However, new TLA+ users would probably write it differenty. Below you
can see the version that is more likely to be written by a specification
writer who has good experience in software engineering:
\* Uniformly advance the clocks and update the drifts.
\* Constructing functions via explicit iteration. More like a program.
NextSlow ==
\E delta \in Nat \ { 0 }:
/\ LET \* @type: (Str > Int, Str) => (Str > Int);
IncrementInLoop(clk, p) ==
[ clk EXCEPT ![p] = @ + delta ]
IN
clocks' = ApaFoldSet(IncrementInLoop, clocks, Proc)
/\ LET \* @type: (<<Str, Str>> > Int, <<Str, Str>>)
\* => <<Str, Str>> > Int;
SubtractInLoop(dft, pair) ==
LET p == pair[1]
q == pair[2]
IN
[ dft EXCEPT ![p, q] = clocks'[p]  clocks'[q] ]
IN
drift' = ApaFoldSet(SubtractInLoop, drift, Proc \X Proc)
The version NextSlow
is less concise than NextFast
, but it is probably easier to
read for a software engineer. Indeed, we update the variable clocks
via
a set fold, which implements an iteration over the set of processes. What makes
it easier to understand for a software engineer is a local update in the
operator IncrementInLoop
. Likewise, the variable drift
is iteratively
updated with the operator SubtractInLoop
.
If ApaFoldSet
looks unfamiliar to you, check the page on folding sets and
sequences.
Although NextSlow
may look more familiar, it is significantly harder for
Apalache to check than NextFast
. To see the difference, we measure
performance of Apalache for several sizes of Proc
: 3, 5, 7, and 10. We do
this by running Apalache for the values of N
equal to 3, 5, 7, 10. To this
end we define several model files called MC_FoldExcept${N}.tla
for
N=3,5,7,10
. For instance, MC_FoldExcept3.tla
looks as follows:
 MODULE MC_FoldExcept3 
Proc == { "p1", "p2", "p3" }
VARIABLES
\* Process clocks
\*
\* @type: Str > Int;
clocks,
\* Drift between pairs of clocks
\*
\* @type: <<Str, Str>> > Int;
drift
INSTANCE FoldExcept
==============================================================================
We run Apalache for different instances of N
:
apalachemc check next=NextSlow inv=DriftInv MC_FoldExcept${N}.tla
apalachemc check next=NextFast inv=DriftInv MC_FoldExcept${N}.tla
The plot below shows the running times for the versions NextSlow
and
NextFast
:
The plot speaks for itself. The version NextFast
is dramatically faster than
NextSlow
for an increasing number of processes. Interestingly, NextFast
is
also more concise. In principle, both NextFast
and NextSlow
describe the
same behavior. However, NextFast
looks higherlevel: It looks like it
computes clocks
and drifts
in parallel, whereas NextSlow
computes
these functions in a loop (though the order of iteration is unknown). Actually,
whether these functions are computed sequentially or in parallel is irrelevant
for our specification, as both NextFast
and NextSlow
describe a single
step of our system! We can view NextSlow
as an implementation of NextFast
,
as NextSlow
contains a bit more computation details.
From the performance angle, the above plot may seem counterintuitive to
software engineers. Indeed, we are simply updating an arraylike data structure
in a loop. Normally, it should not be computationally expensive. However,
behind the scenes, Apalache is producing constraints about all function
elements for each iteration. Intuitively, you can think of it as being fully
copied at every iteration, instead of one element being updated. From this
perspective, the iteration in NextSlow
should clearly be less efficient.
Symbolic Model Checking
This brief introduction to symbolic model checking discusses the following:
 Statespaces and transition systems
 What is a symbolic state?
 What are symbolic traces?
 How do I interpret Apalache counterexamples?
A glossary of notation and definitions can be found below
On statespaces and transition systems
A TLA+ specification defines a triple \((S,S_0,\to)\), called a transition system. \(S\) is the state space, \(S_0\) is the set of initial states \(\left(S_0 \subseteq S\right)\), and \(\to\) is the transition relation, a subset of \(S^2\).
State spaces
The structure of a single state depends on the number of variables a specification declares. For example, if a specification declares
VARIABLE A1, A2, A3, ..., Ak
then a state is a mapping \([A_1 \mapsto a_1, \dots, A_k \mapsto a_k]\), where \(a_i\) represents the value of the variable Ai
, for each \(i = 1,\dots,k\). Here, we represent TLA+ variable names as unique formal symbols, where, for example the TLA+ variable A1
is represented by the formal symbol \(A_1\). By convention, we will use markdownsyntax to refer to objects in TLA+ specifications, and latex notation otherwise.
The state space \(S\) is then the set of all such mappings, i.e. the set of all possible combinations of values that variables may hold.
For brevity, whenever the specification defines exactly one variable, we will treat a state as a single value \(a_1\) instead of the mapping \([A_1 \mapsto a_1]\).
In untyped TLA+, one can think of \(S\) as \(U^{\{A_1,\dots, A_k\}}\), that is, the set of all mappings, which assign a value in \(U\), the universe of all TLA+ values, to each symbol. This set is naturally isomorphic to \(U^k\). In typed TLA+, such as in Apalache, where variable declarations look like:
VARIABLE
\* @type: T1;
A1,
...,
\* @type: Tk;
Ak
\(S\) is additionally restricted, such that for all \(s \in S\) each symbol \(A_i\) maps to a value \(s(A_i) \in V_i\), where \(Vi \subset U\) is the set of all values, which hold the type \(T_i\), for each \(i = 1,\dots,k\). For example, in the specification with
VARIABLE
\* @type: Bool;
A1,
\* @type: Bool;
A2
The state space is \(\mathbb{B}^{\{x,y\}}\) when considering types, since each variable can hold one of two boolean values. In the untyped setting, the state space is infinite, and contains states where, for example, \(A1\) maps to [z \in 1..5 > "a"]
and \(A2\) maps to CHOOSE p \in {}: TRUE
.
As Apalache enforces a type system,the remainder of this document will assume the typed setting.
This assumption does not change any of the definitions.
We will also assume that every specification declares an initialstate predicate Init
, a transitionpredicate Next
and an invariant Inv
(if not specified, assumed to be TRUE
).
For simplicity, we will also assume that the specification if free of constants, resp. that all of the constants have been initialized.
Initial states
The second component, \(S_0\), the set of all initial states, is derived from \(S\) and Init
.
The initial state predicate is a Boolean formula, in which specificationvariables appear as free logic variables.
The operator Init
characterizes a predicate \(P_{S_0} \in \mathbb{B}^S\) in the following way: given a state \(s \in S\), the formula obtained by replacing all occurrences of variable names Ai
in Init
with the values \(s(A_i)\) is a Boolean formula with no free variables (in a welltyped, parseable specification), which evaluates to either TRUE
or FALSE
. We say \(P_{S_0}(s)\) is the evaluation of this formula.
By the subsetpredicate equivalence, we identify the predicate \(P_{S_0}\) with a subset \(S_0\) of \(S\): \(S_0 = \{ s \in S\mid P_{S_0}(s) = TRUE \}\).
For example, given
VARIABLE
\* @type: Int;
x,
\* @type: Int;
y
Init == x \in 3..5 /\ y = 2
we see that \(S = \mathbb{Z}^{\{x,y\}}\) and \(S_0 = \{ [x \mapsto 3, y \mapsto 2], [x \mapsto 4, y \mapsto 2], [x \mapsto 5, y \mapsto 2] \}\).
Transitions
Similar to \(S_0\), \(\to\) is derived from \(S\) and Next
. If \(S_0\) is a singleargument predicate \(S_0 \in \mathbb{B}^S\), then \(\to\) is a relation \(\to \in \mathbb{B}^{S^2}\).
\(\to(s_1,s_2)\) is the evaluation of the formula obtained by replacing all occurrences of variable names Ai
in Next
with the values \(s_1(A_i)\), and all occurrences of Ai'
with \(s_2(A_i)\).
By the same principle of subsetpredicate equivalence, we can treat \(\to\) as a subset of \(S^2\).
As mentioned in the notation section, it is generally more convenient to use the infix notation
\(s_1 \to s_2\) over \(\to(s_1, s_2)\). We say that a state \(s_2\) is a successor of the state \(s_1\) if \(s_1 \to s_2\).
For example, given
VARIABLE
\* @type: Int;
x,
\* @type: Int;
y
Init == x \in 3..5 /\ y = 2
Next == x' \in { x, x + 1 } /\ UNCHANGED y
One can deduce, for any state \([x \mapsto a, y \mapsto b] \in S\), that it has two successors: \([x \mapsto a + 1, y \mapsto b]\) and \([x \mapsto a, y \mapsto b]\) because the following relations hold \([x \mapsto a, y \mapsto b] \to [x \mapsto a + 1, y \mapsto b]\) and \( [x \mapsto a, y \mapsto b] \to [x \mapsto a, y \mapsto b] \).
Lastly, we define traces in the following way: A trace of length \(k\) is simply a sequence of states \(s_0,\dots, s_k \in S\), such that \(s_0 \in S_0\) and \(s_i \to s_{i+1}\) for all \(i\in \{0,\dots,k1\}\). This definition naturally extends to inifinite traces.
For example, the above specification admits the following traces of length 2 (among others): \[ [x \mapsto 3, y \mapsto 2], [x \mapsto 3, y \mapsto 2], [x \mapsto 3, y \mapsto 2] \] \[ [x \mapsto 3, y \mapsto 2], [x \mapsto 4, y \mapsto 2], [x \mapsto 5, y \mapsto 2] \] \[ [x \mapsto 4, y \mapsto 2], [x \mapsto 5, y \mapsto 2], [x \mapsto 5, y \mapsto 2] \]
Reachable states
Using the above definitions, we can define the set of states reachable in exactly \(k\)steps, for \(k \in \mathbb{N}\), denoted by \(R(k)\). We define \(R(0) = S_0\) and for each \(k \in \mathbb{N}\), \[ R(k+1) := \{ t \in S \mid \exists s \in R(k) \ .\ s \to t \} \]
Similarly, we can define the set of states reachable in at most \(k\)steps, denoted \(r(k)\), for \(k \in \mathbb{N}\) by \[ r(k) := \bigcup_{i=0}^k R(i) \]
Finally, we define the set of all reachable states, \(R\), as the (infinite) union of all \(R(k)\), over \(k \in \mathbb{N}\): \[ R := \bigcup_{k \in \mathbb{N}} R(k) \]
For example, given
VARIABLE
\* @type: Int;
x,
\* @type: Int;
y
Init == x \in 1..3 /\ y = 2
Next == x' = x + 1 /\ UNCHANGED y
we can deduce:
\begin{align} R(0) &= r(0) = S_0 = \{[x\mapsto 1, y\mapsto 2],[x\mapsto 2, y\mapsto 2],[x\mapsto 3, y\mapsto 2]\} \\ \\ R(1) &= \{[x\mapsto 2, y\mapsto 2], [x\mapsto 3, y\mapsto 2], [x\mapsto 4, y\mapsto 2]\} \\ r(1) &= \{[x\mapsto 1, y\mapsto 2], [x\mapsto 2, y\mapsto 2], [x\mapsto 3, y\mapsto 2], [x\mapsto 4, y\mapsto 2]\} \\ \\ R(2) &= \{[x\mapsto 3, y\mapsto 2], [x\mapsto 4, y\mapsto 2], [x\mapsto 5, y\mapsto 2]\} \\ r(2) &= \{[x\mapsto 1, y\mapsto 2], [x\mapsto 2, y\mapsto 2], [x\mapsto 3, y\mapsto 2], [x\mapsto 4, y\mapsto 2], [x\mapsto 5, y\mapsto 2]\} \end{align}
and so on. We can express this compactly as: \begin{align} [x\mapsto a, y \mapsto b] \in R(i) &\iff i+1 \le a \le i + 3 \land b = 2 \\ [x\mapsto a, y \mapsto b] \in r(i) &\iff 1 \le a \le i + 3 \land b = 2 \\ [x\mapsto a, y \mapsto b] \in R &\iff 1 \le a \land b = 2 \end{align}
Finite diameters
We say that a transition system has a finite diameter, if there exists a \(k \in N\), such that \(R = r(k)\).
If such an integer exists then the smallest integer \(k\), for which this holds true, is the diameter of the transition system. In other words, if the transition system \((S,S_0,\to)\) has a finite diameter of \(k\), any state that is reachable from a state in \(S_0\), is reachable in at most \(k\) transitions. The example above clearly does not have a finite diameter, since \(R\) is infinite, but \(r(k)\) is finite for each \(k\).
However, the spec
VARIABLE
\* @type: Int;
x
Init == x = 0
Next == x' = (x + 1) % 7
has a finite diameter (more specifically, a diameter of 6), because:
 \(R = \{0,1,\dots,6\}\) (the set of remainders modulo 7), since those are the only values
x'
, which is defined as a% 7
expression, can take.  for any \(k = 0,\dots,5\), it is the case that \(r(k) = \{0,\dots,k\} \ne R\), so the diameter is not in \(\{1,\dots,5\}\)
 for any \(k \ge 6\), \(r(k) = r(6) = R\)
Invariants
Much like Init
, an invariant operator Inv
defines a predicate. However, it is not, in general, the case that Inv
defines a predicate over S
. There are different cases we can consider, discussed in more detail here. For the purposes of this document, we focus on state invariants, i.e. operators which use only unprimed variables and no temporal or trace operators. A state invariant operator Inv
defines a predicate \(I\) over \(S\).
We say that the \(I\) is an invariant in the transition system, if \(R \subseteq I\), that is, for every reachable state \(s_r \in R\), \(I(s_r)\) holds true. If \(R \setminus I\) is nonempty (i.e., there exists a state \(s_r \in R\), such that \(\neg I(s_r)\)), we refer to elements of \(R \setminus I\) as witnesses to invariant violation.
Goals of model checking
The goal of model checking is to determine whether or not \(R \setminus I\) contains a witness. The goal of bounded model checking is to determine, given a bound \(k\), whether or not \(r(k) \setminus I\) contains a witness.
In a transition system with a bounded diameter, one can use bounded model checking to solve the general model checking problem, since \(R \setminus I\) is equivalent to \(r(k) \setminus I\) for a sufficiently large \(k\). In general, if the system does not have a bounded diameter, failing to find a witness in \(r(k) \setminus I\) cannot be used to reason about the absence of witnesses in \(R \setminus I\)!
Explicitstate model checking
The idea behind explicitstate model checking is to simply perform the following algorithm (in pseudocode, \(\leftarrow\) represents assignment):
Compute \(S_0\) and set \(Visited \leftarrow \emptyset, ToVisit \leftarrow S_0\)

While \(ToVisit \ne \emptyset\), pick some \(s \in ToVisit\): 1. If \(\neg I(s)\) then terminate, since a witness is found. 1. If \(I(s)\) then compute \(Successors(s) = \{ t \in S\mid s \to t \}\). Set \begin{align} Visited &\leftarrow Visited \cup \{s\}\\ ToVisit &\leftarrow (ToVisit \cup Successors(s)) \setminus Visited \end{align}

If \(ToVisit = \emptyset\) terminate. \(R = Visited\) and \(I\) is an invariant.
While simple to describe, there are several limitations of this approach in practice. The first limitation is the absence of a termination guarantee. More specifically, this algorithm terminates if and only if \(R\) is finite. For example:
VARIABLE
\* @type: Int;
x
Init == x = 0
Next == x' = x + 1
defines a states space, for which \(R = \mathbb{N}\), so the above algorithm never terminates. Further, in the general case it is difficult or impossible to compute \(S_0\) or the set \(Successors(s)\) defined in the algorithm. As an example, consider the following specification:
VARIABLE x
Successor(n) == IF n % 2 = 0 THEN n \div 2 ELSE 3*n + 1
RECURSIVE kIter(_,_)
kIter(a,k) == IF k <= 0 THEN a ELSE Successor(kIter(a, k1))
ReachesOne(a) == \E n \in Nat: kIter(a,n) = 1
Init == x \in { n \in Nat: ~ReachesOne(n) }
The specification encodes the Collatz conjecture, so computing \(S_0\) is equivalent to proving or disproving the conjecture, which remains an open problem at present. It is therefore unreasonable to expect any model checker to be able to accept such input, despite the fact that the condition is easily describable in firstorder logic.
A similar problem can occur in computing \(Successors(s)\); the relation between variables Ai
(\(s(A_i)\)) and Ai'
(\(s_2(A_i)\)) may be given by means of an implicit function or uncomputable expression.
Therefore, most tools impose the following constraints, which make computing \(S_0\) and \(Successors(s)\) possible without any sort of specialized solver:
The specification must have the shape
VARIABLE A1,...,Ak
Init == /\ A1 \in F1()
/\ A2 \in F2(A1)
...
/\ Ak \in Fk(A1,...,A{k1})
Next == /\ CondN(A1,...,Ak)
/\ A1' \in G1(A1,...,Ak)
/\ A2' \in G2(A1,...,Ak, A1')
...
/\ Ak' \in Gk(A1,...,Ak, A1',...,A{k1}')
or some equivalent form, in which variable values in a state can be iteratively computed, one at a time, by means of an explicit formula, which uses only variables computed so far. For instance,
VARIABLE x,y
Init == /\ x \in 1..0
/\ y \in { k \in 1..10, k > x }
Next == \/ /\ x > 5
/\ x' = x  1
/\ y' = x' + 1
\/ /\ x <= 5
/\ y' = 5  x
/\ x' = x + y'
allows one to compute both \(S_0\) as well as \(Successors(s)\), for any \(s\), by traversing the conjunctions in the syntaximposed order.
However, even in a situation where states are computable, and \(R\) is finite, the size of \(R\) itself might be an issue in practice. We can create very compact specifications with large statespace sizes:
VARIABLE A1,...,Ak
Init == /\ A1 = 0
...
/\ Ak = 0
Next == \/ /\ A1' = (A1 + 1) % C
/\ UNCHANGED <<A2,...,Ak>>
\/ /\ A2' = (A2 + 1) % C
/\ UNCHANGED <<A1,A3,...,Ak>>
...
\/ /\ Ak' = (Ak + 1) % C
/\ UNCHANGED <<A1,...,A{k1}>>
This specification will have \(C^k\) distinct states, despite its rather simplistic behavior.
Explicitstate bounded model checking
Adapting the general explicitstate approach to bounded model checking is trivial, and therefore not particularly interesting. Assume a bound \(k \in \mathbb{N}\) on the length of the traces considered.
Compute \(S_0\) and set \(Visited \leftarrow \emptyset, ToVisit \leftarrow \{ (s,0)\mid s \in S_0 \}\)

While \(ToVisit \ne \emptyset\), pick some \((s,j) \in ToVisit\):
 If \(\neg I(s)\) then terminate, since a witness is found.
 If \(I(s)\) then: \begin{align} Visited &\leftarrow Visited \cup \{(s,j)\} \\ ToVisit &\leftarrow (ToVisit \cup T) \setminus Visited \end{align} where \( T \) equals \(\{(t,j+1)\mid t \in Successors(s)\}\) if \(j < k\) and \(\emptyset\) otherwise

If \(ToVisit = \emptyset\) terminate. \(r(k) = \{v \mid \exists j \in \mathbb{N} \ .\ (v,j) \in Visited\}\) and \(I\) holds in all states reachable in at most \(k\) steps.
A real implementation would, for efficiency reasons, avoid entering the same state via traces of different length, but the basic idea would remain unchanged. Bounding the execution length guarantees termination of the algorithm if \(S_0\) is finite and each state has finitely many successors w.r.t. \(\to\), even if the state space is unbounded in general. However, this comes at a cost of guarantees: while bounded model checking might still find an invariant violation if it can occur within the bound \(k\), it will fail if the shortest possible trace, on which the invariant is violated has a length greater than \(k\).
If the system has a finite diameter, bounded model checking is equivalent to model checking, as long as \(k\) exceeds the diameter.
Symbolic bounded model checking
For a given \(k \in \mathbb{N}\), we want to find a way to determine if \(r(k) \setminus I\) is empty, without testing every single state in \(r(k)\) like in the explicitstate approach.
The key insight behind symbolic model checking is the following: it is often the case that the size of the reachable state space is large, not because of the properties of the specification, but simply because of the constants or sets involved.
Consider the example:
VARIABLE
\* @type: Int;
x
Init == x = 1
Next1 == x' \in 1..9
Next2 == x' \in 1..999999999999
Inv == x < 5
The sets of reachable states defined by each Next
have sizes proportional to the upper bounds of the ranges used. However, to find a violation of the invariant, one merely needs to identify a state \(s\) in which, for example, \(s(x) = 7\), which belongs to both sets.
It is not necessary, or efficient, to loop over elements in the range and test each one against Inv
to find a violation. Depending on the logic fragment Inv
belongs to, there usually exist strategies for finding such violations much faster.
From this perspective, if, for some \(k\), we succeeded in finding a predicate \(P\) over \(S\), such that:
 \(P\) belongs to a logic fragment, for which optimizations exist
 \(P\) has a witness iff a state reachable in at most \(k\) steps violates \(I\): \(\left(\exists s \in S \ .\ P(s)\right) \iff r(k) \setminus I \ne \emptyset\)
we can use specialized techniques within the logical fragment to evaluate \(P\) and find a witness to the violation of \(I\), or else conclude that \(r(k) \subseteq I\).
To do this, it is sufficient to find a predicate \(P_R^l\) encoding \(R(l)\), for each \(l \in \{0,\dots,k\}\), since: \begin{align} s \in r(l) \iff& \lor s \in R(0) \\ &\lor s \in R(1) \\ &\dots \\ &\lor s \in R(l) \end{align}
How does one encode \(P_R^0\)? \[ s \in R(0) \iff s \in S_0 \iff P_{S_0}(s) \]
so \(P_R^0(s) = P_{S_0}(s)\). What about \(P_R^1\)? \begin{align} s \in R(1) &\iff s \in \{ t \in S \mid \exists s_0 \in R(0) \ .\ s_0 \to t \} \\ &\iff \exists s_0 \in R(0) \ .\ s_0 \to s \\ &\iff \exists s_0 \in S \ .\ P_R^0(s_0) \land s_0 \to s \end{align} so \(P_R^1(s) := \exists s_0 \in S \ .\ P_R^0(s_0) \land s_0 \to s\)
continuing this way, we can determine \[ P_R^k(s) := \exists s_{k1} \in S \ .\ P_R^{k1}(s_{k1}) \land s_{k1} \to s \] Which can be expanded to \[ Pk(s) = \exists s_0,\dots,s_{k1} \in S \ .\ P_{S_0}(s_0) \land s_0 \to s_1 \land s_1 \to s_2 \land \dots \land s_{k1} \to s \]
Then, the formula describing invariant violation in exactly \(k\) steps, \(\exists s_k \in R(k) \setminus I\), becomes \[ \exists s_0,\dots,s_k \in S \ .\ P_{S_0}(s_0) \land \neg I(s_k) \land \bigwedge_{i=0}^{k1} s_i \to s_{i+1} \]
The challenge in designing a symbolic model checker is determining, given TLA+ operators Init
, Next
and Inv
, the encodings of \(P_{S_0}, \to, I\) as formulas in logcis supported by external solvers, for example SMT.
Symbolic states
In an explicit approach, the basic unit of computation is a single state \(s \in S\). However, as demonstrated above, symbolic approaches deal with logical formulas. Recall that a state formula, such as Init
is actually a predicate over \(S\), and a predicate is equivalent to a subset of \(S\).
Predicates tend to not distinguish between certain concrete states. For instance, the formula
\(x < 3\) is equally false for both \(x = 7\) and \(x = 70000000\). It is useful to characterize all of the states, in which a predicate evaluates to the same value. This is because we will define symbolic states in terms of equivalence relations:
A predicate \(P\) over \(S\) naturally defines an equivalence relation \(\circledcirc_P\): For \(a,b \in S\), we say that \(a \circledcirc_P b\) holds if \(P(a) = P(b)\).
Proving that this relation satisfies the criteria for an equivalence relation is left as an exercise to the reader.
This equivalence relation has only two distinct equivalence classes, since \(P(s)\) can only be TRUE
or FALSE
.
We can therefore think of predicates in the following way: Each predicate \(P\) slices the
set \(S\) into two disjoint subsets, i.e. the equivalence classes of \(\circledcirc_P\).
An equivalent formulation of the above is saying that each predicate \(P\) defines a quotient space \(S / \circledcirc_P\), of size \(2\).
Recall that we have expressed the set of states \(R(l)\) with the predicate \(P_R^l\), for each \(l \in \{0,\dots,k\}\). By the above, \(P_R^l\) defines an equivalence relation \(\circledcirc_{P_R^l}\) on \(S\), and consequently, two equivalence classes. For notational clarity, we use \(\circledcirc^l\) instead of \(\circledcirc_{P_R^l}\). Each concrete state \(s \in S\) belongs to exactly one equivalence class \(\lbrack s \rbrack_{ \circledcirc^l} \in S / \circledcirc^l\).
The states in \(R(l)\) correspond to the equivalence class in which \(P_R^l\) holds true (i.e. \(s \in R(l) \iff \lbrack s \rbrack_{\circledcirc^l} = \{t \in S \mid P_R^l(t) = TRUE\}\)), and the ones in \(S \setminus R(l)\) correspond to the equivalence class in which \(P_R^l\) is false (i.e. \(s \notin R(l) \iff \lbrack s \rbrack_{\circledcirc^l} = \{t \in S \mid P_R^l(t) = FALSE\}\)).
We define symbolic states in the following way: Given a predicate \(P\) over \(S\), a symbolic state with respect to \(P\) is an element of \(S / \circledcirc_P\), where \(\circledcirc_P\) is the equivalence relation derived from \(P\) (i.e. \(a \circledcirc_P b \iff P(a) = P(b)\)). Recall the subsetpredicate equivalence: in this context, a symbolic state, w.r.t. \(P\) is equivalent to a predicate, specifically, either \(P\) or \(\neg P\).
For example, given
VARIABLE
\* @type: Int;
x,
\* @type: Int;
y
Init == x = 1 /\ y = 1
Next == x' \in 1..5 /\ y \in {0,1}
and the predicate \(P(s) = s(x) < 3\), the symbolic states are \[ \{ [x \mapsto a, y \mapsto b] \mid a,b\in \mathbb{Z} \land a < 3 \} \] and \[ \{ [x \mapsto a, y \mapsto b] \mid a,b\in \mathbb{Z} \land a \ge 3 \} \]
while the symbolic states w.r.t. \(R(0)\) are \[ \{ [x \mapsto 1, y \mapsto 1] \} \] and \[ \{ [x \mapsto a, y \mapsto b] \mid a,b\in \mathbb{Z} \land ( a \ne 1 \lor b \ne 1 )\} \]
If we only care about characterizing invariant violations, the above techniques are sufficient. However, specification invariants are often composed of multiple smaller, independent invariants. For feedback purposes, it can be beneficial to identify, whenever an invariant violation occurs, the precise subinvariant that is the cause. Suppose we are given an invariant \(s(x) > 0 \land s(y) > 0\). The information whether a reachable state has just \(s(x) \le 0\), just \(s(y) \le 0\), or both can help determine problems at the design level.
More generally: often, a predicate \(P\) is constructed as a conjunction of other predicates, e.g. \(P(s) \iff p_1(s) \land \dots \land p_m(s)\). A violation of \(P\) means a violation of (at least) one of \(p_1,\dots,p_m\), but knowing which one enables additional analysis.
A collection of predicates \(p_1,\dots,p_m\) over \(S\) define an equivalence relation \(\circledcirc\lbrack p_1,\dots,p_m\rbrack\)in the following way: For \(a,b \in S\), we say that \( a \circledcirc\lbrack p_1,\dots,p_m\rbrack\ b\) holds if \(p_1(a) = p_1(b) \land \dots \land p_m(a) = p_m(b)\). Clearly, \(\circledcirc\lbrack p_1\rbrack = \circledcirc_{p_1}\).
Since a predicate can only evaluate to one of two values, there exist only two equivalence classes for \(\circledcirc_P\), i.e. only two symbolic states w.r.t. \(P\): one is the set of all states for which \(P\) is TRUE
, and the other is the set of all values for which \(P\) is FALSE
.
In this sense, \(S / \circledcirc_P\) is isomorphic to the set \(\mathbb{B}\).
In the case of \(\circledcirc\lbrack p_1,\dots,p_m\rbrack\), there are \(2^m\) different \(m\)tuples with values from \(\mathbb{B}\), so \(S / \circledcirc\lbrack p_1,\dots,p_m\rbrack\) is isomorphic to \(\mathbb{B}^m\) .
What is the relation between \(\circledcirc\lbrack p_1,\dots,p_m\rbrack\) and \(\circledcirc_P\), where \(P(s) = p_1(s) \land \dots \land p_m(s)\)? Clearly, \(P(s) = TRUE \iff p_1(s) = \dots = p_m(s) = TRUE\). Consequently, there is one equivalence class in \(S / \circledcirc_P\), that is equal to \[ C_1 = \{ s \in S \mid P(s) = TRUE \} \] and one equivalence class in \(S / \circledcirc\lbrack p_1,\dots,p_m\rbrack\) that is equal to \[ C_2 = \{ s \in S \mid p_1(s) = TRUE \land \dots \land p_m(s) = TRUE \} \]
They are one and the same, i.e. \(C_1 = C_2\). The difference is, that splitting \(P\) into \(m\) components \(p_1,\dots,p_m\) splits the other (unique) equivalence class \(C \in \{ c \in S / \circledcirc_P \mid c \ne C_1 \}\) into \(2^m  1\) parts, which are the equivalence classes in \(\{ c \in S/\circledcirc\lbrack p_1,\dots,p_m\rbrack \mid c \ne C_2 \}\).
Consequently, we can also define symbolic states with respect to a set of predicates \(p_1,\dots,p_m\), implicitly conjoined, as elements of \(S / \circledcirc\lbrack p_1,\dots,p_m\rbrack\). Similarly, by the subsetpredicate equivalence, a symbolic state, w.r.t. \(p_1,\dots,p_m\) can be viewed as one of \begin{align} p_1(s) \land p_2(s) \land \dots \land p_m(s) \qquad&= P(s) \\ \neg p_1(s) \land p_2(s) \land \dots \land p_m(s) \qquad&  \\ p_1(s) \land \neg p_2(s) \land \dots \land p_m(s) \qquad&  \\ \dots \qquad& > \text{(as a disjunction)} = \neg P(s) \\ \neg p_1(s) \land \neg p_2(s) \land \dots \land \neg p_{m1}(s) \land p_m(s) \qquad&  \\ \neg p_1(s) \land \neg p_2(s) \land \dots \land \neg p_{m1}(s) \land \neg p_m(s) \qquad&  \\ \end{align}
For example, take \(p_1(s) = s \in R(k)\) and \(p_2(s) = \neg I(s)\). With respect to \(p_1(s) \land p_2(s)\), there are two symbolic states: one corresponds to the set of all states which are both reachable and in which the invariant is violated, while the other corresponds to the set of all states, which are either not reachable, or in which the invariant holds. Conversely, with respect to \(p_1,p_2\), there are four symbolic states: one corresponds to states which are both reachable and violate the invariant, one corresponds to states which are reachable, but which do not violate the invariant, one corresponds to states which are not reachable, but violate the invariant and the last one corresponds to states which are neither reachable, nor violate the invariant.
Symbolic traces
Having defined symbolic states, what is then the meaning of a symbolic trace? Recall, a trace of length \(k\) is simply a sequence of reachable states \(s_0,\dots, s_k \in S\), such that \(s_0 \in S_0\) and \(s_i \to s_{i+1}\). In the symbolic setting, a symbolic trace is a sequence of symbolic states \(C_0,\dots,C_k \subseteq S\), such that \[ C_0 \in S / \circledcirc^0 \land \dots \land C_k \in S / \circledcirc^k \]
and, for each \(i = 0,\dots,k\), it is the case that \(C_i = \{ s \in S \mid P_R^i(s) = TRUE\}\).
In other words, a symbolic trace is the unique sequence of symbolic states, which correspond to the set of explicit states evaluating to TRUE
under each of \(P_R^0,\dots,P_R^k\) respectively.
Recall that \(P_R^{i+1}(s_{i+1})\) was defined as \(\exists s_i \in S \ .\ P_R^i(s_i) \land s_i \to s_{i+1}\). While, in the explicit case, we needed to enforce the condition \(s_i \to s_{i+1}\), in the symbolic case this is already a part of the predicate definition.
For example, consider:
VARIABLE
\* @type: Int;
x
Init == x \in {0,1}
Next == x' = x + 1
a trace of length 2 would be one of \(0,1,2\) or \(1,2,3\). A symbolic trace would be the sequence \[ \{0,1\}, \{1,2\}, \{2,3\} \]
In the case of symbolic states, we were particularly interested in symbolic states with respect to predicates that encoded reachability.
Unlike the case of invariants, where we considered conjunctions of subinvariants, the most interesting scenario w.r.t. traces is when a transition relation is presented as a disjunction of transitions, i.e. when \begin{align} s_1 \to s_2 \iff& \lor t_1(s_1,s_2)\\ & \lor t_2(s_1,s_2)\\ & \dots \\ & \lor t_m(s_1,s_2) \end{align}
At the specification level, this is usually the case when one can nondeterministically choose to perform one of \(m\) actions, and each \(t_1,\dots,t_m\) is an encoding of one such action, which, like \(\to\), translates to a binary predicate over \(S\).
Instead of a single trace \(C_1, \dots, C_k\), where states in \(C_{i+1}\) are reachable from states in \(C_i\) via \(\to\), we want to separate sets of states reachable by each \(t_i\) individually.
Recall that symbolic traces are sequences of symbolic states, implicitly related by \(\to\), since \(R\) is defined in terms of \(\to\). We define a symbolic trace decomposition by \(t_1,\dots,t_m\), in the following way: If \(t_1,\dots,t_m\) are relations, such that \(s_1 \to s_2 \iff \bigvee_{i=1}^m t_i(s_1,s_2)\), the decomposition of a symbolic trace \(X_0,\dots,X_k\) of length \(k\) w.r.t. \(t_1,\dots,t_m\) is a set \( D = \{ Y(\tau) \mid \tau \in \{1,\dots,m\}^{\{1,\dots, k\}} \} \) , such that:
 \(Y(\tau)\) is a partial symbolic trace of length k: \(Y_0(\tau) = X_0, Y_1(\tau),\dots, Y_k(\tau)\)
 For each \(i = 0,\dots,k1\), \(Y_{i+1}\) is the set of all states reachable from \(Y_i\) by the transition fragment \(t_j\), where \(j = \tau(i+1)\): \[ Y_{i+1}(\tau) = \{ s_{i+1} \in X_{i+1} \mid \exists s_i \in Y_i(\tau) \ .\ t_{\tau(i+1)}(s_i,s_{i+1}) \} \]
An interesting property to observe is that, for each \(i=1,\dots,k\), the sets \(Y_i(\tau)\), over all possible \(\tau\), form a decomposition of \(X_i\). Concretely: \[ X_i = \bigcup \left\{ Y_i(\tau)\mid \tau \in \{1,\dots,m\}^{\{1,\dots, k\}} \right\} \]
Less obvious is the fact that, the larger the index \(i\), the finer this decomposition becomes. Consider \(i=1\). Since \(Y_0\) is fixed, there are as many different \(Y_1(\tau)\) components as there are possible values of \(\tau(1)\), i.e. \(m\). As \(Y_2\) depends on \(Y_1\), there are as many different components as there are pairs \((\tau(1),\tau(2))\), i.e. \(m^2\), and so on until \(k\), where there are \(m^k\) possible \(Y_k(\tau)\) sets. In practice, however, many of these sets are empty.
Let us look at an example:
VARIABLE
\* @type: Int;
x
A1 == /\ x > 4
/\ x' = x  1
A2 == /\ x < 7
/\ x' = x + 1
A3 == x' = x
A4 == /\ x = 1
/\ x' = 10
Init == x \in 1..10
Next == \/ A1
\/ A2
\/ A3
\/ A4
The \(\to\) predicate can be decomposed into: \begin{align} t_1(s_1,s_2) &= s_1(x) > 4 \land s_2(x) = s_1(x)  1 \\ t_2(s_1,s_2) &= s_1(x) < 7 \land s_2(x) = s_1(x) + 1 \\ t_3(s_1,s_2) &= s_2(x) = s_1(x) \\ t_4(s_1,s_2) &= s_1(x) = 1 \land s_2(x) = 10 \\ \end{align}
Suppose we fix the length of the trace \(k = 2\). Without considering the decomposition, the symbolic trace is equal to \[ X_0 = \{1,\dots,10\}, X_1 = \{1,\dots,10\}, X_2 = \{1,\dots,10\} \]
Under the decomposition, we have \(m^k = 4^2 = 16\) candidates for \(\tau\). Let us look at \(\tau_1\), for which \(\tau_1(1) = 1, \tau_1(2) = 2\), representing an execution where the action A1
is followed by the action A2
.
If \(Y_0(\tau_1),Y_1(\tau_1),Y_2(\tau_1)\) is a partial trace (i.e. one of the elements in the decomposition \(D\)), then:

\(Y_1(\tau_1) = \{ b \in X_1 \mid \exists a \in Y_0(\tau_1) \ .\ t_{\tau_1(1)}(a,b)\}\) which means \[ Y_1(\tau_1) = \{ b \in \{1,\dots,10\} \mid \exists a \in \{1,\dots,10\} \ .\ a > 4 \land b = a  1 \} = \{4,\dots,9\} \]

\(Y_2(\tau_1) = \{ b \in X_2 \mid \exists a \in Y_1(\tau_1)\ .\ t_{\tau_1(2)}(a,b)\}\) which means \[ Y_2(\tau_1) = \{ b \in \{1,\dots,10\} \mid \exists a \in \{4,\dots,9\} \ .\ a < 7 \land b = a + 1 \} = \{5,\dots,7\} \]
so the partial trace, corresponding to the sequence of actions Init,A1,A2
is
\[
\{1,\dots,10\}, \{4,\dots,9\}, \{5,\dots,7\}
\]
In fact, we can draw a table, representing partial traces corresponding to sequences of actions:
Sequence of actions (after Init )  Partial trace (without \(Y_0\)) 

A1, A1  \(\{4, \dots, 9\}, \{4, \dots, 8\}\) 
A1, A2  \(\{4, \dots, 9\}, \{5, \dots, 7\}\) 
A1, A3  \(\{4, \dots, 9\}, \{4, \dots, 9\}\) 
A1, A4  \(\{4, \dots, 9\}, \emptyset\) 
A2, A1  \(\{2, \dots, 7\}, \{4, \dots, 6\}\) 
A2, A2  \(\{2, \dots, 7\}, \{3, \dots, 7\}\) 
A2, A3  \(\{2, \dots, 7\}, \{2, \dots, 7\}\) 
A2, A4  \(\{2, \dots, 7\}, \emptyset\) 
A3, A1  \(\{1, \dots, 10\}, \{4, \dots, 9\}\) 
A3, A2  \(\{1, \dots, 10\}, \{2, \dots, 7\}\) 
A3, A3  \(\{1, \dots, 10\}, \{1, \dots, 10\}\) 
A3, A4  \(\{1, \dots, 10\}, \{10\}\) 
A4, A1  \(\{10\}, \{9\}\) 
A4, A2  \(\{10\}, \emptyset\) 
A4, A3  \(\{10\}, \{10\}\) 
A4, A4  \(\{10\}, \emptyset\) 
Clearly, the elements in every column (representing the various \(Y_i(\tau)\)), add up to \(X_i = \{1,\dots,10\}\). Also noticeable is the fact that some actions disable others, represented by the fact that some \(Y_2(\tau)\) sets are empty. For example, the action A2
disables A4
, because after A2
, x
cannot hold the value \(1\), which is a precondition for A4
.
Counterexamples in Apalache
Finally, we can interpret Apalache counterexamples in the context of the above definitions. Given an invariant \(I\), a transition system \((S, S_0, \to)\) and an upper bound on executions \(k\), Apalache first finds predicates \(t_1,\dots,t_m\) partitioning \(\to\). Then, it encodes a symbolic trace \(X_0,\dots,X_k\) and its decomposition \(D\). A counterexample in Apalache defines an explicit trace \(s_0,s_1,\dots,s_l \in S\) for some \(l \le k\), as well as a sequence \(t_{\tau(1)}, \dots, t_{\tau(l)}\) (in the comments). The predicate sequence defines a partial trace (of length \(l\)) \(Y_0(\tau),\dots,Y_l(\tau)\) and \(s_0,\dots,s_l\) are chosen such that \(s_i \in Y_i(\tau)\).
Take the following specification and counterexample, for \(k = 10\):
 MODULE example 
EXTENDS Integers
VARIABLE
\* @type: Int;
x
A == /\ x = 1
/\ x' = x + 1
B == /\ x > 1
/\ x' = x + 1
Init == x = 1
Next == \/ A
\/ B
Inv == x < 3
====================
 MODULE counterexample 
EXTENDS test
(* Constant initialization state *)
ConstInit == TRUE
(* Initial state *)
State0 == x = 1
(* Transition 0 to State1 *)
State1 == x = 2
(* Transition 1 to State2 *)
State2 == x = 3
(* The following formula holds true in the last state and violates the invariant *)
InvariantViolation == x >= 3
================================================================================
We can see that, even though \(k=10\), we found a violation in \(l=2\) steps.
Each State{i}
represents one of \(s_0,\dots,s_l\), by explicitly defining variable values in that state (e.g. x = 1 /\ y = 2 /\ z = "A"
).
The comment (* Transition X to StateY *)
outlines which \(t_1,\dots,t_m\) was used to reach \(s_{i+1}\) from \(s_i\) (0indexed).
The shape of \(t_i\) can be found by looking at the file XX_OutTransitionFinderPass.tla
, and will be named Next_si_i
.
In the above case, Transition 0
refers to the one representing A
and Transition 1
refers to the one representing B
.
InvariantViolation
is the negation of the invariant Inv
, and it will hold in State{l}
(in this case, x < 3
does not hold in State2
, where x = 3
).
Notation and definitions
We use the following definitions and conventions:
 Common sets: We use the notation \(\mathbb{Z}\) to refer to the set of all integers, \(\mathbb{B}\) to refer to the set of Booleans \(\{TRUE,FALSE\}\), and \(\mathbb{N}\) to refer to the set of all naturals, i.e. \(\mathbb{N} = \{z \in \mathbb{Z}\mid z \ge 0\}\).
 Function sets: We denote by \(B^A\) the set of all functions from \(A\) to \(B\), i.e. \(f \in B^A \iff f\colon A \to B\).
 Powersets: We denote by \(2^A\) the set of all subsets of a set \(A\), i.e. \(B \subseteq A \iff B \in 2^A\)
 Isomorphisms: Sets \( A \) and \(B\) are called isomorphic, if there exists a bijective function \(b\in B^A\).
 Predicates: Given a set \(T\), a predicate over \(T\) is a function \(P \in \mathbb{B}^T\), that is, a function \(P\), such that \(P(t) \in \mathbb{B}\) for each \(t \in T\).
 Relations: Predicates over \(A \times B\) are called relations. A relation \(R\) over \(T \times T\) is an equivalence relation, if the following holds:
 For all \(t \in T\), it is the case that \(R(t,t)\) (reflexivity).
 For all \(s,t \in T\), \(R(s,t)\) holds if and only if \(R(t,s)\) holds (symmetry).
 For all \(r,s,t \in T\), \(R(r,s) \land R(s,t)\) implies \(R(r,t)\) (transititvity).
 Equivalence classes: An equivalence relation \(R\) over \(T \times T\) defines a function \(E \in (2^T)^T\), such that, for each \(t \in T\), \(E(t) = \{ s \in T\mid R(t,s) \}\). \(E(t)\) is called the equivalence class of \(t\) for \(R\), denoted as \(\lbrack t\rbrack_R\).
 Quotient space: An equivalence relation \(R\) over \(T \times T\) defines a quotient space, denoted \(T / R\), such that \(T / R = \{ \lbrack t\rbrack_R \mid t \in T \} \subseteq 2^T\).
 Subsetpredicate equivalence: For any set \(T\), there exists a natural isomorphism between \(\mathbb{B}^T\) and \(2^T\) (implied by the similarity in notation): Each predicate \(P \in \mathbb{B}^T\) corresponds to the set \(\{ t \in T \mid P(t) = TRUE\} \in 2^T\). For this reason, predicates are often directly identified with the subset they are equivalent to, and we write \(P \subseteq T\) for brevity.
 Infix notation: Given a relation \(R \in \mathbb{B}^{A\times B}\), we commonly write \(a\ R\ b\) instead of \(R(a,b)\) (e.g. \(a > b\) instead of \(>(a,b)\)).
 Cartesian product: Given a set \(T\), we use \(T^2\) to refer to \(T \times T\). \(T^k\), for \(k > 2\) is defined similarly.
Temporal properties and counterexamples
Difficulty: Red trail – Medium
Author: Philip Offtermatt, 2022
In this tutorial, we will show how Apalache can be used to decide temporal
properties that are more general than invariants.
This tutorial will be most useful to you if you have a basic understanding of
linear temporal logic, e.g. the semantics of <>
and []
operators.
See a writeup of temporal operators here.
Further, we assume you are familiar with TLA+, but expert knowledge is not necessary.
As a running example, the tutorial uses a simple example specification, modelling a devious nondeterministic traffic light.
Specifying temporal properties
The traffic light has two main components: A lamp which can be either red or green, and a button which can be pushed to request the traffic light to become green. Consequently, there are two variables: the current state of the light (either green or red), and whether the button has been pushed that requests the traffic light to switch from red to green.
The full specification of the traffic light is here:
TrafficLight.tla.
But don't worry  we will dissect the spec in the following.
In the TLA specification, we declare two variables:
VARIABLES
\* If true, the traffic light is green. If false, it is red.
\* @type: Bool;
isGreen,
\* If true, the button has been pushed to request the light to become green, but the light has
\* not become green since then.
\* If false, the light has become green since the button has last been pushed
\* or the button has never been pushed.
\* @type: Bool;
requestedGreen
Initially, the light is red and green has not yet been requested:
\* The light is initially red, and the button was not pressed.
Init ==
/\ isGreen = FALSE
/\ requestedGreen = FALSE
We have three possible actions:
 The traffic light can switch from red to green,
 The traffic light can switch from green to red, or
 The button can be pushed, thus requesting that the traffic light becomes green.
(*  *)
(* requesting green light *)
\* The switch to green can only be requested when the light is not green, and
\* the switch has not *already* been requested since the light last turned green.
RequestGreen_Guard ==
/\ ~isGreen
/\ ~requestedGreen
RequestGreen_Effect ==
/\ requestedGreen' = TRUE
/\ UNCHANGED << isGreen >>
RequestGreen ==
RequestGreen_Guard /\ RequestGreen_Effect
(*  *)
(* switching to red light *)
\* The light can switch to red at any time if it is currently green.
SwitchToRed_Guard == isGreen
SwitchToRed_Effect ==
/\ isGreen' = FALSE
/\ UNCHANGED << requestedGreen >>
SwitchToRed ==
SwitchToRed_Guard /\ SwitchToRed_Effect
(*  *)
(* switching to green light *)
\* The light can switch to green if it is currently red, and
\* the button to request the switch to green has been pressed.
SwitchToGreen_Guard ==
/\ ~isGreen
/\ requestedGreen
SwitchToGreen_Effect ==
/\ isGreen' = TRUE
/\ requestedGreen' = FALSE
SwitchToGreen ==
SwitchToGreen_Guard /\ SwitchToGreen_Effect
Next ==
\/ RequestGreen
\/ SwitchToRed
\/ SwitchToGreen
In the interest of simplicity, we'll assume that the button cannot be pushed when green is already requested, and that similarly it's not possible to push the button when the light is already green.
Now, we are ready to specify the properties that we are interested in. For example, when green is requested, at some point afterwards the light should actually turn green. We can write the property like this:
RequestWillBeFulfilled ==
[](requestedGreen => <>isGreen)
Intuitively, the property says:
"Check that at all points in time ([]),
if right now, RequestGreen
is true,
then at some future point in time, IsGreen
is true."
Let's run Apalache to check this property:
apalachemc check temporal=RequestWillBeFulfilled TrafficLight.tla
...
The outcome is: NoError
Checker reports no error up to computation length 10
It took me 0 days 0 hours 0 min 2 sec
Total time: 2.276 sec
EXITCODE: OK
This is because our traffic watch is actually
deterministic:
If it is red and green has not been requested,
the only enabled action is RequestGreen
.
If it is red and green has been requested,
only SwitchToGreen
is enabled.
And finally, if the light is green,
only SwitchToRed
is enabled.
However, we want to make our traffic light more devious. We will allow the model to stutter, that is, just let time pass and take no action.
We can write a new next predicate that explicitly allows stuttering like this:
\* @type: <<Bool, Bool>>;
vars == << isGreen, requestedGreen >>
StutteringNext ==
[Next]_vars
Recall that [Next]_vars
is shorthand for Next \/ UNCHANGED vars
. Now, let us try to verify the property once again,
using the modified next predicate:
apalachemc check next=StutteringNext \
temporal=RequestWillBeFulfilled TrafficLight.tla
Step 2: picking a transition out of 3 transition(s) I@18:04:16.132
State 3: Checking 1 state invariants I@18:04:16.150
State 3: Checking 1 state invariants I@18:04:16.164
State 3: Checking 1 state invariants I@18:04:16.175
State 3: Checking 1 state invariants I@18:04:16.186
Check an example state in: /home/user/apalache/docs/src/tutorials/_apalacheout/TrafficLight.tla/20220530T180413_3349613574715319837/counterexample1.tla, /home/user/apalache/docs/src/tutorials/_apalacheout/TrafficLight.tla/20220530T180413_3349613574715319837/MC1.out, /home/user/apalache/docs/src/tutorials/_apalacheout/TrafficLight.tla/20220530T180413_3349613574715319837/counterexample1.json, /home/user/apalache/docs/src/tutorials/_apalacheout/TrafficLight.tla/20220530T180413_3349613574715319837/counterexample1.itf.json E@18:04:16.346
State 3: state invariant 0 violated. E@18:04:16.346
Found 1 error(s) I@18:04:16.347
The outcome is: Error I@18:04:16.353
Checker has found an error I@18:04:16.354
It took me 0 days 0 hours 0 min 2 sec I@18:04:16.354
Total time: 2.542 sec I@18:04:16.354
This time, we get a counterexample.
Let's take a look at /home/user/apalache/docs/src/tutorials/_apalacheout/TrafficLight.tla/20220530T180413_3349613574715319837/counterexample1.tla
.
Let's first focus on the initial state.
(* Initial state *)
(* State0 ==
RequestWillBeFulfilled_init = FALSE
/\ __loop_InLoop = FALSE
/\ __loop_☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ __loop_requestedGreen ⇒ ♢isGreen = TRUE
/\ __loop_♢isGreen = FALSE
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = TRUE
/\ requestedGreen ⇒ ♢isGreen = TRUE
/\ ♢isGreen = FALSE
/\ ♢isGreen_unroll = FALSE
/\ isGreen = FALSE
/\ requestedGreen = FALSE *)
State0 ==
RequestWillBeFulfilled_init = FALSE
/\ __loop_InLoop = FALSE
/\ __loop___temporal_t_1 = FALSE
/\ __loop___temporal_t_2 = TRUE
/\ __loop___temporal_t_3 = FALSE
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
/\ __temporal_t_1 = FALSE
/\ __temporal_t_1_unroll = TRUE
/\ __temporal_t_2 = TRUE
/\ __temporal_t_3 = FALSE
/\ __temporal_t_3_unroll = FALSE
/\ isGreen = FALSE
/\ requestedGreen = FALSE
Two things are notable:
 The initial state formula appears twice, once as a comment and once in TLA+.
 There are way more variables than the two variables we specified.
The comment and the TLA+ specification express the same state, but in the comment, some variable names from the encoding have been replaced
with more humanreadable names.
For example, there is a variable called ☐(requestedGreen ⇒ ♢isGreen)
in the comment,
which is called __temporal_t_1
in TLA+.
In the following, let's focus on the content of the comment, since it's easier to understand what's going on.
There are many additional variables in the counterexample because to check temporal formulas, Apalache uses an encoding that transforms temporal properties to invariants. If you are interested in the technical details, the encoding is described in sections 3.2 and 4 of Biere et al.. However, to understand the counterexample, you don't need to go into the technical details of the encoding. We'll go explain the counterexample in the following.
We will talk about traces in the following. You can find more information about (symbolic) traces here. For the purpose of this tutorial, however, it will be enough to think of a trace as a sequence of states that were encountered by Apalache, and that demonstrate a violation of the property that is checked.
Counterexamples encode lassos
First, it's important to know that for finitestate systems, counterexamples to temporal properties are traces ending in a loop, which we'll call lassos in the following. If you want to learn more about why this is the case, have a look at the book on model checking.
A loop is a partial trace that starts and ends with the same state. A lasso is made up of two parts: A prefix, followed by a loop. It describes a possible infinite execution: first it goes through the prefix, and then repeats the loop forever.
For example, what is a trace that is a counterexample to the property ♢isGreen
?
It's an execution that loops without ever finding a state that satisfies isGreen
.
For example, a counterexample trace might visually look like this:
In contrast, as long as the model checking engine has not found a lasso, there may still exist some future state satisfying isGreen
.
Utilizing auxiliary variables to find lassos
The encoding for temporal properties involves lots of auxiliary variables. While some can be very helpful to understand counterexamples, many are mostly noise.
Let's first understand how Apalache can identify lassos using auxiliary variables.
The auxiliary variable __loop_InLoop
is true in exactly the states belonging to the loop.
Additionally, at the first state of the loop, i.e., when __loop_InLoop
switches from false to true,
we store the valuation of each variable in a shadow copy whose name is prefixed by __loop_
.
Before the first state of the loop, the __loop_
carry arbitrary values.
In our example, it looks like this:
(* State0 ==
...
/\ __loop_InLoop = FALSE
...
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
...
/\ isGreen = FALSE
/\ requestedGreen = FALSE *)
(* State1 ==
...
/\ __loop_InLoop = FALSE
...
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
...
/\ isGreen = FALSE
/\ requestedGreen = TRUE *)
(* State2 ==
...
/\ __loop_InLoop = FALSE
...
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
...
/\ isGreen = FALSE
/\ requestedGreen = TRUE *)
(* State3 ==
...
/\ __loop_InLoop = TRUE
...
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = TRUE
...
/\ isGreen = FALSE
/\ requestedGreen = TRUE *)
So, initially, isGreen
and requestedGreen
are both false.
Further, __loop_InLoop
is false, and the copies of isGreen
and requestedGreen
, which are called
__loop_isGreen
and __loop_requestedGreen
, are equal to the values of isGreen
and requestedGreen
.
From state 0 to state 1, requestedGreen
changes from false to true.
From state 1 to state 2, the system stutters, and the valuation of model variables remains unchanged.
Finally, in state 3 __loop_InLoop
is set to true, which means that
the loop starts in state 2, and the trace from state 3 onward is inside the loop.
However, since state 3 is the last state, this means simply that the trace loops in state 2.
Since the loop starts, the copies of the system variables are also set to the values of the variables in state 2,
so __loop_isGreen = FALSE
and __loop_requestedGreen = TRUE
.
The lasso in this case can be visualized like this:
It is also clear why this trace violates the property:
requestedGreen
holds in state 1, but isGreen
never holds,
so in state 1 the property requestedGreen => <>isGreen
is violated.
Auxiliary variables encode evaluations of subformulas along the trace
Next, let us discuss the other auxiliary variables that are introduced by Apalache to check the temporal property. These extra variables correspond to parts of the temporal property we want to check. These are the following variables with their valuations in the initial state:
(* State0 ==
RequestWillBeFulfilled_init = FALSE
...
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = TRUE
/\ requestedGreen ⇒ ♢isGreen = TRUE
/\ ♢isGreen = FALSE
/\ ♢isGreen_unroll = FALSE
...
There are three groups of variables:
 Variables that look like formulas, e.g.
☐(requestedGreen ⇒ ♢isGreen)
 Variables that look like formulas and end with
_unroll
, e.g.☐(requestedGreen ⇒ ♢isGreen)_unroll
 The variable
RequestWillBeFulfilled_init
.
Let's focus on the non_unroll
variables that look like formulas
first.
Recall that the temporal property we want to check is [](requestedGreen => <>isGreen)
.
That's also the name of one of the variables: The value of the variable
☐(requestedGreen ⇒ ♢isGreen)
tells us whether starting in the current state, the
formula [](requestedGreen => <>isGreen)
holds. Since we are looking at a counterexample to this formula, it is not
surprising that the formula does not hold in the initial state of the counterexample.
Similarly, the variable requestedGreen ⇒ ♢isGreen
tells us whether
the property requestedGreen ⇒ ♢isGreen
holds at the current state.
It might be surprising to see that the property holds
but recall that in state 0, requestedGreen = FALSE
, so the implication is satisfied.
Finally, we have the variable ♢isGreen
, which is false, telling
us that along the execution, isGreen
will never be true.
You might already have noticed the pattern of which formulas appear as variables.
Take our property [](requestedGreen => <>isGreen)
.
The syntax tree of this formula looks like this:
For each node of the syntax tree where the formula contains a temporal operator,
there is an auxiliary variable.
For example, there would be auxiliary variables for the formulas []isGreen
and (<>isGreen) /\ ([]requestedGreen)
, but not for the formula isGreen /\ requestedGreen
.
As mentioned before, the value of an auxiliary variable in a state tells us whether from that state, the corresponding subformula is true. In this particular example, the formulas that correspond to variables in the encoding are filled with orange in the syntax tree.
What about the _unroll
variables? There is one _unroll
variable for each immediate application of a temporal operator in the formula.
For example, ☐(requestedGreen ⇒ ♢isGreen)_unroll
is the unrollvariable for the
leading box operator.
To illustrate why these are necessary, consider the formula
[]isGreen
. To decide whether this formula holds in the last state of the loop, the algorithm needs to know whether
isGreen
holds in all states of the loop. So it needs to store this information when it traverses the loop.
That's why there is an extra variable, which stores whether isGreen
holds on all states of the loop, and Apalache can access this information when it explores the last state of the loop.
Similarly, the unrollvariable ♢isGreen_unroll
holds true
if there is a state on the loop such that isGreen
is true.
Let us take a look at the valuations of ☐(requestedGreen ⇒ ♢isGreen)_unroll
along our counterexample to see this.
(* State0 ==
...
/\ __loop_InLoop = FALSE
...
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = TRUE
...
(* State1 ==
...
/\ __loop_InLoop = FALSE
...
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = TRUE
...
(* State2 ==
...
/\ __loop_InLoop = FALSE
...
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = TRUE
...
(* State3 ==
...
/\ __loop_InLoop = TRUE
...
/\ ☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ ☐(requestedGreen ⇒ ♢isGreen)_unroll = FALSE
...
So in the last state, ☐(requestedGreen ⇒ ♢isGreen)_unroll
is not true, since ☐(requestedGreen ⇒ ♢isGreen)
does not hold in state 2, which is on the loop.
Similar to the __loop_
copies for model variables,
we also introduce copies for all (temporal) subformulas, e.g., __loop_☐(requestedGreen ⇒ ♢isGreen)
for ☐(requestedGreen ⇒ ♢isGreen)
.
These fulfill the same function as the __loop_
copies for the
original variables of the model, i.e., retaining the state of variables from the first state of the loop, e.g.,
(* State0 ==
...
/\ __loop_☐(requestedGreen ⇒ ♢isGreen) = FALSE
/\ __loop_requestedGreen ⇒ ♢isGreen = TRUE
/\ __loop_♢isGreen = FALSE
/\ __loop_isGreen = FALSE
/\ __loop_requestedGreen = FALSE
Finally, let's discuss RequestWillBeFulfilled_init
.
This variable is an artifact of the translation for temporal properties.
Intuitively, in any state, the variable will be true if the variable encoding the formula RequestWillBeFulfilled
is true in the first state.
A trace is a counterexample if RequestWillBeFulfilled
is false in the first state, so RequestWillBeFulfilled_init
is false,
and a loop satisfying requirements on the auxiliary variables is found.
Further reading
In this tutorial, we learned how to specify temporal properties in Apalache, and how to read counterexamples for such properties.
If you want to dive deeper into the encoding, it is formally explained in sections 3.2 and 4 of Biere et al.. To understand why this encoding was chosen, you can read the ADR on temporal properties. Finally, if you want to go into the nittygritty details and see the encoding in action, you can look at the intermediate TLA specifying the encoding.
Run
apalachemc check next=StutteringNext \
writeintermediate=yes temporal=RequestWillBeFulfilled TrafficLight.tla
You will get intermediate output in a folder named like
_apalache_out/TrafficLight/TIMESTAMP/intermediate/
.
There, take a look at 0X_OutTemporalPass.tla
.
Overview
How to write type annotations
Revision: August 24, 2022
Important updates:

Version 0.29.0: The new syntax for records and variants is enabled by default (previously, enabled with
features=rows
). For the transition period, the old type syntax can be activated withfeatures=norows
. See Recipe 9 on transitioning to Type System 1.2. 
Version 0.25.10: This HOWTO introduces new syntax for type aliases. See Type Aliases in ADR002.

Version 0.25.9: This HOWTO introduces new syntax for record types and variants, which is currently under testing. This syntax is activated via the option
features=rows
. See Type System 1.2 in ADR002. 
Version 0.23.1: The example specification uses recursive operators, which were removed in version 0.23.1.

Version 0.15.0: This HOWTO discusses how to write type annotations for the type checker Snowcat, which is used in Apalache since version 0.15.0 (introduced in 2021).
This HOWTO gives you concrete steps to extend TLA+ specifications with type annotations. You can find the detailed syntax of type annotations in ADR002. The first rule of writing type annotations:
Do not to write any annotations at all, until the type checker Snowcat is asking you to write a type annotation.
Of course, there must be an exception to this rule. You have to write type annotations for CONSTANTS and VARIABLES. This is because Snowcat infers types of declarations in isolation instead of analyzing the whole specification. The good news is that the type checker finds the types of many operators automatically.
Recipe 1: Annotating variables
Consider the example HourClock.tla from Specifying Systems:
 MODULE HourClock 
\* This is a local copy of the example from Specifying Systems:
\* https://github.com/tlaplus/Examples/blob/master/specifications/SpecifyingSystems/RealTime/HourClock.tla
EXTENDS Naturals
VARIABLE
\* @type: Int;
hr
HCini == hr \in (1 .. 12)
HCnxt == hr' = IF hr # 12 THEN hr + 1 ELSE 1
HC == HCini /\ [][HCnxt]_hr
TypeOK == hr \in (1 .. 12)

THEOREM HC => []HCini
==============================================================
Without thinking much about the types, run the type checker:
$ apalachemc typecheck HourClock.tla
The type checker complains about not knowing the type of the variable hr
:
...
Typing input error: Expected a type annotation for VARIABLE hr
...
Annotate the type of variable hr
as below. Note carefully that the type
annotation should be between the keyword VARIABLE
and the variable name.
This is because variable declarations may declare several variables at once.
In this case, you have to write one type annotation per name.
VARIABLE
\* @type: Int;
hr
Run the type checker again. You should see the following message:
...
> Running Snowcat .::.
> Your types are purrfect!
> All expressions are typed
...
Recipe 2: Annotating constants
Consider the example Channel.tla from Specifying Systems:
 MODULE Channel 
\* This is a typed version of the example from Specifying Systems:
\* https://github.com/tlaplus/Examples/blob/master/specifications/SpecifyingSystems/FIFO/Channel.tla
EXTENDS Naturals
CONSTANT Data
VARIABLE chan
TypeInvariant == chan \in [val : Data, rdy : {0, 1}, ack : {0, 1}]

Init == /\ TypeInvariant
/\ chan.ack = chan.rdy
Send(d) == /\ chan.rdy = chan.ack
/\ chan' = [chan EXCEPT !.val = d, !.rdy = 1  @]
Rcv == /\ chan.rdy # chan.ack
/\ chan' = [chan EXCEPT !.ack = 1  @]
Next == (\E d \in Data : Send(d)) \/ Rcv
Spec == Init /\ [][Next]_chan

THEOREM Spec => []TypeInvariant
=======================================================================
Run the type checker:
$ apalachemc typecheck Channel.tla
The type checker does not know the type of the variable chan
:
Typing input error: Expected a type annotation for VARIABLE chan
According to TypeInvariant
, the variable chan
is a record that has three
fields: val
, rdy
, and ack
. The field val
ranges over a set Data
,
which is actually defined as CONSTANT
. In principle, we can annotate the
constant Data
with a set of any type, e.g., Set(Int)
or Set(BOOLEAN)
.
Since the specification is not using any operators over Data
except equality,
we can use an uninterpreted type as a type for set elements, e.g.,
we can define Data
to have the type Set(DATUM)
. Uninterpreted types are
always written in CAPITALS. Now we can annotate Data
and chan
as follows:
CONSTANT
\* @type: Set(DATUM);
Data
VARIABLE
\* @type: { val: DATUM, rdy: Int, ack: Int };
chan
Note carefully that the type annotation should be between the keyword
CONSTANT
and the constant name. This is because constant declarations may
declare several constants at once. In this case, you have to write one type
annotation per name.
Have a look at the type of chan
:
\* @type: { val: DATUM, rdy: Int, ack: Int };
The type of chan
is a record that has three fields: field val
of type
DATUM
, field rdy
of type Int
, field ack
of type Int
. The record type syntax is similar to dictionary syntax from programming languages (e.g. Python). We made it different
from TLA+'s syntax for records [ val > v, rdy > r, ack > a ]
and record sets [ val: V, rdy: R, ack: A ]
, to avoid confusion between
types and values.
Run the type checker again. You should see the following message:
$ apalachemc typecheck ChannelTyped.tla
...
> Running Snowcat .::.
> Your types are purrfect!
> All expressions are typed
Recipe 3: Annotating operators
Check the example CarTalkPuzzle.tla from the repository of TLA+
examples. This example has 160 lines of code, so we do not inline it here.
By running the type checker as in previous sections, you should figure out
that the constants N
and P
should be annotated with the type Int
.
Annotate N
and P
with Int
and run the type checker:
$ apalachemc typecheck CarTalkPuzzle.tla
Now you should see the following error:
[CarTalkPuzzle.tla:52:3252:35]: Cannot apply f to the argument x() in f[x()].
[CarTalkPuzzle.tla:50:152:53]: Error when computing the type of Sum
Although the error message may look confusing, the reason is simple: The type
checker cannot figure out whether the operator Sum
expects a sequence
or a function of integers as its first parameter. By looking carefully at
the definition of Sum
, we can see that it expects: (1) a function from
integers to integers as its first parameter, (2) a set of integers
as its second parameter, and (3) an integer as a result. Hence, we annotate
Sum
as follows:
RECURSIVE Sum(_,_)
\* type: (Int > Int, Set(Int)) => Int;
Sum(f,S) ==
...
Note that the annotation has to be written between RECURSIVE Sum(_, _)
and
the definition of Sum
. This might change later, see Issue 578 at tlaplus.
After providing the type checker with the annotation for Sum
, we get one
more type error:
[CarTalkPuzzle.tla:160:23160:26]: Cannot apply B to the argument x in B[x].
[CarTalkPuzzle.tla:160:7160:37]: Error when computing the type of Image
This time the type checker cannot choose between two options for the second
parameter of Image
: It could be a function, or a sequence. We help the
type checker by writing that the second parameter should be a function
of integers to integers, that is, Int > Int
:
\* @type: (Set(Int), Int > Int) => Set(Int);
Image(S, B) == {B[x] : x \in S}
This time the type checker can find the types of all expressions:
...
> Running Snowcat .::.
> Your types are purrfect!
> All expressions are typed
...
Recipe 4: Using variants in heterogenous sets
Check the example TwoPhase.tla from the repository of TLA+ examples (you will also need TCommit.tla, which is imported by TwoPhase.tla). This example has 176 lines of code, so we do not inline it here.
As you probably expected, the type checker complains about not knowing
the types of constants and variables. As for constant RM
, we opt for using
an uninterpreted type that we call RM
. That is:
CONSTANT
\* @type: Set(RM);
RM \* The set of resource managers
By looking at the spec, it is easy to guess the types of the variables
rmState
, tmState
, and tmPrepared
:
VARIABLES
\* @type: RM > Str;
rmState, \* $rmState[rm]$ is the state of resource manager RM.
\* @type: Str;
tmState, \* The state of the transaction manager.
\* @type: Set(RM);
tmPrepared, \* The set of RMs from which the TM has received $"Prepared"$
\* messages.
The type of the variable msgs
is less obvious. We can check the original
(untyped) definitions of TPTypeOK
and Message
to get an idea about the
type of msgs
:
Message ==
({[type > t, rm > r]: t \in {"Prepared"}, r \in RM }
\union
{[type > t] : t \in {"Commit", "Abort"}})
TPTypeOK ==
...
/\ msgs \in SUBSET Message
From these (untyped) definitions, you can see that msgs
is a set that
contains records of two types: { type: Str }
and { type: Str, rm: RM }
.
This seems to be problematic, as we have to mix in two records types in a
single set, which requires us to specify its only type.
To this end, we have to use the Variants module, which is distributed with
Apalache. For reference, check the Chapter on variants. First, we declare a
type alias for the type of messages in a separate file called
TwoPhaseTyped_typedefs.tla
:
 MODULE TwoPhaseTyped_typedefs 
(*
@typeAlias: message = Commit(NIL)  Abort(NIL)  Prepared(RM);
*)
TwoPhaseTyped_aliases == TRUE
======================================================================
Usually, we place type aliases in a separate file for when we have to use the same type alias in different specifications, e.g., the specification and its instance for model checking.
With the type alias MESSAGE
, we specify that a message is a variant type,
that is, it can represent three kinds of different values:

A value tagged with
Commit
. Since we do not require the variant to carry any value here, we simply declare that the value has the uninterpreted typeNIL
. This is simply a convention, we could use any type in this case. 
A value tagged with
Abort
. Similar toCommit
, we are using theNIL
type. 
A value tagged with
Prepared
. In this case, the value is of importance. We are using the valueRM
, that is, the (uninterpreted) type of a resource manager.
Once we have specified the variant type, we introduce three constructors, one per variant option:
\* @type: $message;
MkCommit == Variant("Commit", "0_OF_NIL")
\* @type: $message;
MkAbort == Variant("Abort", "0_OF_NIL")
\* @type: RM => $message;
MkPrepared(rm) == Variant("Prepared", rm)
Since the values carried by the Commit
and Abort
messages are not
important, we use the uninterpeted value "0_OF_NIL"
. This is merely a
convention. We could use any value of type NIL
. Importantly, the operators
MkAbort
, MkCommit
, and MkPrepared
all produce values of type MESSAGE
,
which makes it possible to add them to a single set of messages.
Now it should be clear how to specify the type of the variable msgs
:
\* @type: Set($message);
msgs
We run the type checker once again:
$ apalachemc typecheck TwoPhaseTyped.tla
...
> All expressions are typed
Type checker [OK]
As you can see, variants require quite a bit of boilerplate. If you can simply
introduce a set of records of the same type, this is usually a simpler
solution. For instance, we could partition msgs
into three subsets: the
subset of Commit
messages, the subset of Abort
messages, and the subset of
Prepared
messages. See the discussion in Idiom 15.
Recipe 5: functions as sequences
Check the example Queens.tla from the repository of TLA+ examples. It has 85 lines of code, so we do not include it here. Similar to the previous sections, we annotate constants and variables:
CONSTANT
\* @type: Int;
N \** number of queens and size of the board
...
VARIABLES
\* @type: Set(Seq(Int));
todo,
\* @type: Set(Seq(Int));
sols
After having inspected the type errors reported by Snowcat, we annotate the
operators Attacks
, IsSolution
, and vars
as follows:
\* @type: (Seq(Int), Int, Int) => Bool;
Attacks(queens,i,j) ==
...
\* @type: Seq(Int) => Bool;
IsSolution(queens) ==
...
\* @type: <<Set(Seq(Int)), Set(Seq(Int))>>;
vars == <<todo,sols>>
Now we run the type checker and receive the following type error:
[Queens.tla:35:4435:61]: The operator IsSolution of type ((Seq(Int)) => Bool) is applied to arguments of incompatible types in IsSolution(queens):
Argument queens should have type Seq(Int) but has type (Int > Int). E@11:07:53.285
[Queens.tla:35:135:63]: Error when computing the type of Solutions
Let's have a closer look at the problematic operator definition of Solutions
:
Solutions ==
{ queens \in [1..N > 1..N]: IsSolution(queens) }
This looks interesting: IsSolution
expects a sequence, whereas
Solutions
produces a set of functions. This is obviously not a
problem in untyped TLA+. In fact, it is a wellknown idiom: Construct a
function by using the function set operator, and then apply sequence operators to it.
In Apalache we have to explicitly write that a function should be reinterpreted
as a sequence. To this end, we have to use the operator FunAsSeq
from the
module Apalache.tla. Hence, we add Apalache
to the EXTENDS
clause and
apply the operator FunAsSeq
as follows:
EXTENDS Naturals, Sequences, Apalache
...
Solutions ==
LET Queens == { FunAsSeq(queens, N, N): queens \in [1..N > 1..N] } IN
{queens \in Queens : IsSolution(queens)}
This time the type checker can find the types of all expressions:
> Running Snowcat .::.
> Your types are purrfect!
> All expressions are typed
Recipe 6: type aliases
Type aliases can be used to provide a concise label for complex types, or to clarify the intended meaning of a simple types in the given context.
Type aliases are declared with the @typeAlias
annotation, as follows:
\* @typeAlias: aliasNameInCamelCase = <type>;
For example, suppose we have annotated some constants as follows:
CONSTANTS
\* @type: Set(PERSON);
Missionaries,
\* @type: Set(PERSON);
Cannibals
If we continue annotating other declarations in the specification, we will see
that the type Set(PERSON)
is used frequently. Type aliases let us provide a
shortcut.
By convention, we introduce all type aliases by annotating an operator called
<PREFIX>_typedefs
, where the <PREFIX>
is replaced with a unique prefix to
prevent name clashes across different modules. Typically <PREFIX>
is just the
module name. For the MissionariesAndCannibalsTyped.tla example, we have:
\* @typeAlias: persons = Set(PERSON);
MissionariesAndCannibals_typedefs = TRUE
Having defined the type alias, we can use it in other definitions anywhere else in the file:
CONSTANTS
\* @type: $persons;
Missionaries,
\* @type: $persons;
Cannibals
VARIABLES
\* @type: Str;
bank_of_boat,
\* @type: Str > $persons;
who_is_on_bank
Surely, we did not gain much by writing $persons
instead of Set(PERSON)
.
But if your specification has complex types (e.g., records), aliases may help
you in minimizing the burden of specification maintenance. If you add one
more field to the record type, it suffices to change the definition of the type
alias, instead of changing the record type everywhere.
For more details on the design and usage, see Type Aliases in ADR002.
Recipe 7: Multiline annotations
A type annotation may span over multiple lines. You may use both the (* ... *)
syntax as well as the singleline syntax \* ...
. All three examples below
are accepted by the parser:
VARIABLES
(*
@type: Int
=> Bool;
*)
f,
\* @type:
\* Int
\* => Bool;
g,
\* @type("Int
\* => Bool
\* ")
h
Note that the parser removes the leading strings " \*"
from the annotations,
similar to how multiline strings are treated in modern programming languages.
Recipe 8: Comments in annotations
Sometimes, it helps to document the meaning of type components. Consider the following example from Recipe 5:
\* @type: (Seq(Int), Int, Int) => Bool;
Attacks(queens,i,j)
If you think that an explanation of the arguments would help, you can do that as follows:
(*
@type:
(
// the column of an nth queen, for n in the sequence domain
Seq(Int),
// the index (line number) of the first queen
Int,
// the index (line number) of the second queen
Int
) => Bool;
*)
Attacks(queens,i,j)
You don't have to do that, but if you feel that types can also help you in documenting your specification, you have this option.
Recipe 9: Migrate from Type System 1 to Type System 1.2
As explained in ADR002, Type System 1.2 (TS1.2) differs from Type System 1 (TS1) as follows:

TS1 allows one to mix records of varying domains, as long as the records agree on the types of the common fields. Hence, record access is not enforced by the type checker and thus is errorprone.

TS1 is using the syntax
[ field_n: T_1, ..., field_n: T_n ]
, which is sometimes confused with the TLA+ expression[ field_n: e_1, ..., field_n: e_n ]
. 
TS1.2 is using the syntax
{ field_n: T_1, ..., field_n: T_n }
for record types and the syntaxTag_1(T_1)  ...  Tag_n(T_n)
for variant types. 
TS1.2 differentiates between records of different domains and does not allow the specification writer to mix them. As a result, TS1.2 can catch incorrect record access. Instead of mixing records, TS1.2 allows one to mix Variants.

TS1.2 supports Row polymorphism and thus lets the user write type annotations over records and variants, whose shape is only partiallydefined. For example,
{ foo: Int, bar: Bool, a }
defines a record type that has at least two fields (that is,foo
of typeInt
andbar
of typeBool
), but may have more fields, which are captured with the row variablea
.
Case 1: plain records
Many specifications are using plain records. For instance, they do not assign records of different domains to the same variable. Nor do they mix records of different domains in the same set. Plenty of specifications fall into this class.
For example, check Recipe 2. In this recipe, the variable chan
is
always carrying a record with the domain { "val", "rdy", "ack" }
.
In this case, all you have to do is to replace the old record types of the form
[ field_n: T_1, ..., field_n: T_n ]
with the new record types of the form { field_n: T_1, ..., field_n: T_n }
. That is, replace [
and ]
with {
and
}
, respectively.
Case 2: mixed records
Some specifications are using mixed records, which are similar to unions in C.
For example, check Recipe 4. In this recipe, the variable
tmPrepared
is a set that contains records of different domains. For instance,
tmPrepared
may be equal to:
{ [ type > "Commit" ], [ type > "Prepared", rm > "0_OF_RM" ] }
In this case, you have two choices:

Partition the single variable into multiple variables, see Idiom 15.

Introduce variant types, see Recipe 4.
Known issues
Annotations of LOCAL operators
In contrast to all other cases, a local operator definition does require
a type annotation before the keyword LOCAL
, not after it. For example:
\* @type: Int => Int;
LOCAL LocalInc(x) == x + 1
This may change later, when the tlaplus Issue 578 is resolved.
How to use uninterpreted types
This HOWTO explains what uninterpreted types are in the context of Apalache's type system, outlined in ADR002, and where/how to use them.
What are uninterpreted types?
It is often the case, when writing specifications, that inputs (CONSTANTS
) describe a collection of values, where the only relevant property is that all of the values are considered unique. For instance, TwoPhase.tla defines
CONSTANT RM \* The set of resource managers
however, for the purposes of specification analysis, it does not matter if we instantiate
RM = 1..3
or RM = {"a","b","c"}
, because the only operators applied to elements of RM
are polymorphic in the type of the elements of RM
.
For this reason, Apalache supports a special kind of type annotation: uninterpreted types. The type checker Snowcat makes sure that a value belonging to an uninterpreted type is only ever passed to polymorphic operators, and, importantly, that it is never compared to a value of any other type.
When to use uninterpreted types?
For efficiency reasons, you should use uninterpreted types whenever a CONSTANT
or value represents (an element of) a collection of unique identifiers, the precise value of which does not influence the properties of the specification.
On the other hand, if, for example, the order of values matters, identifiers should likely be 1..N
and hold type Int
instead of an uninterpreted type, since Int
values can be passed to the nonpolymorphic <,>,>=,<=
operators.
How to annotate uninterpreted types
Following ADR002, an annotation with an uninterpreted type looks exactly like an annotation with a type alias:
\* @type: UTNAME;
where UTNAME
matches the pattern [AZ_][AZ09_]*
, and is not a type alias defined elsewhere.
How to introduce values belonging to an uninterpreted type
Apalache uses the following conventionbased naming scheme for values of uninterpreted types:
"identifier_OF_TYPENAME"
where:
TYPENAME
is the uninterpreted type to which this value belongs, matching the pattern[AZ_][AZ09_]*
, andidentifier
is a unique identifier within the uninterpreted type, matching the pattern[azAZ09_]+
.
Example: "1_OF_UT"
is a value belonging to the uninterpreted type UT
, as is "2_OF_UT"
. These two values are distinct by definition. On the contrary,
"1_OF_ut"
does not meet the criteria for a value belonging to an uninterpreted type ( lowercase ut
is not a valid identifier for an uninterpreted type), so it is treated as a string value.
Note: Values matching the pattern "FRESH[09]+_OF_TYPENAME"
are reserved for internal use, to allow Apalache to construct fresh values.
Uninterpreted types, Str
, and comparisons
Importantly, while both strings and values belonging to uninterpreted types are introduced using the "..."
notation, they are treated as having distinct, incomparable types.
Examples:
 The following expression is typeincorrect:
"abc" = "bc_OF_A" \* Cannot compare values of types Str and A
 The following expression is typeincorrect:
"1_OF_A" = "1_OF_B" \* Cannot compare values of types A and B
 The following expressions are typecorrect:
\* Can compare 2 values of type A. "1_OF_A" = "2_OF_A" \* = FALSE, identifiers are different "1_OF_A" = "1_OF_A" \* = TRUE, identifiers are the same
Getting Started
Apalache is a symbolic model checker for the specification language TLA+. As such, it is a recent alternative to the explicitstate model checker TLC.
Apalache vs. TLC
Whereas TLC enumerates the states produced by the behaviors of a TLA+ specification, Apalache translates the verification problem to a set of logical constraints. These constraints are solved by an SMT solver, for example, by Microsoft Z3. That is, Apalache operates on formulas (i.e., symbolically), not by enumerating states one by one (i.e., state enumeration).
Shall I use Apalache or TLC?
Depending on the specification you wrote, either TLC or Apalache may be more efficient in checking it. While TLC is a mature tool, Apalache is still experimental, so be prepared to use the commandline and to help us discover bugs.
Assumptions
Apalache is working under the following assumptions:
 As in TLC, all specification parameters are fixed and finite, i.e., the system state is initialized with integers, finite sets, and functions of finite domains and codomains.
 As in TLC, all data structures evaluated during an execution are finite, e.g., a system specification cannot operate on the set of all integers.
 Only finite executions of bounded length are analyzed.
Conventions
Throughout this manual, we use the following conventions:
APALACHE_HOME
is used as shorthand for the path to a local copy of the Apalache source code.
Installation
There are three ways to run Apalache:
 Prebuilt package: download a prebuilt package and run it in the JVM.
 Docker: download and run a Docker image.
 Build from source: build Apalache from sources and run the compiled package.
If you just want to try the tool, we recommend using the prebuilt package.
System requirements
Memory: Apalache uses Microsoft Z3 as a backend SMT solver, and the required memory largely depends on Z3. We recommend to allocate at least 4GB of memory for the tool.
Prebuilt Packages
You need to download and install a Java Virtual Machine first. We recommend version 17 of the Eclipse Temurin or Zulu builds of OpenJDK.
Once you have installed Java, download the latest release and unpack into a directory of your choice. Depending on your OS, you have two options.
Option 1: Linux, macOS. You can run the script ./bin/apalachemc
, or,
better, add the ./bin
directory to your PATH
and run apalachemc
.
Option 2: Windows. You have to run Java directly with
java.exe jar ./lib/apalache.jar <args>
The arguments <args>
are explained in Running the Tool.
If you would like to contribute a commandline script for running Apalache in Windows, please open a pull request.
Using a docker image
We publish Docker images for every release 😎
NOTE: Running Apalache through a docker application image complicates configuration of the tool considerably. Unless you have a pressing need to use the docker image, we recommend using one of our prebuilt releases.
Docker lets you to run the Apalache tool from inside an isolated container. The only dependency required to run Apalache is the a suitable JVM, and the container supplies this. However, you must already have docker installed.
To get the latest Apalache image, issue the command:
docker pull ghcr.io/informalsystems/apalache
Running the docker image
To run an Apalache image, issue the command:
$ docker run rm v <yourspecdirectory>:/var/apalache ghcr.io/informalsystems/apalache <args>
The following docker parameters are used:

rm
to remove the container on exit 
v <yourspecdirectory>:/var/apalache
bindmounts<yourspecdirectory>
into/var/apalache
in the container. This is necessary for Apalache to access your specification and the modules it extends. From the user perspective, it works as if Apalache was executing in<yourspecdirectory>
. In particular the tool logs are written in that directory.When using SELinux, you might have to use the modified form of
v
option:v <yourspecdirectory>:/var/apalache:z

informalsystems/apalache
is the APALACHE docker image name. By default, thelatest
stable version is used; you can also refer to a specific tool version, e.g.,informalsystems/apalache:0.6.0
orinformalsystems/apalache:main

<args>
are the tool arguments as described in Running the Tool.
We provide a convenience wrapper for this docker command in
script/rundocker.sh
. Assuming you've downloaded the Apalache source code into
a directory located at APALACHE_HOME
, you can run the latest
image via the
script by running:
$ $APALACHE_HOME/script/rundocker.sh <args>
To specify a different image, set APALACHE_TAG
like so:
$ APALACHE_TAG=foo $APALACHE_HOME/script/rundocker.sh <args>
Setting an alias
If you are running Apalache on Linux 🐧 or MacOS 🍏, you can define this handy alias in your rc file, which runs Apalache in docker while sharing the working directory:
###### using the latest stable
$ alias apalache='docker run rm v $(pwd):/var/apalache ghcr.io/informalsystems/apalache'
###### using the latest main
$ alias apalache='docker run rm v $(pwd):/var/apalache ghcr.io/informalsystems/apalache:main'
Using the development branch of Apalache
The development of Apalache proceeds at a quick pace and we cut releases weekly.
Please refer to the changelog and the manual on the main
development
branch for a report of the newest features. Since we cut releases weekly, you
should have access to all the latest features in the last week by using the
latest
tag. However, if you wish to use the very latest developments as they
are added throughout the week, you can run the image with the main
tag: just
type ghcr.io/informalsystems/apalache:main
instead of
ghcr.io/informalsystems/apalache
everywhere.
Do not forget to pull the docker image from time to time:
docker pull ghcr.io/informalsystems/apalache:main
Run it with the following command:
$ docker run rm v <yourspecdirectory>:/var/apalache ghcr.io/informalsystems/apalache:main <args>
To create an alias pointing to the main
version:
$ alias apalache='docker run rm v $(pwd):/var/apalache ghcr.io/informalsystems/apalache:main'
Building an image
For an end user there is no need to build an Apalache image. If you like to
produce a modified docker image, take into account that it will take about 30
minutes for the image to get built, due to compilation times of Microsoft Z3. To
build a docker image of Apalache, issue the following command in
$APALACHE_HOME
:
$ docker image build t apalache:0.7.0 .
Building from source
 Install
git
.  Install the Eclipse Temurin or Zulu builds of OpenJDK 17.
 Apalache currently requires Scala 2.13.8, see this table of compatible JDK versions.
 We recommend OpenJDK 17, the latest LTS release.
 Install sbt.
 On Debian Linux or Ubuntu, follow this guide
 On Arch:
sudo pacman Syu sbt
 On macOS / Homebrew:
brew install sbt
 Clone the git repository:
git clone https://github.com/informalsystems/apalache.git
.  Change into the project directory:
cd apalache
.  Install direnv and run
direnv allow
, or use another way to set up the shell environment.  To build and package Apalache for development purposes, run
make
. To skip running the tests, you can run
make package
.
 To skip running the tests, you can run
 To build and package Apalache for releases and enduser distribution, run
make dist
.  The distribution package will be built to
./target/universal/apalache<VERSION>
, and you can move this wherever you'd like, and ensure that the<distpackagelocation>/bin
directory is added to yourPATH
.
Running from within the Apalache repo
If you prefer to keep the built package inside of the Apalache source
repository, you have three options after running make
:
 Add the
./bin
directory in the source repository to yourPATH
, which will makeapalchemc
available.  Install direnv and run
direnv allow
, which will putapalchemc
in your path when you are inside of the Apalache repo directory.  Run
./bin/apalachemc
directly.
Running the Tool
Optin statistics programme: if you optin for statistics collection (off by default), then every run of Apalache
will submit anonymized statistics to
tlapl.us
. See the details in TLA+ Anonymized Execution Statistics.
Apalache supports several modes of execution. You can run it with the help
option,
to see the complete list of modes and their invocation commands:
$ apalachemc help
The most important commands are as follows:

parse
reads a TLA+ specification with the SANY parser and flattens it by instantiating all modules. It terminates successfully, if there are no parse errors. The input specification toparse
may be given in standard TLA+ format, or in the JSON serialization format, while the outputs are produced in both formats. 
typecheck
performs all of the operations ofparse
and additionally runs the type checker Snowcat to infer the types of all expressions in the parsed specification. It terminates successfully, if there are no type errors. 
simulate
performs all of the operations oftypecheck
and additionally runs the model checker in simulation mode, which randomly picks a sequence of actions and checks the invariants for the subset of all executions which only admit actions in the selected order. It terminates successfully, if there are no invariant violations. This command usually checks randomized symbolic runs much faster than thecheck
command. 
check
performs all of the operations oftypecheck
and then runs the model checker in bounded model checking mode, which checks invariants for all executions, the length of which does not exceed the value specified by thelength
parameter. It terminates successfully, if there are no invariant violations. 
test
performs all of the operations ofcheck
in a mode that is designed to test a single action.
1. Model checker and simulator commandline parameters
1.1. Model checker commandline parameters
The model checker can be run as follows:
$ apalachemc check [config=filename] [init=Init] [cinit=ConstInit] \
[next=Next] [inv=Inv1,...,Invn] [length=10] \
[temporal=TemporalProp1,...,TemporalPropn] \
[algo=(incrementaloffline)] \
[discarddisabled] [nodeadlock] \
[tuningoptionsfile=filename] [tuningoptions=key1=val1:...:keyn=valn] \
[smtencoding=(oopsla19arrays)] \
[outdir=./path/to/dir] \
[writeintermediate=(truefalse)] \
[configfile=./path/to/file] \
[profiling=false] \
[outputtraces=false] \
<myspec>.tla
The arguments are as follows:

General parameters:
config
specifies the TLC configuration fileinit
specifies the initialization predicate,Init
by defaultnext
specifies the transition predicate,Next
by defaultcinit
specifies the constant initialization predicate, optionalinv
specifies the invariants to check, as a comma separated list, optionallength
specifies the maximal number ofNext
steps, 10 by defaulttemporal
specifies the temporal properties to check, as a comma separated list, optional

Advanced parameters:
algo
lets you to choose the search algorithm:incremental
is using the incremental SMT solver,offline
is using the nonincremental (offline) SMT solversmtencoding
lets you choose how the SMT instances are encoded:oopsla19
(default) uses QF_UFNIA, andarrays
(experimental) andfunArrays
(experimental) use SMT arrays with extensionality. This parameter can also be set via theSMT_ENCODING
environment variable. See the alternative SMT encoding using arrays for details.discarddisabled
does a precheck on transitions and discard the disabled ones at every step. If you know that many transitions are always enabled, set it to false. Sometimes, this precheck may be slower than checking the invariant. Default: true.maxerror <n>
instructs the search to stop aftern
errors, see Enumeration of counterexamples. Default: 1.view <name>
sets the state view to<name>
, see Enumeration of counterexamples.nodeadlock
disables deadlockchecking, whendiscarddisabled=false
is on. Whendiscarddisabled=true
, deadlocks are found in any case.tuningoptionsfile
specifies a properties file that stores options for fine tuningtuningoptions
can pass and/or override these fine tuning options on the command lineoutdir
set location for outputting any generated logs or artifacts,./_apalacheout
by defaultwriteintermediate
iftrue
, then additional output is generated. See Detailed output.false
by defaultrundir=DIRECTORY
write all outputs directly into the specifiedDIRECTORY
configfile
a file to use for loading configuration parameters. This will prevent Apalache from looking for any local.apalache.cfg
file.profiling
(Bool): This flag governs the creation ofprofilerules.txt
used in profiling. The file is only created ifprofiling
is set toTrue
. Settingprofiling
toFalse
is incompatible with thesmtprof
flag. The default isFalse
.outputtraces
: save an example trace for each symbolic run, default:false
Options passed with tuningoptions
have priority over options passed with tuningoptionsfile
.
If an initialization predicate, transition predicate, or invariant is specified both in the configuration file, and on the command line, the command line parameters take precedence over those in the configuration file.
In case conflicting arguments are passed for the same parameter, the following precedence order is followed:
 Commandline value
 Environment variable value
 Configuration file value
1.2. Simulator commandline parameters
The simulator can be run as follows:
$ apalachemc simulate
[allcheckeroptions] [maxrun=NUM] <myspec>.tla
The arguments are as follows:

Special parameters:
maxrun=NUM
: but produce up toNUM
simulation runs (unlessmaxerror
errors have been found), default:100
1.3. Supplying JVM arguments
You can supply JVM argument to be used when running Apalache by setting the
environment variable JVM_ARGS
. For example, to change the JVM heap size from
the default (4096m
) to 1G
invoke Apalache as follows:
JVM_ARGS="Xmx1G" apalachemc <args>
If you are running Apalache via docker directly (instead of using the script in
$APALACHE_HOME/script/rundocker.sh
), you'll also need to expose the
environment variable to the docker container:
$ JVM_ARGS="Xmx1G" docker run e JVM_ARGS rm v <yourspecdirectory>:/var/apalache ghcr.io/informalsystems/apalache <args>
To track memory usage with: jcmd <pid> VM.native_memory summary
, you can set
JVM_ARGS="XX:NativeMemoryTracking=summary"
1.4. Bounded model checking
By default, Apalache performs bounded model checking, that is, it encodes a
symbolic execution of length k
and a violation of a state invariant in SMT:
/\ Init[v_0/v]
/\ Next[v_0/v, v_1/v'] /\ Next[v_1/v, v_2/v'] /\ ... /\ Next[v_{k1}/v, v_k/v']
/\ ~Inv[v_0/v] \/ ~Inv[v_1/v] \/ ... \/ ~Inv[v_k/v]
Here an expression Inv[v_i/v]
means that the state variables v
are replaced with their copies v_i
for the
state i
. Likewise, Next[v_i/v,v_{i+1}/v']
means that the state variables v
are replaced with their copies v_i
for the state i
, whereas the state
variables v'
are replaced with their copies
v_{i+1}
for the state i+1
.
Bounded model checking is an incomplete technique. If Apalache finds a bug in this symbolic execution (by querying z3), then it reports a counterexample. Otherwise, it reports that no bug was found up to the given length. If a bug needs a long execution to get revealed, bounded model checking may miss it!
1.5. Checking an inductive invariant
To check executions of arbitrary lengths, one usually finds a formula that satisfies the two following properties:
/\ Init => TypeOK /\ IndInv
/\ TypeOK /\ IndInv /\ Next => TypeOK' /\ IndInv'
In normal words: (1) The initial states satisfy the constraint TypeOK /\ IndInv
, and (2) whenever the specification makes a step when starting in a state that satisfies TypeOK /\ IndInv
, it
ends up in a state that again satisfies TypeOK /\ IndInv
.
Note that we usually check IndInv
in conjunction with TypeOK
, as we have to constrain the variable values. In
the y2k
example, our inductive invariant is actually constraing the variables. In fact, such an inductive invariant is
usually called TypeOK
.
To check an inductive invariant IndInv
in Apalache, you run two commands that check the above two formulas:

IndInit: Check that the initial states satisfy
IndInv
:$ apalachemc check init=Init inv=IndInv length=0 <myspec>.tla

IndNext: Check that
Next
does not drive us outside ofIndInv
:$ apalachemc check init=IndInv inv=IndInv length=1 <myspec>.tla
Usually, you look for an inductive invariant to check a safety predicate. For
example, if you have found an inductive invariant IndInv
and want to check a
safety predicate Safety
, you have to run Apalache once again:

IndProp: Check that all states captured with
IndInv
satisfy the predicateSafety
:$ apalachemc check init=IndInv inv=Safety length=0 <myspec>.tla
It may happen that your inductive invariant IndInv
is too weak and it
violates Safety
. In this case, you would have to add additional constraints to IndInv
.
Then you would have to check the queries IndInit, IndNext, and IndProp again.
2. Examples
2.1. Checking safety up to 20 steps
$ cd test/tla
$ apalachemc check length=20 inv=Safety y2k_override.tla
This command checks, whether Safety
can be violated in 20 specification steps. If Safety
is not violated, your spec
might still have a bug that requires a computation longer than 20 steps to manifest.
2.2. Checking an inductive invariant:
$ cd test/tla
$ apalachemc check length=0 init=Init inv=Inv y2k_override.tla
$ apalachemc check length=1 init=Inv inv=Inv y2k_override.tla
The first call to apalache checks, whether the initial states satisfy the invariant. The second call to apalache checks, whether a single specification step satisfies the invariant, when starting in a state that satisfies the invariant. ( That is why these invariants are called inductive.)
2.3. Using a constant initializer:
$ cd test/tla
apalachemc check cinit=ConstInit length=20 inv=Safety y2k_cinit.tla
This command checks, whether Safety
can be violated in 20 specification steps. The constants are initialized with the
predicate
ConstInit
, defined in y2k_cinit.tla
as:
ConstInit == BIRTH_YEAR \in 0..99 /\ LICENSE_AGE \in 10..99
In this case, Apalache finds a safety violation, e.g., for
BIRTH_YEAR=89
and LICENSE_AGE=10
. A complete counterexample is printed in counterexample.tla
.
The final lines in the file clearly indicate the state that violates the invariant:
State14 ==
/\ BIRTH_YEAR = 89
/\ LICENSE_AGE = 10
/\ hasLicense = TRUE
/\ year = 0
(* The following formula holds true in the last state and violates the invariant *)
InvariantViolation == hasLicense /\ year  BIRTH_YEAR < LICENSE_AGE
3. Module lookup
Apalache uses the SANY parser, which is the standard parser of TLC and the TLA+ Toolbox. By default, SANY is looking for modules (in this order) in
 The current working directory.
 The directory containing the main TLA+ file passed on the CLI.
 A small Apalache standard library (bundled from
$APALACHE_HOME/src/tla
).  The Java package
tla2sany.StandardModules
(usually provided by thetla2tools.jar
that is included in the Java classpath).
Note: To let TLA+ Toolbox and TLC know about the Apalache modules, include
$APALACHE_HOME/src/tla
in the lookup directories, as explained by Markus Kuppe for
the TLA+ Community Modules.
4. Detailed output
The location for detailed output is determined by the value of the outdir
parameter, which specifies the path to a directory into which all Apalache
runs write their outputs (see configuration instructions).
Each run will produce a unique subdirectory inside its "namespace", derived from
the file name of the specification, using the following convention
yyyyMMddTHHmmss_<UNIQUEID>
.
For an example, consider using the default location of the rundir
for a run of
Apalache on a spec named test.tla
. This will create a directory structuring matching following pattern:
./_apalacheout/
└── test.tla
└── 20211105T225455_810261790529975561
The default value for the outdir
is the _apalacheout
directory in the
current working directly. The subdirectory test.tla
is derived from the name
of the spec on which the tool was run, and the runspecific subdirectory
20211105T225455_810261790529975561
gives a unique location to write the
all the outputs produced by the run.
The tool only writes important messages on stdout, but a detailed log can be
found in the detailed.log
in the runspecific subdirectory.
The directory also includes a file run.txt
, reporting the command line
arguments used during the run.
Additionally, if the parameter writeintermediate
is set to true
(see
configuration instructions) each pass of the model checker produces
intermediate TLA+ files in the runspecific subdirectory:
 File
outparser.tla
is produced as a result of parsing and importing into the intermediate representation, Apalache TLA IR.  File
outconfig.tla
is produced as a result of substituting CONSTANTS, as described in Setting up specification parameters.  File
outinline.tla
is produced as a result of inlining operator definitions andLETIN
definitions.  File
outpriming.tla
is produced as a result of replacing constants and variables inConstInit
andInit
with their primed versions.  File
outvcgen.tla
is produced as a result of extracting verification conditions, e.g., invariants to check.  File
outprepro.tla
is produced as a result of running all preprocessing steps.  File
outtransition.tla
is produced as a result of finding assignments and symbolic transitions.  File
outopt.tla
is produced as a result of expression optimizations.  File
outanalysis.tla
is produced as a result of analysis, e.g., marking Skolemizable expressions and expressions to be expanded.
5. Parsing and prettyprinting
If you'd like to check that your TLA+ specification is syntactically correct, without running the model checker, you can run the following command:
$ apalachemc parse <myspec>.tla
In this case, Apalache performs the following steps:

It parses the specification with SANY.

It translates SANY semantic nodes into Apalache IR .

If the
writeintermediate
flag is given, it prettyprints the IR into the output directory (see Detailed output).
You can also write output to a specified location by using the output
flag.
E.g.,
$ apalachemc parse output=result.json <myspec>.tla
will write the IR to the file result.json
.
reads a TLA+ specification with the SANY parser and flattens it by
An example TLA+ specification
We introduce a TLA+ specification that we use to discuss features of Apalache in
the following sections. Its source code can be found in
test/tla/y2k.tla
:
 MODULE y2k 
(*
* A simple specification of a year counter that is subject to the Y2K problem.
* In this specification, a registration office keeps records of birthdays and
* issues driver's licenses. As usual, a person may get a license, if they
* reached a certain age, e.g., age of 18. The software engineers never thought
* of their program being used until the next century, so they stored the year
* of birth using only two digits (who would blame them, the magnetic tapes
* were expensive!). The new millenium came with new bugs.
*
* This is a made up example, not reflecting any real code.
* To learn more about Y2K, check: https://en.wikipedia.org/wiki/Year_2000_problem
*
* Igor Konnov, January 2020
*)
EXTENDS Integers
CONSTANT
\* @type: Int;
BIRTH_YEAR, \* the year to start with, between 0 and 99
\* @type: Int;
LICENSE_AGE \* the minimum age to obtain a license
ASSUME(BIRTH_YEAR \in 0..99)
ASSUME(LICENSE_AGE \in 1..99)
VARIABLE
\* @type: Int;
year,
\* @type: Bool;
hasLicense
Age == year  BIRTH_YEAR
Init ==
/\ year = BIRTH_YEAR
/\ hasLicense = FALSE
NewYear ==
/\ year' = (year + 1) % 100 \* the programmers decided to use two digits
/\ UNCHANGED hasLicense
IssueLicense ==
/\ Age >= LICENSE_AGE
/\ hasLicense' = TRUE
/\ UNCHANGED year
Next ==
\/ NewYear
\/ IssueLicense
\* The somewhat "obvious" invariant, which is violated
Safety ==
hasLicense => (Age >= LICENSE_AGE)
Setting up specification parameters
Similar to TLC, Apalache requires the specification parameters to be restricted to finite values. In contrast to TLC, there is a way to initialize parameters by writing a symbolic constraint, see Section 5.3.
Using INSTANCE
You can set the specification parameters, using the standard INSTANCE
expression of TLA+. For instance, below is the example
y2k_instance.tla
,
which instantiates y2k.tla
:
 MODULE y2k_instance 
(*
* Another way to instantiate constants for apalache is to
* use INSTANCE.
*)
VARIABLE
\* @type: Int;
year,
\* @type: Bool;
hasLicense
INSTANCE y2k WITH BIRTH_YEAR < 80, LICENSE_AGE < 18
=============================================================================
The downside of this approach is that you have to declare the variables of the extended specification. This is easy with only two variables, but can quickly become unwieldy.
Convention over configuration
Alternatively, you can extend the base module and use overrides:
 MODULE y2k_override 
(*
* One way to instantiate constants for apalache is to use the OVERRIDE prefix.
*)
EXTENDS y2k
OVERRIDE_BIRTH_YEAR == 80
OVERRIDE_LICENSE_AGE == 18
=============================================================================
ConstInit predicate
This approach is similar to the Init
operator, but applied to the
constants. We define a special operator, e.g., called ConstInit
. For
instance, below is the example
y2k_cinit.tla
:
 MODULE y2k_cinit 
(*
* Another way to instantiate constants for apalache is give it constraints
* on the constants.
*)
EXTENDS y2k
ConstInit ==
/\ BIRTH_YEAR \in 0..99
/\ LICENSE_AGE \in 10..99
=============================================================================
To use ConstInit
, pass it as the argument to apalachemc
. For instance, for
y2k_cinit
, we would run the model checker as follows:
$ cd $APALACHE_HOME/test/tla
$ apalachemc check inv=Safety \
length=20 cinit=ConstInit y2k_cinit.tla
Parameterized initialization
As a bonus of this approach, Apalache allows one to check a specification over a bounded set of parameters. For example:
CONSTANT N, Values
ConstInit ==
/\ N \in 3..10
/\ Values \in SUBSET 0..4
/\ Values /= {}
The model checker will try the instances for all the combinations of
the parameters specified in ConstInit
, that is, in our example, it will
consider N \in 3..10
and all nonempty value sets that are subsets of 0..4
.
Limitation
ConstInit
should be a conjunction of assignments and possibly of additional
constraints on the constants. For instance, you should not write N = 10 \/ N = 20
. However, you can write N \in {10, 20}
.
TLC configuration file
We support configuring Apalache via TLC configuration files; these files are produced automatically by TLA Toolbox, for example. TLC configuration files allow one to specify which initialization predicate and transition predicate to employ, which invariants to check, as well as to initialize specification parameters. Some features of the TLC configuration files are not supported yet. Check the manual page on "Syntax of TLC Configuration Files".
Behavior in versions >=0.25.0:
Apalache never loads a TLC configuration file, unless a filename is passed
via the option config=<filename>
. If a filename is passed but the file
does not exist, Apalache reports an error.
Behavior in versions <0.25.0:
If you are checking a file <myspec>.tla
, and the file <myspec>.cfg
exists in
the same directory, it will be picked up by Apalache automatically. You can also
explicitly specify which configuration file to use via the config
option.
Principles of Symbolic Model Checking with Apalache
In order to take advantage of Apalache's symbolic model checking, there are a few principles one must bear in mind when writing TLA.
Note that Apalache requires type annotations. Check the Snowcat tutorial and HOWTO on annotations.
Topics:
 Assignments and symbolic transitions
 Folding sets and sequences
 Invariants: State, Action, Trace
 Enumeration of counterexamples
 The Apalache Module
 Naturals module
Assignments and symbolic transitions
Let us go back to the example
test/tla/y2k.tla
and run apalachemc
against
test/tla/y2k_override.tla
while instructing Apalache to write intermediate output files:
$ apalachemc check writeintermediate=true y2k_override.tla
We can check the detailed output of the TransitionFinderPass
in the file
_apalacheout/y2k_override.tla/<timestamp>/intermediate/<pass>_OutTransitionFinderPass.tla
, where
<timestamp>
looks like 20211201T120741_1998641578103809179
, and <pass>
is a twodigit number like 08
:
 MODULE 09_OutTransition 
EXTENDS Integers, Sequences, FiniteSets, TLC, Apalache
VARIABLE
(*
@type: Int;
*)
year
VARIABLE
(*
@type: Bool;
*)
hasLicense
(*
@type: (() => Bool);
*)
ASSUME(80 \in 0 .. 99)
(*
@type: (() => Bool);
*)
ASSUME(18 \in 1 .. 99)
(*
@type: (() => Bool);
*)
Init_si_0000 == year' := 80 /\ hasLicense' := FALSE
(*
@type: (() => Bool);
*)
Next_si_0000 == year' := ((year + 1) % 100) /\ hasLicense' := hasLicense
(*
@type: (() => Bool);
*)
Next_si_0001 == year  80 >= 18 /\ hasLicense' := TRUE /\ year' := year
================================================================================
As you can see, the model checker did two things:
 It has translated several expressions that look like
x' = e
intox' := e
. For instance, you can seeyear' := 80
andhasLicense' := FALSE
inInit_si_0000
. We call these expressions assignments.  It has factored the operator
Next
into two operatorsNext_si_0000
andNext_si_0001
. We call these operators symbolic transitions.
Pure TLA+ does not have the notions of assignments and symbolic
transitions. However, TLC sometimes treats expressions x' = e
and x' \in S
as if they were assigning a value to the variable x'
. TLC does so
dynamically, during the breadthfirst search. Apalache looks statically for assignments
among the expressions x' = e
and x' \in S
.
When factoring out operators into symbolic transitions, Apalache splits the
action operators Init
and Next
into disjunctions (e.g., A_0 \/ ... \/ A_n
),
represented in the concrete syntax as a sequence of operator definitions of the
form
A$0 == ...
...
A$n == ...
The main contract between the assignments and symbolic transitions is as follows:
For every variable
x
declared withVARIABLE
, there is exactly one assignment of the formx' := e
in every symbolic transitionA_n
.
If Apalache cannot find expressions with the above properties, it fails.
Consider the example
test/tla/Assignments20200309.tla
:
 MODULE Assignments20200309 
VARIABLE
\* @type: Int;
a
\* this specification fails, as it has no expression
\* that can be treated as an assignment
Init == TRUE
Next == a' = a
Inv == FALSE
===============
Run the checker with:
apalachemc check Assignments20200309.tla
Apalache reports an error as follows:
...
PASS #9: TransitionFinderPass
To understand the error, [check the
manual](https://apalache.informal.systems/docs/apalache/principles/assignments.html):
Assignment error: No assignments found for: a
It took me 0 days 0 hours 0 min 1 sec
Total time: 1.88 sec
EXITCODE: ERROR (255)
More details
Check Assignments and Symbolic Transitions in Depth.
Folding Sets and Sequences
Folds are an efficient replacement for recursive operators and functions.
Apalache natively implements two operators users might be familiar with from the community modules or functional programming. Those operators are ApaFoldSet
and ApaFoldSeqLeft
. This brief introduction to fold operators highlights the following:
 What are the semantics of fold operators?
 How do I use these operators in Apalache?
 Should I use folding or recursion?
 Examples of common operators defined with folds
Syntax
The syntax of the fold operators is as follows:
\* @type: ( (a, b) => a, a, Set(b) ) => a;
ApaFoldSet( operator, base, set )
\* @type: ( (a, b) => a, a, Seq(b) ) => a;
ApaFoldSeqLeft( operator, base, seq )
Semantics of fold operators
Folding refers to iterative application of a binary operator over a collection. Given an operator Op
, a base value b
and a collection of values C
, the definition of folding Op
over C
starting with b
depends on the type of the collection C
.
Semantics of ApaFoldSeqLeft
In the case of folding over sequences, C
is a sequence <<a_1, ..., a_n>>
. Then, ApaFoldSeqLeft( Op, b, C )
is defined as follows:
 If
C
is empty, thenApaFoldSeqLeft( Op, b, <<>> ) = b
, regardless ofOp
 If
C
is nonempty, we establish a recursive relation between folding overC
and folding overTail(C)
in the following way:ApaFoldSeqLeft( Op, b, C ) = ApaFoldSeqLeft( Op, Op(b, Head(C)), Tail(C) )
.
Semantics of ApaFoldSet
In the case of folding over sets, C
is a set {a_1, ..., a_n}
. Then, ApaFoldSet( Op, b, C )
is defined as follows:
 If
C
is empty, thenApaFoldSet( Op, b, {} ) = b
, regardless ofOp
 If
C
is nonempty, we establish a recursive relation between folding overC
and folding over some subset ofC
in the following way:ApaFoldSet( Op, b, C ) = ApaFoldSet( Op, Op(b, x), C \ {x} )
, wherex
is some arbitrary member ofC
(e.g.x = CHOOSE y \in C: TRUE
). Note that Apalache does not guarantee a deterministic choice ofx
, unlike what usingCHOOSE
would imply.
Note that the above are definitions of a left fold in the literature. Apalache does not implement a right fold.
For example, if C
is the sequence <<x,y,z>>
, the result is equal to Op( Op( Op(b, x), y), z)
. If C = {x,y}
, the result is either Op( Op(b, x), y)
or Op( Op(b, y), x)
. Because the order of elements selected from a set is not predefined, users should be careful, as the result is only uniquely defined in the case that the operator is both associative (Op(Op(a,b),c) = Op(a,Op(b,c))
) and commutative (Op(a,b) = Op(b,a)
).
For example, consider the operator Op(p,q) == 2 * p + q
, which is noncommutative, and the set S = {1,2,3}
. The value of ApaFoldSet(Op, 0, S)
depends on the order in which Apalache selects elements from S:
Order  ApaFoldSet value 

1 > 2 > 3  11 
1 > 3 > 2  12 
2 > 1 > 3  13 
2 > 3 > 1  15 
3 > 1 > 2  16 
3 > 2 > 1  17 
Because Apalache does not guarantee deterministic choice in the order of iteration, users should treat all of the above results as possible outcomes.
Using fold operators in Apalache
As shown by the type signature, Apalache permits a very general form of folding, where the types of the collection elements and the type of the base element/returntype of the operator do not have to match. Again, we urge users to exercise caution when using ApaFoldSet
with an operator, for which the types a
and b
are different, as such operators cannot be commutative or associative, and therefore the result is not guaranteed to be unique and predictable.
The other component of note is operator
, the name (not definition) of some binary operator, which is available in this context. The following are examples of valid uses of folds:
PlusOne(p,q) == p + q + 1
X == ApaFoldSet( PlusOne, 0, {1,2,3} ) \* X = 9
X == LET Count(p,q) == p + 1 IN ApaFoldSeqLeft( Count, 0, <<1,2,3>> ) \* X = 3
while these next examples are considered invalid:
\* LAMBDAS in folds are not supported by Apalache
\* Define a LETIN operator Plus(p,q) == p + q instead
X == ApaFoldSet( LAMBDA p,q: p + q, 0, {1,2,3} )
\* Builtin operators cannot be called by name in Apalache
\* Define a LETIN operator Plus(p,q) == p + q instead
X == ApaFoldSet( + , 0, {1,2,3} )
Local LET definitions can also be used as closures:
A(x) == LET PlusX(p,q) == p + q + x IN ApaFoldSeqLeft( PlusX, 0, <<1,2,3>> )
X == A(1) \* X = 9
Folding VS recursion
While TLA+ allows users to write arbitrary recursive operators, they are, in our experience, mostly used to implement collection traversals. Consider the following implementations of a Max
operator, which returns the largest element of a sequence:
\* Max(<<>>) = inf, but integers are unbounded in TLA+,
\* so there is no natural minimum like MIN_INT in programming languages
CONSTANT negInf
RECURSIVE MaxRec(_)
MaxRec(seq) == IF seq = <<>>
THEN negInf
ELSE LET tailMax == MaxRec(Tail(seq))
IN IF tailMax > Head(seq)
THEN tailMax
ELSE Head(seq)
MaxFold(seq) == LET Max(p,q) == IF p > q THEN p ELSE q
IN ApaFoldSeqLeft( Max, negInf, seq )
The first advantage of the fold implementation, we feel, is that it is much more clear and concise. It also does not require a termination condition, unlike the recursive case.
One inherent problem of using recursive operators with a symbolic encoding, is the inability to estimate termination.
While it may be immediately obvious to a human, that MaxRec
terminates after no more than Len(seq)
steps, automatic termination analysis is, in general, a rather complex and incomplete form of static analysis.
Apalache addresses this by finitely unrolling recursive operators and requires users to provide unroll limits (UNROLL_LIMIT_MaxRec == ...
), which serve as a static upper bound to the number of recursive reentries, because in general, recursive operators may take an unpredictable number of steps (e.g. computing the Collatz sequence) or never terminate at all.
Consider a minor adaptation of the above example, where the author made a mistake in implementing the operator:
RECURSIVE MaxRec(_)
MaxRec(seq) == IF seq = <<>>
THEN negInf
ELSE LET tailMax == MaxRec( seq ) \* forgot Tail!
IN IF tailMax > Head(seq)
THEN tailMax
ELSE Head(seq)
Now, MaxRec
never terminates, but spotting this error might not be trivial at a glance. This is where we believe folds hold the second advantage: ApaFoldSet
and ApaFoldSeqLeft
always terminate in Cardinality(set)
or Len(seq)
steps, and each step is simple to describe, as it consists of a single operator application.
In fact, the vast majority of the traditionally recursive operators can be equivalently rewritten as folds, for example:
RECURSIVE Cardinality(_)
Cardinality(set) == IF set = {}
THEN 0
ELSE LET x == CHOOSE y \in set: TRUE
IN 1 + Cardinality( set \ {x} )
CardinalityFold(set) == LET Count(p,q) == p + 1 \* the value of q, the set element, is irrelevant
IN ApaFoldSet( Count, 0, set )
Notice that, in the case of sets, picking an arbitrary element x
, to remove from the set at each step, utilizes the CHOOSE
operator. This is a common trait shared by many operators that implement recursion over sets.
Since the introduction of folds, the use of CHOOSE
in Apalache is heavily discouraged as it is both inefficient, as well as nondeterministic (unlike how CHOOSE
is defined in TLA+ literature). For details, see the discussion in issue 841.
So the third advantage of using folds is the ability to, almost always, avoid using the CHOOSE
operator.
The downside of folding, compared to general recursion, is the inability to express nonprimitively recursive functions. For instance, one cannot define the Ackermann function, as a fold. We find that in most specifications, this is not something the users would want to implement anyway, so in practice, we believe it is almost always better to use fold over recursive functions.
Folding VS quantification and CHOOSE
Often, folding can be used to select a value from a collection, which could alternatively be described by a predicate and selected with CHOOSE
. Let us revisit the MaxFold
example:
MaxFold(seq) == LET Max(p,q) == IF p > q THEN p ELSE q
IN ApaFoldSeqLeft( Max, negInf, seq )
The foldless case could, instead of using recursion, compute the maximum as follows:
MaxChoose(seq) ==
LET Range == {seq[i] : i \in DOMAIN seq}
IN CHOOSE m \in Range : \A n \in Range : m >= n
The predicatebased approach might result in a more compact specification, but that is because specifications have no notion of execution or complexity. Automatic verification tools, such as Apalache, the job of which includes finding witnesses to predicates, can work much faster with the fold approach. The reason is that evaluating CHOOSE x \in S : \A y \in S: P(x,y)
is quadratic in the size of S
(in a symbolic approach this is w.r.t. the number of constraints). For each candidate x
, the entire set S
must be tested for P(x,_)
. On the other hand, the fold approach is linear in the size of S
, since each element is visited exactly once.
In addition, the fold approach admits no undefined behavior. If, in the above example, seq
was an empty sequence, the value of the computed maximum depends on the value of CHOOSE x \in {}: TRUE
, which is undefined in TLA+, while the foldbased approach allows the user to determine behavior in that scenario (via the initial value).
Our general advice is to use folds over CHOOSE
with quantified predicates wherever possible, if you're willing accept a very minor increase in specification size in exchange for a decrease in Apalache execution time, or, if you wish to avoid CHOOSE
over empty sets resulting in undefined behavior.
Examples: The versatility of folds
Here we give some examples of common operators, implemented using folds:
 MODULE FoldDefined 
EXTENDS Apalache
\* Sum of all values of a set of integers
Sum(set) == LET Plus(p,q) == p + q IN ApaFoldSet( Plus, 0, set )
\* Reimplementation of UNION setOfSets
BigUnion(setOfSets) == LET Union(p,q) == p \union q IN ApaFoldSet( Union, {}, setOfSets )
\* Reimplementation of SelectSeq
SelectSeq(seq, Test(_)) == LET CondAppend(s,e) == IF Test(e) THEN Append(s, e) ELSE s
IN ApaFoldSeqLeft( CondAppend, <<>>, seq )
\* Quantify the elements in S matching the predicate P
Quantify(S, P(_)) == LET CondCount(p,q) == p + IF P(q) THEN 1 ELSE 0
IN ApaFoldSet( CondCount, 0, S )
\* The set of all values in seq
Range(seq) == LET AddToSet(S, e) == S \union {e}
IN LET Range == ApaFoldSeqLeft( AddToSet, {}, seq )
\* Finds the the value that appears most often in a sequence. Returns elIfEmpty for empty sequences
Mode(seq, elIfEmpty) == LET ExtRange == Range(seq) \union {elIfEmpty}
IN LET CountElem(countersAndCurrentMode, e) ==
LET counters == countersAndCurrentMode[1]
currentMode == countersAndCurrentMode[2]
IN LET newCounters == [ counters EXCEPT ![e] == counters[e] + 1 ]
IN IF newCounters[e] > newCounters[currentMode]
THEN << newCounters, e >>
ELSE << newCounters, currentMode >>
IN ApaFoldSeqLeft( CountElem, <<[ x \in ExtRange > 0 ], elIfEmpty >>, seq )[2]
\* Returns TRUE iff fn is injective
IsInjective(fn) ==
LET SeenBefore( seenAndResult, e ) ==
IF fn[e] \in seenAndResult[1]
THEN [ seenAndResult EXCEPT ![2] = FALSE ]
ELSE [ seenAndResult EXCEPT ![1] = seenAndResult[1] \union {fn[e]} ]
IN ApaFoldSet( SeenBefore, << {}, TRUE >>, DOMAIN fn )[2]
================================
For the sake of comparison, we rewrite the above operators using recursion, CHOOSE
or quantification:
 MODULE NonFoldDefined 
EXTENDS Apalache
RECURSIVE Sum(_)
Sum(S) == IF S = {}
THEN 0
ELSE LET x == CHOOSE y \in S : TRUE
IN x + Sum(S \ {x})
RECURSIVE BigUnion(_)
BigUnion(setOfSets) == IF setOfSets = {}
THEN {}
ELSE LET S == CHOOSE x \in setOfSets : TRUE
IN S \union BigUnion(setOfSets \ {x})
RECURSIVE SelectSeq(_,_)
SelectSeq(seq, Test(_)) == IF seq = <<>>
THEN <<>>
ELSE LET tail == SelectSeq(Tail(seq), Test)
IN IF Test( Head(seq) )
THEN <<Head(seq)>> \o tail
ELSE tail
RECURSIVE Quantify(_,_)
Quantify(S, P(_)) == IF S = {}
THEN 0
ELSE LET x == CHOOSE y \in S : TRUE
IN (IF P(x) THEN 1 ELSE 0) + Quantify(S \ {x}, P)
RECURSIVE Range(_)
Range(seq) == IF seq = <<>>
THEN {}
ELSE {Head(seq)} \union Range(Tail(seq))
Mode(seq, elIfEmpty) == IF seq = <<>>
THEN elIfEmpty
ELSE LET numOf(p) == Quantify( DOMAIN seq, LAMBDA q: q = p )
IN CHOOSE x \in Range(seq): \A y \in Range(seq) : numOf(x) >= numOf(y)
IsInjective(fn) == \A a,b \in DOMAIN fn : fn[a] = fn[b] => a = b
================================
In most cases, recursive operators are much more verbose, and the operators using CHOOSE
and/or quantification mask double iteration (and thus have quadratic complexity).
For instance, the evaluation of the foldless IsInjective
operator actually requires the traversal of all domain pairs, instead of the single domain traversal with fold.
In particular, Mode
, the most verbose among the folddefined operators, is still very readable (most LETIN operators are introduced to improve readability, at the cost of verbosity) and quite efficient, as its complexity is linear w.r.t. the length of the sequence (the mode could also be computed directly, without a subcall to Range
, but the example would be more difficult to read), unlike the variant with CHOOSE
and \A
, which is quadratic.
Invariants: state, action, and trace
Until recently, Apalache only supported checking of state invariants. A state invariant is a predicate over state variables and constants. State invariants are, by far, the most common ones. Recently, we have added support for action invariants and trace invariants. Action properties were highlighted by Hillel Wayne; they can be checked with action invariants. Trace invariants let us reason about finite executions.
State invariants
You have probably seen state invariants before. Consider the following specification.
 MODULE Invariants 
EXTENDS Integers, Sequences, FiniteSets
VARIABLES
\* @typeAlias: set = Set(Int);
\* @typeAlias: state = { In: $set, Done: $set, Out: $set };
\* @type: $set;
In,
\* @type: $set;
Done,
\* @type: $set;
Out
\* @type: <<$set, $set, $set>>;
vars == <<In, Done, Out>>
Init ==
/\ \E S \in SUBSET (1..5):
/\ Cardinality(S) > 2
/\ In = S
/\ Done = {}
/\ Out = {}
Next ==
\/ \E x \in In:
/\ In' = In \ { x }
/\ Done' = Done \union { x }
/\ Out' = Out \union { 2 * x }
\/ In = {} /\ UNCHANGED vars
\* state invariants that reason about individual states
StateInv ==
Done \intersect In = {}
BuggyStateInv ==
Done \subseteq In
We let you guess what this specification is doing. As for its properties, it contains two state invariants:
 Predicate
StateInv
that statesDone \intersect In = {}
, and  Predicate
BuggyStateInv
that statesDone \subseteq In
.
We call these predicates state invariants, as we expect them to hold in every state of an execution. To check, whether these invariants hold true, we run Apalache as follows:
$ apalache check inv=StateInv Invariants.tla
...
Checker reports no error up to computation length 10
...
$ apalache check inv=BuggyStateInv Invariants.tla
...
State 1: state invariant 0 violated. Check the counterexample in: counterexample.tla, MC.out, counterexample.json
...
The standard footprint: By default, Apalache checks executions of length up to 10 steps.
Action invariants
Let's have a look at two other predicates in Invariants.tla
:
\* action invariants that reason about transitions (consecutive pairs of states)
ActionInv ==
\/ In = {}
\/ \E x \in Done':
Done' = Done \union { x }
BuggyActionInv ==
Cardinality(In') = Cardinality(In) + 1
Can you see a difference between ActionInv
& BuggyActionInv
and StateInv
& BuggyStateInv
?
You have probably noticed that ActionInv
as well as BuggyActionInv
use
unprimed variables and primed variables. So they let us reason about two
consecutive states of an execution. They are handy for checking specification
progress. Similar to state invariants, we can check, whether action invariants
hold true by running Apalache as follows:
$ apalache check inv=ActionInv Invariants.tla
...
Checker reports no error up to computation length 10
...
$ apalache check inv=BuggyActionInv Invariants.tla
...
State 0: action invariant 0 violated. Check the counterexample in: counterexample.tla, MC.out, counterexample.json
...
There is no typo in the CLI arguments above: You pass action invariants the same way as you pass state invariants. Preprocessing in Apalache is clever enough to figure out, what kind of invariant it is dealing with.
Trace invariants
Let's have a look at the following two predicates in Invariants.tla
:
\* trace invariants that reason about executions
\* @type: Seq($state) => Bool;
TraceInv(hist) ==
\/ hist[Len(hist)].In /= {}
\* note that we are using the last state in the history and the first one
\/ { 2 * x: x \in hist[1].In } = hist[Len(hist)].Out
\* @type: Seq($state) => Bool;
BuggyTraceInv(hist) ==
\/ hist[Len(hist)].In /= {}
\* note that we are using the last state in the history and the first one
\/ { 3 * x: x \in hist[1].In } = hist[Len(hist)].Out
These predicates are quite different from state invariants and action
invariants. Both TraceInv
and BuggyTraceInv
accept the parameter hist
,
which store the execution history as a sequence of records. Having the
execution history, you can check plenty of interesting properties. For
instance, you can check, whether the result of an execution somehow matches the
input values.
$ apalache check inv=TraceInv Invariants.tla
...
Checker reports no error up to computation length 10
...
$ apalache check inv=BuggyTraceInv Invariants.tla
...
State 3: trace invariant 0 violated. Check the counterexample in: counterexample.tla, MC.out, counterexample.json
...
Trace invariants are quite powerful. You can write down temporal properties as trace invariants. However, we recommend to use trace invariants for testing, as they are too powerful. For verification, you should use temporal properties.
Enumerating counterexamples
By default, Apalache stops whenever it finds a property violation. This is true for the commands that are explained in the Section on Running the Tool. Sometimes, we want to produce multiple counterexamples; for instance, to generate multiple tests.
Consider the following TLA+ specification:
 MODULE View2 
EXTENDS Integers
VARIABLES
\* @type: Int;
x
Init ==
x = 0
A ==
x' = x + 1
B ==
x' = x  1
C ==
x' = x
Next ==
A \/ B \/ C
Inv ==
x = 0
We can run Apalache to check the state invariant Inv
:
$ apalache check inv=Inv View2.tla
Apalache quickly finds a counterexample that looks like this:
...
(* Initial state *)
State0 == x = 0
(* Transition 0 to State1 *)
State1 == x = 1
...
Producing multiple counterexamples. If we want to see more examples of invariant violation, we can ask Apalache to produce up to 50 counterexamples:
$ apalache check inv=Inv maxerror=50 View2.tla
...
Found 20 error(s)
...
Whenever the model checker finds an invariant violation, it reports a
counterexample to the current symbolic execution and proceeds with the next action.
For instance, if the symbolic execution Init \cdot A \cdot A
has a concrete
execution that violates the invariant Inv
, the model checker would print this
execution and proceed with the symbolic execution Init \cdot A \cdot B
. That
is why the model checker stops after producing 20 counterexamples.
The option maxerror
is similar to the option continue
in TLC, see TLC
options. However, the space of counterexamples in Apalache may be infinite,
e.g., when we have integer variables, so maxerror
requires an upper bound
on the number of counterexamples.
Partitioning counterexamples with view abstraction.
Some of the produced counterexamples are not really interesting. For
instance, counterexample5.tla
looks like follows:
(* Initial state *)
State0 == x = 0
(* Transition 1 to State1 *)
State1 == x = 1
(* Transition 1 to State2 *)
State2 == x = 2
(* Transition 0 to State3 *)
State3 == x = 1
Obviously, the invariant is violated in State1
already, so states State2
and State3
are not informative. We could write a trace
invariant to enforce invariant violation only in the
last state. Alternatively, the model checker could enforce the constraint that
the invariant holds true in the intermediate states. As invariants usually
produce complex constraints and slow down the model checker, we leave the
choice to the user.
Usually, the specification author has a good idea of how to partition states
into interesting equivalence classes. We let you specify this partitiong by declaring
a view abstraction, similar to the VIEW
configuration option in TLC.
Basically, two states are considered to be similar, if they have the same view.
In our example, we compute the state view with the operator View1
:
\* @type: <<Bool, Bool>>;
View1 ==
<<x < 0, x > 0>>
Hence, the states with x = 1
and x = 25
are similar, because their view has the
same value <<FALSE, TRUE>>
. We can also define the view of an execution, simply
as a sequence of views of the execution states.
Now we can ask Apalache to produce up to 50 counterexamples again. This time we tell it to avoid the executions that start with the view of an execution that produced an error earlier:
$ apalache check inv=Inv maxerror=50 view=View1 View2.tla
...
Found 20 error(s)
...
Now counterexample5.tla
is more informative:
(* Initial state *)
State0 == x = 0
(* Transition 2 to State1 *)
State1 == x = 0
(* Transition 2 to State2 *)
State2 == x = 0
(* Transition 0 to State3 *)
State3 == x = 1
Moreover, counterexample6.tla
is intuitively a mirror version of counterexample5.tla
:
(* Initial state *)
State0 == x = 0
(* Transition 2 to State1 *)
State1 == x = 0
(* Transition 2 to State2 *)
State2 == x = 0
(* Transition 0 to State3 *)
State3 == x = 1
By the choice of the view, we have partitioned the states into three
equivalence classes: x < 0
, x = 0
, and x > 0
. It is often useful to write
a view as a tuple of predicates. However, you can write arbitrary TLA+ expressions.
A view is just a mapping from state variables to the set of values that can be
computed by the view expressions.
We are using this technique in modelbased testing. If you have found another interesting application of this technique, please let us know!
The Apalache Module
Similar to the TLC
module, we provide a module called Apalache
, which can be found in Apalache.tla. Many of
the operators in that module are used internally by Apalache, when rewriting a TLA+ specification. It is useful
to read the comments to the operators defined in Apalache.tla, as they will help you in understanding
the detailed output produced by the tool.
See the detailed description of the Apalache operators.
Naturals
If you look carefully at the HOWTO on
annotations, you will find that
there is no designated type for naturals. Indeed, one can just use the type
Int
, whenever a natural number is required. If we introduced a special type
for naturals, that would cause a lot of confusion for the type checker. What
would be the type of the literal 42
? That depends on, whether you extend
Naturals
or Integers
. And if you extend Naturals
and later somebody else
extends your module and also Integers
, should be the type of 42
be an
integer?
Apalache still allows you to extend Naturals
. However, it will treat all
numberlike literals as integers. This is consistent with the view that the
naturals are a subset of the integers, and the integers are a subset of the
reals. Classically, one would not define subtraction for naturals. However,
the module Naturals
defines binary minus, which can easily drive a variable
outside of Nat
. For instance, see the following example:
 MODULE NatCounter 
EXTENDS Naturals
VARIABLE
\* @type: Int;
x
Init == x = 3
\* a natural counter can go below zero, and this is expected behavior
Next == x' = x  1
Inv == x >= 0
========================================================================
Given that you will need the value Int
for a type annotation, it probably
does not make a lot of sense to extend Naturals
in your own specifications,
as you will have to extend Integers
for the type annotation too.
Apalache configuration
Apalache supports configuration of some parameters governing its behavior.
Application configuration is loaded from the following four sources:
 Command line arguments
 Environment variables
 A local configuration file
 The global configuration file
The order of precedence of the sources follows their numbering: i.e., and any configuration set in an earlier numbered source overrides a configuration set in a later numbered source.
Command line arguments and environment variables
To view the available command line arguments, run Apalache with the help
flag and consult the section on Running the Tool for more
details.
Some parameters configurable via the command line are also configurable via environment variables. These parameters are noted in the CLI's inline help. If a parameter is configured both through a CLI argument and an environment variable, then the CLI argument always takes precedence.
Configuration files
File format and supported parameters
Local configuration files support JSON and the JSON superset HOCON.
Here's an example of a valid configuration for commonly used parameters, along with their default values:
common {
# Directory in which to write all log files and records of each run
outdir = "${PWD}/_apalacheout"
# Whether or not to write additional files, that report on intermediate
# processing steps
writeintermediate = false
# Whether or not to write general profiling data into the `outdir`
profiling = false
# Fixed directory into which generated files are written (absent by default)
# rundir = ~/myrundir
}
A ~
found at the beginning of a file path will expanded into the value set for
the user's home directory.
Details on the effect of these parameters can be found in Running the Tool.
Local configuration file
You can specify a local configuration file explicitly via the configfile
command line argument. If this is not provided, then Apalache will look for the
nearest .apalache.cfg
file, beginning in the current working directory and
searching up through its parents.
Parameters configured in the local configuration file will be overridden by values set via CLI arguments or environment variables, and will override parameters configured via the global configuration file.
Global configuration file
The final fallback for configuration parameters is the global configuration file
named $HOME/.tlaplus/apalache.cfg
.
TLA+ Anonymized Execution Statistics
Apalache participates in the optional anonymized statistics programme along with TLA+ Toolbox, TLC (which is part of the Toolbox), and Visual Studio Code Plugin for TLA+.
The statistics collection is never enabled by default. You have to optin
for the programme either in TLA+ Toolbox, or in Apalache. When statistics
collection is enabled by the user, it is submitted to tlapl.us
via the
util.ExecutionStatisticsCollector, which is part of tla2tools.jar
. Apalache
accesses this class in at.forsyte.apalache.tla.Tool.
As explained in anonymized statistics programme, if you never create the file
$HOME/.tlaplus/esc.txt
, then the statistics is not submitted to tlapl.us
.
If you optin for the programme and later remove the file, then the statistics
will not be submitted too.
Why do we ask you to help us
There are several reasons:

Although our project is open source, developing Apalache is our main job. We are grateful to Informal Systems for supporting us and to TU Wien, Vienna Science and Technology Fund, and Inria Nancy/LORIA, who supported us in the past. It is easier to convince our decision makers to continue the development, if we have clear feedback on how many people use and need Apalache.

We would like to know which features you are using most, so we can focus on them.

We would like to know which operating systems and Java versions need care and better be included in automated test suites.
How to optin and optout
To optin in the statistics collection, execute the following command:
./apalachemc config enablestats=true
As a result of this command, a random identifier is written in
$HOME/.tlaplus/esc.txt
. This identifier is used by the execution statistics
code.
To optout from the statistics collection, execute the following command:
./apalachemc config enablestats=false
What exactly is submitted to tlapl.us
You can check the daily log at execstats.tlapl.us.
The following data is submitted for each run, if you have opted in:
 Total number of CPU cores and cores assigned (the latter is 1 for now, but will change soon)
 Java heap memory size (in Megabytes)
 Apalache version (semantic version + build)
 Command mode:
parse
,check
, ortypecheck
 Name, version, and architecture of the OS
 Vendor, version, and architecture of JVM
 Timestamp + salt (a random number to make time less precise)
 An installation ID that is stored in
$HOME/.tlaplus/esc.txt
Profiling Your Specification
Warning: Profiling only works in the incremental SMT mode, that is, when the model checker is run
with algo=incremental
, or without the option
algo
specified.
As Apalache translates the TLA+ specification to SMT, it often defeats our intuition about the standard bottlenecks that one learns about when running TLC. For instance, whereas TLC needs a lot of time to compute the initial states for the following specification, Apalache can check the executions of length up to ten steps in seconds:
 MODULE powerset 
EXTENDS Integers
VARIABLE S
Init ==
/\ S \in SUBSET (1..50)
/\ 3 \notin S
Next ==
\/ \E x \in S:
S' = S \ {x}
\/ UNCHANGED S
Inv ==
3 \notin S
=========================================================================
Apalache has its own bottlenecks. As it's using the SMT solver z3,
we cannot precisely profile your TLA+ specification. However, we can profile
the number of SMT variables and constraints that Apalache produces for different
parts of your specification. To activate this profiling mode, use the option
smtprof
:
apalache check smtprof powerset.tla
The profiling data is written in the file profile.csv
in the run
directory:
# weight,nCells,nConsts,nSmtExprs,location
4424,2180,2076,28460,powerset.tla:11:513:18
4098,2020,1969,12000,powerset.tla:12:912:20
4098,2020,1969,12000,powerset.tla:12:1412:20
...
The meaning of the columns is as follows:

weight
is the weight of the expression. Currently it is computed asnCells + nConsts + sqrt(nSmtExprs)
. We may change this formula in the future. 
nCells
is the number of arena cells that are created during the translation. Intuitively, the cells are used to keep the potential shapes of the data structures that are captured by the expression. 
nConsts
is the number of SMT constants that are produced by the translator. 
nSmtExprs
is the number of SMT expressions that are produced by the translator. We also include all subexpressions, when counting this metric. 
location
is the location in the source code where the expression was found, indicated by the file name correlated with a range ofline:column
pairs.
To visualize the profiling data, you can use the script script/heatmap.py
:
$APALACHE_HOME/script/heatmap.py profile.csv heatmap.html
The produced file heatmap.html
looks as follows:
The heatmap may give you an idea about the expression that are hard for Apalache. The following picture highlights one part of the Raft specification that produces a lot of constraints:
Five minutes of theory
You can safely skip this section
Given a TLA+ specification, with all parameters fixed, our model checker performs the following steps:

It automatically extracts symbolic transitions from the specification. This allows us to partition the action
Next
into a disjunction of simpler actionsA_1, ..., A_n
. 
Apalache translates operators
Init
andA_1, ..., A_n
to SMT formulas. This allows us to explore bounded executions with an SMT solver (we are using Microsoft's Z3). For instance, a sequence ofk
stepss_0, s_1, ..., s_k
, all of which execute actionA_1
, is encoded as a formulaRun(k)
that looks as follows:
[[Init(s_0)]] /\ [[A_1(s_0, s_1)]] /\ ... /\ [[A_1(s_(k1), s_k)]]
To find an execution of length k
that violates an invariant Inv
, the tool
adds the following constraint to the formula Run(k)
:
[[~Inv(s_0)]] \/ ... \/ [[~Inv(s_k)]]
Here, [[_]]
is the translator from TLA+ to SMT. Importantly, the values for
the states s_0
, ..., s_k
are not enumerated as in TLC, but have to be found
by the SMT solver.
If you would like to learn more about theory behind Apalache, check the paper delivered at OOPSLA19.
Syntax of TLC Configuration Files
Author: Igor Konnov, 2020
To see how Apalache can use TLC configuration files, check the page on Specification Parameters and Running the Tool.
This file presents the syntax of
TLC configuration files
in EBNF and
comments on the treatment of its sections in
Apalache. A detailed discussion
on using the config files with TLC can be found in Leslie Lamport's
Specifying Systems,
Chapter 14 and in
Current Versions of the TLA+ Tools.
In particular, the TLA+ specification of TLC configuration files
is given in Section 14.7.1. The standard parser can be found in
tlc2.tool.impl.ModelConfig
.
As the configuration files have simple syntax, we implement our own parser in
Apalache.
// The configuration file is a nonempty sequence of configuration options.
config ::=
options+
// Possible options, in no particular order, all of them are optional.
// Apalache expects Init after Next, or Next after Init.
options ::=
Init
 Next
 Specification
 Constants
 Invariants
 Properties
 StateConstraints
 ActionConstraints
 Symmetry
 View
 Alias
 Postcondition
 CheckDeadlock
// Set the initialization predicate (over unprimed variables), e.g., Init.
Init ::=
"INIT" ident
// Set the next predicate (over unprimed and primed variables), e.g., Next.
Next ::=
"NEXT" ident
// Set the specification predicate, e.g., Spec.
// A specification predicate usually looks like Init /\ [][Next]_vars /\ ...
Specification ::=
"SPECIFICATION" ident
// Set the constants to specific values or substitute them with other names.
Constants ::=
("CONSTANT"  "CONSTANTS") (replacement  assignment)*
// Replace the constant in the lefthand side
// with the identifier in the righthand side.
replacement ::=
ident "<" ident
// Replace the constant in the lefthand side
// with the constant expression in the righthand side.
assignment ::=
ident "=" constExpr
// A constant expression that may appear in
// the righthand side of an assignment.
constExpr ::=
modelValue
 integer
 string
 boolean
 "{" "}"
 "{" constExpr ("," constExpr)* "}"
// The name of a model value, see Section 14.5.3 of Specifying Systems.
// A model value is essentially an uninterpreted constant.
// All model values are distinct from one another. Moreover, they are
// not equal to other values such as integers, strings, sets, etc.
// Apalache treats model values as strings, which it declares as
// uninterpreted constants in SMT.
modelValue ::= ident
// An integer (no bitwidth assumed)
integer ::=
<string matching regex [09]+>
 "" <string matching regex [09]+>
// A string, starts and ends with quotes,
// a restricted set of characters is allowed (preUTF8 era, Paxon scripts?)
string ::=
'"' <string matching regex [azAZ09_~!@#\$%^&*+=(){}[\],:;`'<>.?/ ]*> '"'
// A Boolean literal
boolean ::= "FALSE"  "TRUE"
// Set an invariant (over unprimed variables) to be checked against
// every reachable state.
Invariants ::=
("INVARIANT"  "INVARIANTS") ident*
// Set a temporal property to be checked against the initial states.
// Temporal properties reason about finite or infinite computations,
// which are called behaviors in TLA+. Importantly, the computations
// originate from the initial states.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
Properties ::=
("PROPERTY"  "PROPERTIES") ident*
// Set a state predicate (over unprimed variables)
// that restricts the state space to be explored.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
StateConstraints ::=
("CONSTRAINT"  "CONSTRAINTS") ident*
// Set an action predicate (over unprimed and primed variables)
// that restricts the transitions to be explored.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
ActionConstraints ::=
("ACTION_CONSTRAINT"  "ACTION_CONSTRAINTS") ident*
// Set the name of an operator that produces a set of permutations
// for symmetry reduction.
// See Section 14.3.3 in Specifying Systems.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
Symmetry ::=
"SYMMETRY" ident
// Set the name of an operator that produces a state view
// (some form of abstraction).
// See Section 14.3.3 in Specifying Systems.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
View ::=
"VIEW" ident
// Whether the tools should check for deadlocks.
// APALACHE IGNORES THIS CONFIGURATION OPTION.
CheckDeadlock ::=
"CHECK_DEADLOCK" ("FALSE"  "TRUE")
// Recent feature: https://lamport.azurewebsites.net/tla/currenttools.pdf
// APALACHE IGNORES THIS CONFIGURATION OPTION.
Postcondition ::=
"POSTCONDITION" ident
// Very recent feature: https://github.com/tlaplus/tlaplus/issues/485
// APALACHE IGNORES THIS CONFIGURATION OPTION.
Alias ::=
"ALIAS" ident
// A TLA+ identifier, must be different from the above keywords.
ident ::=
<string matching regex [azAZ_]([azAZ09_])*>
The new type checker Snowcat
WARNING: Snowcat is our type checker starting with Apalache version 0.15.0. If you are using Apalache prior to version 0.15.0, check the chapter on old type annotations.
How to write type annotations
Check the HOWTO. You can find detailed syntax of type annotations in ADR002.
How to run the type checker
The type checker can be run as follows:
$ apalache typecheck [inferpoly=<bool>] <myspec>.tla
The arguments are as follows:
 General parameters:
inferpoly
controls whether the type checker should infer polymorphic types. As many specs do not need polymorphism, you can set this option tofalse
. The default value istrue
.
There is not much to explain about running the tool. When it successfully finds the types of all expressions, it reports:
> Running Snowcat .::..
> Your types are great!
...
Type checker [OK]
When the type checker finds an error, it explains the error like that:
> Running Snowcat .::.
[QueensTyped.tla:42:4442:61]: Mismatch in argument types. Expected: (Seq(Int)) => Bool
[QueensTyped.tla:42:1442:63]: Error when computing the type of Solutions
> Snowcat asks you to fix the types. Meow.
Type checker [FAILED]
Here is the list of the TLA+ language features that are currently supported by Apalache, following the Summary of TLA+.
Safety vs. Liveness
At the moment, Apalache is able to check state invariants, action invariants, temporal properties, trace invariants, as well as inductive invariants. (See the page on invariants in the manual.) To check liveness/temporal properties, we employ a livenesstosafety transformation.
Language
ModuleLevel constructs
Construct  Supported?  Milestone  Comment 

EXTENDS module  ✔    A few standard modules are not supported yet (Bags) 
CONSTANTS C1, C2  ✔    Either define a ConstInit operator to initialize the constants, use a .cfg file, or declare operators instead of constants, e.g., C1 == 111 
VARIABLES x, y, z  ✔    
ASSUME P  ✔ / ✖    Parsed, but not propagated to the solver 
F(x1, ..., x_n) == exp  ✔ / ✖    Every application of F is replaced with its body. Recursive operators not supported after 0.23.1. From 0.16.1 and later, for better performance and UX, use ApaFoldSet and ApaFoldSeqLeft . 
f[x ∈ S] == exp  ✔ / ✖    Recursive functions not supported after 0.23.1. From 0.16.1 and later, for better performance and UX, use ApaFoldSet and ApaFoldSeqLeft . 
INSTANCE M WITH ...  ✔ / ✖    No special treatment for ~> , \cdot , ENABLED 
N(x1, ..., x_n) == INSTANCE M WITH...  ✔ / ✖    Parameterized instances are not supported 
THEOREM P  ✔ / ✖    Parsed but not used 
LOCAL def  ✔    Replaced with local LETIN definitions 
The constant operators
Logic
Operator  Supported?  Milestone  Comment 

/\ , \/ , ~ , => , <=>  ✔    
TRUE , FALSE , BOOLEAN  ✔    
\A x \in S: p , \E x \in S : p  ✔    
CHOOSE x \in S : p  ✖    Partial support prior to version 0.16.1. From 0.16.1 and later, use Some , ApaFoldSet , or ApaFoldSeqLeft . See #841. 
CHOOSE x : x \notin S  ✖    Not supported. You can use records or a default value such as 1. 
\A x : p, \E x : p  ✖    Use bounded quantifiers 
CHOOSE x : p  ✖   
Sets
Note: only finite sets are supported. Additionally, existential
quantification over Int
and Nat
is supported, as soon as it can be
replaced with a constant.
Operator  Supported?  Milestone  Comment 

= , /= , \in , \notin , \intersect , \union , \subseteq , \  ✔    
{e_1, ..., e_n}  ✔    
{x \in S : p}  ✔    
{e : x \in S}  ✔    
SUBSET S  ✔    Sometimes, the powersets are expanded 
UNION S  ✔    Provided that S is expanded 
Functions
Operator  Supported?  Milestone  Comment 

f[e]  ✔    
DOMAIN f  ✔    
[ x \in S ↦ e]  ✔    
[ S > T ]  ✔    Supported, provided the function can be interpreted symbolically 
[ f EXCEPT ![e1] = e2 ]  ✔   
Records
Use type annotations to help the model checker in finding the right types. Note that our type system distinguishes records from general functions.
Operator  Supported?  Milestone  Comment 

e.h  ✔    
r[e]  ✔/✖    Provided that e is a constant expression. 
[ h1 ↦ e1, ..., h_n ↦ e_n]  ✔    
[ h1 : S1, ..., h_n : S_n]  ✔    
[ r EXCEPT !.h = e]  ✔   
Tuples
Use type annotations to help the model checker in finding the right types. Note that our type system distinguishes records from general functions.
Operator  Supported?  Milestone  Comment 

e[i]  ✔ / ✖    Provided that i is a constant expression 
<< e1, ..., e_n >>  ✔    Use a type annotation to distinguish between a tuple and a sequence. 
S1 \X ... \X S_n  ✔    
[ t EXCEPT ![i] = e]  ✔/✖    Provided that i is a constant expression 
Strings and numbers
Construct  Supported?  Milestone  Comment 

"c1...c_n"  ✔    A string is always mapped to a unique uninterpreted constant 
STRING  ✖    It is an infinite set. We cannot handle infinite sets. 
d1...d_n  ✔    As long as the SMT solver (Z3) accepts that large number 
d1...d_n.d_n+1...d_m  ✖    Technical issue. We will implement it upon a user request. 
Miscellaneous Constructs
Construct  Supported?  Milestone  Comment 

IF p THEN e1 ELSE e2  ✔    Provided that both e1 and e2 have the same type 
CASE p1 > e1 [] ... [] p_n > e_n [] OTHER > e  ✔    Provided that e1, ..., e_n, e have the same type 
CASE p1 > e1 [] ... [] p_n > e_n  ✔    Provided that e1, ..., e_n have the same type 
LET d1 == e1 ... d_n == e_n IN e  ✔  All applications of d1 , ..., d_n are replaced with the expressions e1 , ... e_n respectively. LETdefinitions without arguments are kept in place.  
multiline /\ and \/  ✔   
The Action Operators
Construct  Supported?  Milestone  Comment 

e'  ✔    
[A]_e  ✔    
< A >_e  ✔    
ENABLED A  ✖    Has to be specified manually 
UNCHANGED <<e_1, ..., e_k>>  ✔    Always replaced with e_1' = e_1 /\ ... /\ e_k' = e_k 
A ∙ B  ✖   
The Temporal Operators
Construct  Supported?  Milestone  Comment 

[]F  ✔    
<>F  ✔    
WF_e(A)  ✖    Has to be specified manually (see ENABLED) 
SF_e(A)  ✖    Has to be specified manually (see ENABLED) 
F ~> G  ✔    Always replaced with [](F => <>G) 
F +> G  ✖    
\EE x: F  ✖    
\AA x: F  ✖   
The model checker assumes that the specification has the form Init /\ [][Next]_e
. Other than that, temporal operators
may only appear in temporal properties, not in e.g. actions.
Standard modules
Integers and Naturals
For the moment, the model checker does not differentiate between integers and naturals. They are all translated as integers in SMT.
Operator  Supported?  Milestone  Comment 

+ ,  , * , <= , >= , < , >  ✔    These operators are translated into integer arithmetic of the SMT solver. Linear integer arithmetic is preferred. 
\div , %  ✔    Integer division and modulo 
a^b  ✔ / ✖    Provided a and b are constant expressions 
a..b  ✔ / ✖    Sometimes, a..b needs a constant upper bound on the range. When Apalache complains, use {x \in A..B : a <= x /\ x <= b} , provided that A and B are constant expressions. 
Int , Nat  ✔ / ✖    Supported in \E x \in Nat: p and \E x \in Int: p , if the expression is not located under \A and ~ . We also support assignments like f' \in [S > Int] and tests f \in [S > Nat] 
/  ✖    Real division, not supported 
Sequences
Operator  Supported?  Milestone  Comment 

<<...>>  ✔  Often needs a type annotation.  
Head , Tail , Len``, SubSeq, Append, \o, f[e]`  ✔    
EXCEPT  ✔  
SelectSeq  ✔    Not as efficient, as it could be, see #1350. 
Seq(S)  ✖    Use Gen of Apalache to produce bounded sequences 
FiniteSets
Operator  Supported?  Milestone  Comment 

IsFinite  ✔    Always returns true, as all the supported sets are finite 
Cardinality  ✔    Try to avoid it, as Cardinality(S) produces O(n^2) constraints in SMT for cardinality n 
TLC
Operator  Supported?  Milestone  Comment 

a :> b  ✔    A singleton function <<a, b>> 
f @@ g  ✔    Extends function f with the domain and values of function g but keeps the values of f where domains overlap 
Other operators  Dummy definitions for spec compatibility 
Reals
Not supported, not a priority
Recursive operators and functions
While TLA+ allows the use of recursive operators and functions, we have decided to no longer support them in Apalache from version 0.23.1
onward, and suggest the use of fold operators, described in Folding sets and sequences instead:
* Similar to Skolem, this has to be done carefully. Apalache automatically
* places this hint by static analysis.
*)
ConstCardinality(__cardExpr) == __cardExpr
(**
THEN __v
ELSE LET __w == CHOOSE __x \in __S: TRUE IN
LET __T == __S \ {__w} IN
ApaFoldSet(__Op, __Op(__v,__w), __T)
These operators are treated by Apalache in a more efficient manner than
recursive operators. They always take at most S
or Len(seq)
steps to evaluate and require no additional annotations.
Note that the remainder of this section discusses only recursive operators, for brevity. Recursive functions share the same issues.
The problem with recursive operators
In the preprocessing phase, Apalache replaces every application of a userdefined operator with its body. We call this process "operator inlining". This obviously cannot be done for recursive operators, since the process would never terminate. Additionally, even if inlining wasn't problematic, we would still face the following issues when attempting to construct a symbolic encoding:

A recursive operator may be nonterminating (although a nonterminating operator is useless in TLA+);

A terminating call to an operator may take an unpredictable number of iterations.
A note on (2): In practice, when one fixes specification parameters (that is,
CONSTANTS
), it is sometimes possible to find a bound on the number of operator iterations. For instance, consider the following specification:
 MODULE Rec6 
EXTENDS Integers
N == 5
VARIABLES
\* @type: Set(Int);
set,
\* @type: Int;
count
RECURSIVE Sum(_)
Sum(S) ==
IF S = {}
THEN 0
ELSE LET x == CHOOSE y \in S: TRUE IN
x + Sum(S \ {x})
UNROLL_DEFAULT_Sum == 0
UNROLL_TIMES_Sum == N
Init ==
/\ set = {}
/\ count = 0
Next ==
\E x \in (1..N) \ set:
/\ count' = count + x
/\ set' = set \union {x}
Inv == count = Sum(set)
=======================================
It is clear that the expression Sum(S)
requires Cardinality(S)
steps of recursive computation. Moreover, as the unspecified invariant set \subseteq 1..N
always holds for this specification, every call Sum(set)
requires up to N
recursive steps.
The previously supported approach
Previously, when it was possible to find an upper bound on the number of iterations of an operator Op
, such as N
for Sum
in the example above, Apalache would unroll the recursive operator up to this bound.
Two additional operators, UNROLL_DEFAULT_Op
and UNROLL_TIMES_Op
, were required, for instance:
UNROLL_DEFAULT_Sum == 0
UNROLL_TIMES_Sum == N
With the operators UNROLL_DEFAULT_Op
and UNROLL_TIMES_Op
,
Apalache would internally replace the definition of Op
with an operator OpInternal
, that had the following property:
OpInternal
was nonrecursive If computing
Op(x)
required a recursion stack of depth at mostUNROLL_TIMES_Op
, thenOpInternal(x) = Op(x)
 Otherwise,
OpInternal(x)
would return the value, which would have been produced by the computation ofOp(x)
, if all applications ofOp
while the recursion stack height wasUNROLL_TIMES_Op
returnedUNROLL_DEFAULT_Op
instead of the value produced by another recursive call toOp
Unsurprisingly, (3) caused a lot of confusion, particularly w.r.t. the meaning of the value UNROLL_DEFAULT_Op
. Consider the following example:
RECURSIVE Max(_)
Max(S) ==
IF S = {}
THEN 0
ELSE
LET x == CHOOSE v \in S: TRUE IN
LET maxRest == Max(S \ {x})
IN IF x < maxRest THEN maxRest ELSE x
As computing Max(S)
requires S
recursive calls, there is no static upper bound to the recursion stack height that works for all set sizes. Therefore, if one wanted to use this operator in Apalache, one would have to guess (or externally compute) a value N
, such that, in the particular specification, Max
would never be called on an argument, the cardinality of which exceeded N
, e.g.
UNROLL_TIMES_Max = 2
In this case, Apalache would produce something equivalent to
MaxInternal(S) ==
IF S = {}
THEN 0
ELSE
LET x1 == CHOOSE v \in S: TRUE IN
LET maxRest1 ==
IF S \ {x1} = {}
THEN 0
ELSE
LET x2 == CHOOSE v \in S \ {x1}: TRUE IN
LET maxRest2 ==
IF S \ {x1, x2} = {}
THEN 0
ELSE
LET x3 == CHOOSE v \in S \ {x1, x2}: TRUE IN
LET maxRest3 == UNROLL_DEFAULT_Max
IN IF x3 < maxRest3 THEN maxRest3 ELSE x3
IN IF x2 < maxRest2 THEN maxRest2 ELSE x2
IN IF x1 < maxRest1 THEN maxRest1 ELSE x1
In this case, MaxInternal({1,42}) = 42 = Max({1,42})
, by property (2) as expected, but MaxInternal(1..10)
can be any one of
3..10 \union {UNROLL_DEFAULT_Max}
(depending on the value of
UNROLL_DEFAULT_Max
and the order in which elements are selected by CHOOSE
), by property (3).
So how does one select a sensible value for UNROLL_DEFAULT_Op
? The problem is, one generally cannot.
In the Max
case, one could pick a "very large" N
and then assume that Max
computation has "failed" (exceeded the UNROLL_TIMES_Max
recursion limit) if the result was ever equal to N
, though "very large" is of course subjective and gives absolutely no guarantees that Max
won't be called on a set containing some element M > N
.
As the recursion becomes more complex (e.g. nonprimitive or nontail), attempting to implement a sort of "monitor" via default values quickly becomes impractical, if not impossible.
Fundamentally though, it is very easy to accidentally either introduce spurious invariant violations, or hide actual invariant violations by doing this. For instance, in a specification with
UNROLL_DEFAULT_Max = 42
UNROLL_TIMES_Max = 2
Inv == \A n \in 10..20: Max(1..n) = 42
Apalache could "prove" Inv
holds, as it would rewrite this Inv
to
\A n \in 10..20: MaxInternal(1..n) = 99
and MaxInternal(1..n)
evaluates to 99
for all n \in 3..99
(and might still evaluate to 99
for n > 99
, based on the CHOOSE
order), despite the fact that Max(1..n) = n
in the mathematical sense.
Consider now the much simpler alternative:
NonRecursiveMax(S) ==
LET Max2(a,b) == IF a < b THEN b ELSE a IN
ApaFoldSet(Max2, 0, S)
In this case, the user doesn't have to think about defaults (aside from the emptyset case), or recursion, as ApaFoldSet
ensures S
step "iteration". As an additional benefit, one also doesn't need to use CHOOSE
to select elements this way.
So ultimately, the reasons for abandoning support for recursive operators boils down to the following:
 In the vast majority of cases, equivalent functionality can be achieved by using
ApaFoldSet
orApaFoldSeqLeft
UNROLL_TIMES_Op
is hard to determine, or doesn't exist statically,UNROLL_DEFAULT_Op
is hard to determine, Apalache doesn't have runtime evaluation of recursion, so it can't natively determine when a call to a recursive
Op
would have required more thanUNROLL_TIMES_Op
steps  The use of recursive operators produces unpredictable results, particularly when used in invariants
Known issues
This page collects known issues that were reported by the users.
Deadlock detection
Deadlock detection is imprecise. It may report false negatives, see Issue 711.
Affected versions: <= 0.15.x
Planned fix: Issue 712
Updating records with excess fields
Given a record with a type declaration specifying n
fields, if that record is
given more than n
fields and the specification includes an EXCEPT
expression
that updates the record, Apalache may be unable to check the specification.
In the following example, the variable m
is given the type of a record with
1
field (namely a
), but it is then assigned to a record with 2
fields
(namely a
and foo
).
VARIABLE
\* @type: [a: Int];
m
Init == m = [a > 0, foo > TRUE]
Next ==
\/ m' = m
\/ m' = [m EXCEPT !.a = 0]
Given the current (unsound) typing discipline Apalache uses for records, this
specification is not considered incorrectly typed. However, due to the update
using EXCEPT
in the Next
operator, the specification cannot be checked.
Affected versions: <= 0.15.x
Planned fix: Issue 401
Workaround
Add the foo
field to the variable's type signature:
VARIABLE
\* @type: [a: Int, foo: Bool];
m
Init == m = [a > 0, foo > TRUE]
Next ==
\/ m' = m
\/ m' = [m EXCEPT !.a = 0]
Integer ranges with nonconstant bounds
When using an integer range a..b
, where a
or b
aren't constant (or cannot be simplified to a constant), the current encoding fails (see Issue 425):
 MODULE Example 
EXTENDS Integers
VARIABLE
\* @type: Int;
x
\* @type: (Int) => Set(Int);
1to(n) == 1..n
Init == x = 1
Next == x' = x + 1
Inv == 1 \in { m: a \in 1to(x) }
====================
Affected versions: All
Planned fix: Not in the near future
Workaround
Pick constant bounds Nmin
and Nmax
, such that Nmin <= a <= b <= Nmax
, then use
range(a,b) == { x \in Nmin..Nmax: a <= x /\ x <= b }
instead of a..b
.
Using Seq(S)
The operator Seq(S)
produces an infinite set of unbounded sequences. Hence, Apalache is not able to do anything about
this set. Consider the following snippet:
\E s \in Seq({ 1, 2, 3 }):
seq' = s
Affected versions: All
Planned fix: Not in the near future
Workaround
If you know an upper bound on the length of sequences you need, which is often the case when checking one model, you can work around this issue by using Apalache.Gen:
EXTENDS Apalache
...
LET s == Gen(10) IN
/\ \A i \in DOMAIN s:
s[i] \in { 1, 2, 3 }
/\ seq' = s
In the above example, we instruct Apalache to introduce an unrestricted sequence that contains up to 10 elements; this
is done with Gen
. We further restrict the sequence to contain the elements of { 1, 2, 3 }
.
However, note that our workaround only works for bounded sequences, whereas
Seq({ 1, 2, 3 })
is the set of all sequences whose elements come from { 1, 2, 3 }
.
Antipatterns
This page collects known antipaterns (APs) when writing TLA+ for Apalache. In this context, APs are syntactic forms or specification approaches that, for one reason or another, have particularly slow/complex encodings for the target model checker. For a pattern to be an AP, there must exist a known, equivalent, efficient pattern.
Often, APs arise from a user's past experiences with writing TLA+ for TLC, or from a direct translation of imperative OOP code into TLA+, as those follow a different paradigm, and therefore entail different cost evaluation of which expressions are slow/complex and which are not.
CHOOSE
based recursion
Often, operators that represent operations over sets have the following shape:
RECURSIVE F(_)
F(S) ==
IF S == {}
THEN v
ELSE
LET e == CHOOSE x \in S: TRUE
IN G(F( S \ {e} ), e )
For example, one can implement min/max
operators this way:
RECURSIVE min(_)
min(S) ==
IF S == {}
THEN Infinity
ELSE
LET e == CHOOSE x \in S: TRUE
IN LET minOther == min( S \ {e} )
IN IF e < minOther THEN e ELSE minOther
Apalache dislikes the use of the above, for several reasons. Firstly, since the operator is RECURSIVE
, Apalache does not support it after version 0.23.1. In earlier versions Apalache requires a predefined upper bound on unrolling, which means that the user must know, ahead of time, what the largest S
is, for any set S
, to which this operator is ever applied.
In addition, computing F
for a set S
of size n = S
requires n
encodings of a CHOOSE
operation, which can be considerably expensive in Apalache.
Lastly, Apalache also needs to encode all of the the n
intermediate sets, S \ {e1}
, (S \ {e1}) \ {e2}
, ((S \ {e1}) \ {e2}) \ {e3}
, and so on.
The AP above can be replaced by a very simple pattern:
F(S) == ApaFoldSet( G, v, S )
ApaFoldSet
(and ApaFoldSeqLeft
) were introduced precisely for these scenarios, and should be used over RECURSIVE + CHOOSE
in most cases.
Incremental computation
Often, users introduce an expression Y
, which is derived from another expression X
(Y == F(X)
, for some F
). Instead of defining Y
directly, in terms of the properties it possesses, it is possible to define all the intermediate steps of transforming X
into Y
: "X
is slightly changed into X1
(e.g. by adding one element to a set, or via EXCEPT
), which is changed into X2
, etc. until Xn = Y
". Doing this in Apalache is almost always a bad idea, if a direct characterization of Y
exists.
Concretely, the following constructs are APs:
 Incremental
EXCEPT
G ==
LET F(g, x) == [g EXCEPT ![x] = A(x)]
IN ApaFoldSet(F, f, S)
 Incremental
\union
R ==
LET F(T, e) == T \union {A(e)}
IN ApaFoldSet(F, S0, S)
 Chained
@@/:>
f == ( k1 :> A(k1) ) @@ ( k2 :> A(k2) ) @@ ... @@ ( kn :> A(kn) )
For example:
f == [ x \in 1..20 > 0 ]
Y ==
LET F(g, x) == [g EXCEPT ![x] = x * x]
IN ApaFoldSet(F, f, 7..12 )
TLC likes these sorts of operations, because it manipulates programminglanguage objects in its own implementation. This makes it easy to construct temporary mutable objects, manipulate them (e.g. via forloops) and garbagecollect them after they stop being useful. For constraintbased approaches, like Apalache, the story is different. Not only are these intermediate steps not directly useful (since Apalache is not modeling TLA+ expressions as objects in Sacala), they actually hurt performance, since they can generate a significant amount of constraints, which are all about describing data structures (e.g. two functions being almost equal, except at one point). Essentially, Apalache is spending its resources not on statespace exploration, but on instate value computation, which is not its strong suit. Below we show how to rewrite these APs.
 Incremental
EXCEPT
: Replace
G ==
LET F(g, x) == [g EXCEPT ![x] = A(x)]
IN ApaFoldSet(F, f, S)
with
G ==
[ x \in DOMAIN f >
IF x \in S
THEN A(x)
ELSE f[x]
]
 Incremental
\union
: Replace
R ==
LET F(T, e) == T \union {A(e)}
IN ApaFoldSet(F, S0, S)
with
R == S0 \union { A(e): e \in S }
 Iterated
@@/:>
: Replace
f == ( k1 :> A(k1) ) @@ ( k2 :> A(k2) ) @@ ... @@ ( kn :> A(kn) )
with
f == [ k \in {k1,...,kn} > A(k) ]
Preprocessing in APALACHE
Before translating a specification into SMT, apalache
performs a number of
preprocessing steps:
Inliner
: replaces every call to a userdefined operator with the operator's body
 replaces every call to a letin defined operator of arity at least 1 with the operator's body
PrimingPass
: adds primes to variables inInit
andConstInit
(required byTransitionPass
)VCGen
: extracts verification conditions from the invariant candidate.Desugarer
: removes syntactic sugar like shorthand expressions inEXCEPT
.Normalizer
: rewrites all expressions in negationnormal form.Keramelizer
: translates TLA+ expressions into the kernel language KerA.ExprOptimizer
: statically computes select expressions (e.g. record field access from a known record)ConstSimplifier
: propagates constants
Keramelizer
Keramelizer rewrites TLA+ expressions into KerA. For many TLA+ expressions this translation is clear, however, some expressions cannot be easily translated. Below we discuss such expressions and the decisions that we have made.
References
 Leslie Lamport. Specifying Systems: The TLA+ Language and Tools for Hardware and Software Engineers. AddisonWesley Professional, 2004.
Parameters for fine tuning
The parameters for fine tuning can be passed to the checker in a properties
file. Its name is given with the commandline option tuningoptionsfile=my.properties
.
This file supports variable substitution, e.g., ${x}
is replaced with the
value of x
, if it was previously declared.
Alternatively, you can pass the tuning options right in the commandline by
passing the option tuningoptions
that has the following format:
```
tuningoptions=key1=val1
tuningoptions=key1=val1:key2=val2
...
```
The following options are supported:
Randomization
smt.randomSeed=<int>
passes the random seed to z3
(via z3
's parameters
sat.random_seed
and smt.random_seed
).
Timeouts
search.smt.timeout=<seconds>
defines the timeout to the SMT solver in seconds.
The default value is 0
, which stands for the unbounded timeout. For instance,
the timeout is used in the following cases: checking if a transition is enabled,
checking an invariant, checking for deadlocks. If the solver times out, it
reports 'UNKNOWN', and the model checker reports a runtime error.
Invariant mode
search.invariant.mode=(beforeafter)
defines the moment when the invariant is
checked. In the after
mode, all transitions are first translated, one of them
is picked nondeterministically and then the invariant is checked. Although this
mode reduces the number of SMT queries, it usually requires more memory than the
before
mode. In the before
mode, the invariant is checked for every enabled
transition independently. The before
mode may drastically reduce memory
consumption, but it may take longer than the after
mode, provided that
Apalache has enough memory. The default mode is before
.
Guided search
Preliminaries
In the following, step 0 corresponds to the initialization with Init
, step 1 is the first step with Next
, etc.
Transition filter
search.transitionFilter=<regex>
. Restrict the choice of symbolic transitions
at every step with a regular expression. The regular expression should recognize
words of the form s>t
, where s
is a step number and t
is a transition
number.
For instance,
search.transitionFilter=(0>01>52>(23)[39]>.*[19][09]+>.*)
requires to start with the 0th transition, continue with the 5th transition,
then execute either the 2nd or the 3rd transition and after that execute
arbitrary transitions until the length.
Note that there is no direct correspondence between the transition numbers and
the actions in the TLA+ spec. To find the numbers, run Apalache with
writeintermediate=true
and check the transition numbers in
_apalacheout/<MySpec>.tla/*/intermediate/XX_OutTransitionFinderPass.tla
: the
0th transition is called Next_si_0000
, the 1st transition is called
Next_si_0001
, etc.
Invariant filter
search.invariantFilter=<regex>
. Check the invariant only at the steps that
satisfy the regular expression. The regular expression should recognize words of
the form s>ki
, where s
is a step number, k
is an invariant kind ("state"
or "action"), and i
is an invariant number.
For instance, search.invariantFilter=10>.*15>state020>action1
tells the
model checker to check
 all invariants only after exactly 10 steps have been made,
 the first state invariant only after exactly 15 steps, and
 the second action invariant after exactly 20 steps.
Trace invariants are checked regardless of this filter.
Note that there is no direct correspondence between invariant numbers and the
operators in a TLA+ spec. Rather, the numbers refer to verification conditions
(i.e., broken up parts of a TLA+ invariant operator). To find these numbers, run
Apalache with writeintermediate=true
and check the invariant numbers in
_apalacheout/<MySpec>.tla/*/intermediate/XX_OutVCGen.tla
. The 0th state
invariant is called VCInv_si_0
, the 1st state invariant is called
VCInv_si_1
, and so on. For action invariants, the declarations are named
VCActionInv_si_0
, VCActionInv_si_1
etc.
This option is useful, e.g., for checking consensus algorithms, where the decision cannot be revoked. So instead of checking the invariant after each step, we can do that after the algorithm has made a good number of steps. You can also use this option to check different parts of an invariant on different machines to speed up turnaround time.
Translation to SMT
Short circuiting
rewriter.shortCircuit=(falsetrue)
. When rewriter.shortCircuit=true
, A \/ B
and A /\ B
are translated to SMT as ifthenelse expressions, e.g., (ite A true B)
. Otherwise, disjunctions and conjunctions are directly translated to
(or ...)
and (and ...)
respectively. By default,
rewriter.shortCircuit=false
.
Assignments in Apalache
Any run of Apalache requires an operator name as the value for the parameter
next
(by default, this value is "Next"
). We refer to this operator as the
transition operator (or transition predicate).
Actions, Slices and Minimal Actions
Actions
In TLA+, an action is any Booleanvalued expression or operator, that
contains primed variables (e.g. Next
). For the sake of this definition,
assume UNCHANGED x
is just syntactic sugar for x' = x
. Intuitively,
actions are used to define the values of state variables after a transition,
for example:
VARIABLE x
...
Next == x' = x + 1
The state transition described by Next
is fairly obvious; if x
has the
value of 4
in the current state, it will have the value of 5
in any
successor state. This brings us to the first natural requirement by Apalache:
the transition operator must be an action.
Successor State Encodings
Unfortunately, the notion of an action is too broad to be a sufficient requirement for the transition operator. Consider this slight modification of the above example:
VARIABLE x, y (* new variable *)
...
Next == x' = x + 1
Just as in the first example, the expression x' = x + 1
is, by definition, an
action. However, since the second example defines a state variable y
, this
action is no longer a sufficient description of a relation between a current
state and a successor state; it does not determine a successor value y'
.
This brings us to the second requirement: the transition operator must allow
Apalache to directly encode the relation between two successive states. This
captures two subrequirements: firstly, we disallow transition operators which
fail to specify the value of one or more variables in the successor states,
like the one in the example above. Secondly, we also disallow transition
operators where the value of a successor state variable is determined only by
implicit equations. Consider the following two cases:
VARIABLE y
...
A == y' = 1
B == y' * y'  2 * y' + 1 = 0
Using some basic math, we see that action B
can be equivalently written as
(y'  1)*(y'  1) = 0
, so it describes the exact same successor state, in
which the new value of y
is 1
. What makes it different from action A
is
the fact that this is far from immediately obvious from the syntax. The fact
that there happened to be a closedform solution for which gave us an integer
value for y'
, is a lucky coincidence, as B
could have been, for example,
y' * y' + 1 = 0
, with no real roots. To avoid cases like this, we require
that transition operators explicitly declare the values of state variables in
successor states.
We call syntactic forms, which explicitly represent successor state values,
assignment candidates. An assignment candidate for x
is a TLA+ expression
that has one of the following forms:
x' = e
,x' \in S
,UNCHANGED x
, orx' := e
(note that:=
is the operator defined in Apalache.tla)
So to reformulate the second requirement: the transition operator must contain at least one assignment candidate for each variable declared in the specification.
Control Flow: Minimal and Compound Actions
When writing nontrivial specifications, authors often end up with something similar to the following:
EventA == ...
EventB == ...
...
Next == \/ EventA
\/ EventB
Specifically, EventA
and EventB
often represent mutually exclusive
possibilities of execution. Just like before, the basic definition of an
action is not sufficient to explain the relation of EventA
or EventB
and
Next
; if EventA
is an action and EventB
is an action, then Next
is also
an action. To more accurately describe this scenario, we observe that the
operator or ( \/
) sometimes serves as a kind of parallel composition operator
(
) in process algebra  it connects two (or more) actions into a larger
one.
There are only two operators in TLA+ that could be considered controlflow
operators in this way, the or (\/
) operator and the ifthenelse operator.
We distinguish their uses as action and as value operators:
A == x = 1 \/ x = 2 (* arguments are not actions *)
B == x' = 1 \/ x' = 2 (* arguments are actions *)
Simply put, if all arguments to an operator \/
are actions, then that
operator is an actionor, otherwise it is a valueor. Similarly, if both the
THEN _
and ELSE _
subexpressions of ifthenelse are actions, it is an
actionITE, otherwise it is a valueITE (in particular, a valueITE can be
nonBoolean).
Using these two operators we can define the following terms: A minimal action is an action which contains no actionor and no actionITE. Conversely, a compound action is an action which contains at least one actionor or at least one actionITE.
Slices
Given a transition operator, which is most commonly a compound action, we can decompose it into as many minimal actions as possible. We call this process slicing and the resulting minimal actions slices. This allows us to write transition operators in the following equivalent way:
Next == \/ Slice1
\/ Slice2
...
\/ SliceN
Where each Slice[i]
is a minimal action.
The details of slicing are nuanced and depend on operators other than or (\/
)
and ifthenelse, but we give two examples here:
If a formula A
has the shape A1 \/ ... \/ An
(where A1
, ... An
are
actions), then a slice of A has the shape Si
, where Si
is a slice of some
Ai
.
If a formula A
has the shape IF p THEN A1 ELSE A2
(where A1
, A2
are
actions), then a slice of A has the shape p /\ S1
or \neg p /\ S2
, where
S1
is a slice of A1
and S2
is a slice of S2
.
Slices allow us to formulate the final requirement: the transition operator must be such, that we can select one assignment candidate for each variable in each of its slices (minimal actions) as an assignment. The process and conditions of selecting assignments from assignment candidates is described in the next section.
Assignments and Assignment Candidates
Recall, an assignment candidate for x
is a TLA+ expression that has one of the following forms:
x' = e
,x' \in S
,UNCHANGED x
, orx' := e
(note that:=
is the operator defined inApalache.tla
)
While a transition operator may contain multiple assignment candidates for the same variable, not all of them are chosen as assignments by Apalache. The subsections below describe how the assignments are selected.
Minimality
Assignments aren't spurious; each variable must have at least one assignment per transition operator, but no more than necessary to satisfy all of the additional constraints below (i.e. no more than one assignment per slice).
If all possible slices fail to assign one or more variables, an error, like the one below, is reported:
Assignment error: No assignments found for: x, z
Such errors are usually the result of adding a VARIABLE
without any
accompanying TLA+ code relating to it. The case where at least one transition,
but not all of them, fails to assign a variable is shown below.
Syntax Order
For the purpose of evaluating assignments, Apalache considers the lefttoright
syntax order of andoperator (/\
) arguments. Therefore, as many assignments
as possible are selected from the first (w.r.t. syntax order) argument of and
(/\
), then from the second, and so on.
Example:
Next == x' = 1 /\ x' = 2
In the above example, x' = 1
would be chosen as an assignment to x
, over
x' = 2
.
Assignmentbeforeuse Convention
If, in the syntax order defined above, an expression containing a primed
variable x'
syntactically precedes an assignment to x
, the assignment
finder throws an exception of the following shape:
Assignment error: test.tla:10:1610:17: x' is used before it is assigned.
notifying the user of any variables used before assignment. In particular,
righthandsides of assignment candidates ( e.g. x' + 2
in y' = x' + 2
)are
subject to this restriction as well. Consider:
A == x' > 0 /\ x' = 1
B == y' = x' + 2 /\ x' = 1
In A
, the expression x' > 0
precedes any assignment to x
and in B
,
while y' = x' + 2
is an assignment candidate for y
, it precedes any
assignment to x
, so both expressions are inadmissible (and would trigger
exceptions).
Note that this only holds true if A
(resp. B
) is chosen as the transition
operator. If A
is called inside another transition operator, for example in
Next == x' = 1 /\ A
, no error is reported.
Balance
In cases of the oroperator (\/
), all arguments must have assignments for the
same set of variables. In particular, if one argument contains an assignment
candidate and another does not, such as in this example:
\/ y = 1
\/ y' = 2
the assignment finder will report an error, like the one below:
Assignment error: test.tla:10:1510:19: Missing assignments to: y
notifying the user of any variables for which assignments exist in some, but
not all, arguments to \/
. Note that if we correct and extend the above
example to
/\ \/ y' = 1
\/ y' = 2
/\ y' = 3
the assignments to y
would be y' = 1
and y' = 2
, but not y' = 3
;
minimality prevents us from selecting all three, the syntax order constraint
forces us to select assignments in y' = 1 \/ y' = 2
before y' = 3
and
balance requires that we select both y' = 1
and y' = 2
. On the other hand,
if we change the example to
/\ y' = 3
/\ \/ y = 1
\/ y' = 2
the only assignment has to be y' = 3
. While one of the disjuncts is an
assignment candidate and the other is not, the balance requirement is not
violated here, since neither disjunct is chosen as an assignment.
Similar rules apply to ifthenelse: both the THEN _
and ELSE _
branch must
assign the same variables, however, the IF _
condition is ignored when
determining assignments.
Assignmentfree Expressions
Not all expressions may contain assignments. While Apalache permits the use of
all assignment candidates, except ones defined with :=
(details
here), inside other expressions, some of these candidates will never
be chosen as assignments, based on the syntactic restrictions outlined below:
Given a transition operator A
, based on the shape of A
, the following holds:
 If
A
has the shapeA_1 /\ ... /\ A_n
, then assignments are selected fromA_1, ... , A_n
sequentially, subject to the syntaxorder rule.  If
A
has the shapeA_1 \/ ... \/ A_n
, then assignments are selected in allA_1, ... , A_n
independently, subject to the balance rule.  If
A
has the shapeIF p THEN A_1 ELSE A_2
, then:p
may not contain assignments. Any assignment candidates inp
are subject to the assignmentbeforeuse rule. Assignments are selected in both
A_1
andA_n
independently, subject to the balance rule.
 If
A
has the shape\E x \in S: A_1
, then:S
may not contain assignments. Any assignment candidates inS
are subject to the assignmentbeforeuse rule. Assignments are selected in
A_1
 In any other case,
A
may not contain assignments, however, any assignment candidates inA
are subject to the assignmentbeforeuse rule.
Examples:
A == /\ x' = 2
/\ \E s \in { t \in 1..10 : x' > t }: y' = s
Operator A
contains assignments to both x
and y
; while x' > t
uses
x'
, it does not violate the assignmentbeforeuse rule, since the assignment
to x
precedes the expression, w.r.t. syntax order.
(* INVALID *)
B == \E s \in { t \in 1..10 : x' > t }: y' = s
In operator B
, the assignment to x
is missing, therefore x' > t
produces
an error, as it violates assignmentbeforeuse.
C == /\ x' = 1
/\ IF x' = 0 /\ 2 \in {x', x' + 2, 0}
THEN y' = 1
ELSE y' = 2
The case in C
is similar to A
; conditions of the ifthenelse operator may
not contain assignments to x
, so x' = 0
can never be one, but they may use
x'
, since a preceding expression (x' = 1
) qualifies as an assignment.
(* INVALID *)
D == IF x' = 0
THEN y' = 1
ELSE y' = 2
The operator D
produces an error, for the same reason as B
; even though x' = 0
is an assignment candidate, ifconditions are assignmentfree, so x' = 0
cannot be chosen as an assignment to x
.
(* INVALID *)
E == /\ x' = 2
/\ \A s \in { t \in 1..10 : x' > t }: y' = s
Lastly, while E
looks almost identical to A
, the key difference is that
expressions under universal quantifiers may not contain assignments. Therefore,
y' = s
is not an assignment to y
and thus violates assignmentbeforeuse.
Users may choose, but aren't required, to use manual assignments x' := e
in
place of x' = e
. While the use of this operator does not change Apalache's
internal search for assignments (in particular, using manual assignment
annotations is not a way of circumventing the syntax order requirement), we
encourage the use of manual assignments for clarity.
Unlike other shapes of assignment candidates, whenever a manual assignment is used in a position where the assignment candidate would not be chosen as an assignment (either within assignmentfree expressions or in violation of, for example, the syntax order rule) an error, like one of the two below, is reported:
Assignment error: test.tla:10:1210:18: Manual assignment is spurious, x is already assigned!
or
Assignment error: test.tla:10:1510:21: Illegal assignment inside an assignmentfree expression.
The benefit of using manual assignments, we believe, lies in synchronizing the user's and the tool's understanding of where assignments happen. This helps prevent unexpected results, where the user's expectations or intuition regarding assignment positions are incorrect.
Note: To use manual assignments where the assignment candidate has the shape of
x' \in S
use \E s \in S: x' := s
.
KerA: kernel logic of actions
See TLA+ model checking made symbolic.
TLA+ Language Reference Manual 📗
In this manual, we summarize our knowledge about TLA+ and about its treatment with the Apalache model checker. This is not the manual on Apalache, which can be found in Apalache manual. The TLA+ Video Course by Leslie Lamport is an excellent starting point, if you are new to TLA+. For a comprehensive description and philosophy of the language, check Specifying Systems and the TLA+ Home Page. There are plenty of interesting talks on TLA+ at TLA Channel of Markus Kuppe. This manual completely ignores Pluscal  a higherlevel language on top of TLA+. If you are interested in learning Pluscal, check LearnTla.com by Hillel Wayne.
Contents
 The standard operators of TLA+ 🔌
 Userdefined operators 💡
 Modules and instances: MODULE, EXTENDS and INSTANCES ✂
The standard operators of TLA+
In this document, we summarize the standard TLA+ operators in a form that is similar to manuals on programming languages. The purpose of this document is to provide you with a quick reference, whenever you are looking at the Summary of TLA. The TLA+ Video Course by Leslie Lamport is an excellent starting point, if you are new to TLA+. For a comprehensive description and philosophy of the language, check Specifying Systems and the TLA+ Home Page. You can find handy extensions of the standard library in Community Modules.
We explain the semantics of the operators under the lenses of the Apalache model checker. Traditionally, the emphasis was put on the temporal operators and action operators, as they build the foundation of TLA. We focus on the "+" aspect of the language, which provides you with a language for writing a single step by a state machine. This part of the language is absolutely necessary for writing and reading system specifications. Moreover, we treat equally the "core" operators of TLA+ and the "library" operators: This distinction is less important to the language users than to the tool developers.
In this document, we present the semantics of TLA+, as if it was executed on a computer that is equipped with an additional device that we call an oracle. Most of the TLA+ operators are understood as deterministic operators, so they can be executed on your computer. A few operators are nondeterministic, so they require the oracle to resolve nondeterminism, see Control Flow and Nondeterminism. This is one of the most important features that makes TLA+ distinct from programming languages. Wherever possible, we complement the English semantics with code in Python. Although our semantics are more restrictive than the denotational semantics in Chapter 16 of Specifying Systems, they are very close to the treatment of TLA+ by the model checkers: Apalache and TLC. Our relation between TLA+ operators and Python code bears some resemblance to SALT and PlusPy.
Here, we are using the ASCII notation of TLA+, as this is what you type. We give the nice LaTeX notation in the detailed description. The translation table between the LaTeX notation and ASCII can be found in Summary of TLA.
The "+" Operators in TLA+
Booleans 🚥
Good old Booleans. Learn more...
 Boolean algebra:
 Boolean set:
BOOLEAN
Control flow and nondeterminism 🔀
Hidden powers of TLA+. Learn more...
 Nondeterminism with
A_1 \/ ... \/ A_n
 Nondeterminism with
\E x \in S: P
 Nondeterminism with
IF p THEN e_1 ELSE e_2
 Nondeterminism with
CASE
andCASEOTHER
Deterministic conditionals 🚕
You need them less often than you think. Learn more...
 Deterministic
IFTHENELSE
 Deterministic
CASE
andCASEOTHER
Integers 🔢
Unbounded integers like in Python. Learn more...
Strings 🔡
String constants. You learned it!
 String literals, e.g.,
"hello"
and"TLA+ is awesome"
. In Apalache, the literals have the type
Str
.
 In Apalache, the literals have the type
 Set of all finite strings:
STRING
. In Apalache, the set
STRING
has the typeSet(Str)
.
 In Apalache, the set
Sets 🍣
Like frozen sets in Python, but cooler Learn more...
 Set constructors:
 Enumeration:
{ e_1, ..., e_n }
 Filter:
{ x \in S: p }
 Map:
{ e: x \in S }
 Powers:
SUBSET S
andUNION S
 Enumeration:
 Set algebra:
 Union:
S \union T
(alsoS \cup T
),  Intersection:
S \intersect T
(alsoS \cap T
),  Difference:
S \ T
 Union:
 Set predicates:
 Membership:
x \in S
andx \notin S
,  Subsets:
S \subseteq T
,  Finiteness:
IsFinite
 Membership:
 Cardinality of a finite set:
Cardinality
Logic 🐙
How logicians write loops. Learn more...
 Equality:
=
and/=
(also#
)  Bounded quantifiers:
\A x \in S: p
and\E x \in S: p
 Unbounded quantifiers:
\A x: p
and\E x: p
 Choice:
CHOOSE x \in S: p
andCHOOSE x: p
Functions 💹
Like frozen dictionaries in Python, but cooler. Learn more...
 Function constructor:
[ x \in S > e ]
 Set of functions:
[S > T]
 Function application:
f[e]
 Function replacement:
[ f EXCEPT ![e_1] = e_2 ]
 Function domain:
DOMAIN f
Records 📚
Records like everywhere else. Learn more...
 Record constructor:
[ h_1 > e_1, ..., h_n > e_n ]
 Set of records:
[ h_1: S_1, ..., h_n: S_n ]
 Access by field name:
e.h
 Records are functions. All operators of functions are supported.
Tuples 📐
Well, tuples, indexed with 1, 2, 3... Learn more...
 Tuple constructor:
<< e_1, ..., e_n >>
 Cartesian product:
S_1 \X ... \X S_n
(alsoS_1 \times ... \times S_n
)  Tuples are functions. All operators of functions are supported.
Sequences 🐍
Functions that pretend to be lists, indexed with 1, 2, 3,...
 Add to end:
Append(s, e)
 First and rest:
Head(s)
andTail(s)
 Length:
Len(s)
 Concatenation:
s \o t
(alsos \circ t
)  Subsequence:
SubSeq(s, i, k)
 Sequence filter:
SelectSeq(s, Test)
 Set of finite sequences over
S
:Seq(S)
 Sequences are functions. All operators of functions and tuples are supported.
Bags 👜
 TBD
Reals 🍭
Like "reals" in your math classes, not floating point

All operators of
Integers
but interpreted over reals 
a / b
,Real
,Infinity
Naturals 🐾
If you are Indiana Jones...
 All operators of
Integers
except: unary minusa
andInt
The "A" Operators in TLA+
Action operators 🏃
Taking a step
 Prime:
e'
 Preservation:
UNCHANGED e
 Stuttering:
[A]_e
and<A>_e
 Action enablement:
ENABLED A
 Sequential composition:
A \cdot B
The "TL" Operators in TLA+
Temporal operators 🔜
Talking about computations, finite and infinite
 Always:
[]F
 Eventually:
<>F
 Weak fairness:
WF_e(A)
 Strong fairness:
SF_e(A)
 Leadsto:
F ~> G
 Guarantee:
F +> G
 Temporal hiding:
\EE x: F
 Temporal universal quantification:
\AA x: F
Booleans
You find these operators in every programming language and every textbook on logic. These operators form propositional logic.
Constants
TLA+ contains three special constants: TRUE
, FALSE
, and BOOLEAN
.
The constant BOOLEAN
is defined as the set {FALSE, TRUE}
.
In Apalache, TRUE
, FALSE
, and BOOLEAN
have the types Bool
, Bool
,
and Set(Bool)
, respectively.
A note for settheory purists: In theory, TRUE
and FALSE
are also sets, but
in practice they are treated as indivisible values. For instance, Apalache and
TLC will report an error, if you try to treat FALSE
and TRUE
as sets.
Operators
Warning: Below, we discuss Boolean operators in terms of the way they are usually
defined in programming languages. However, it is important to understand that the
disjunction operator F \/ G
induces a nondeterministic effect when F
or G
contain
the prime operator ('
), or when they are used inside the initialization predicate Init
.
We discuss this effect Control Flow and Nondeterminism.
And (conjunction)
Notation: F /\ G
or F \land G
LaTeX notation:
Arguments: Two or more arbitrary expressions.
Apalache type: (Bool, Bool) => Bool
Effect: The binary case F /\ G
evaluates to:

TRUE
, if bothF
andG
evaluate toTRUE
. 
FALSE
, ifF
evaluates toFALSE
, orF
evaluates toTRUE
andG
evaluates toFALSE
.
The general case F_1 /\ ... /\ F_n
can be understood by evaluating
the expression F_1 /\ (F_2 /\ ... /\ (F_{n1} /\ F_n)...)
.
Determinism: Deterministic, if the arguments are deterministic. Otherwise, the possible effects of nondeterminism of each argument are combined. See Control Flow and Nondeterminism.
Errors: In pure TLA+, the result is undefined if either conjunct evaluates to a nonBoolean value (the evaluation is lazy). In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
TRUE /\ TRUE \* TRUE
FALSE /\ TRUE \* FALSE
TRUE /\ FALSE \* FALSE
FALSE /\ FALSE \* FALSE
FALSE /\ 1 \* FALSE in TLC, type error in Apalache
1 /\ FALSE \* error in TLC, type error in Apalache
Example in Python:
>>> True and True
True
>>> False and True
False
>>> True and False
False
>>> False and False
False
>>> False and 1 # 1 is cast to True
False
>>> 1 and False # 1 is cast to True
False
Special syntax form: To minimize the number of parentheses, conjunction can be written in the indented form:
/\ F_1
/\ G_1
...
/\ G_k
/\ F_2
...
/\ F_n
Similar to scopes in Python, the TLA+ parser groups the expressions according
to the number of spaces in front of /\
. The formula in the above example
is equivalent to:
F_1 /\ (G_1 /\ ... /\ G_k) /\ F_2 /\ ... /\ F_n
Or (disjunction)
Notation: F \/ G
or F \lor G
LaTeX notation:
Arguments: Two or more Boolean expressions.
Apalache type: (Bool, Bool) => Bool
Effect:
The binary case F \/ G
evaluates to:

FALSE
, if bothF
andG
evaluate toFALSE
. 
TRUE
, ifF
evaluates toTRUE
, orF
evaluates toFALSE
andG
evaluates toTRUE
.
The general case F_1 \/ ... \/ F_n
can be understood by evaluating
the expression F_1 \/ (F_2 \/ ... \/ (F_{n1} \/ F_n)...)
.
Determinism: deterministic, if the arguments may not update primed variables. If the arguments may update primed variables, disjunctions may result in nondeterminism, see Control Flow and Nondeterminism.
Errors: In pure TLA+, the result is undefined, if a nonBoolean argument is involved in the evaluation (the evaluation is lazy). In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
TRUE \/ TRUE \* TRUE
FALSE \/ TRUE \* TRUE
TRUE \/ FALSE \* TRUE
FALSE \/ FALSE \* FALSE
TRUE \/ 1 \* TRUE in TLC, type error in Apalache
1 \/ TRUE \* error in TLC, type error in Apalache
Example in Python:
>>> True or True
True
>>> False or True
True
>>> True or False
True
>>> False or False
False
Special syntax form: To minimize the number of parentheses, disjunction can be written in the indented form:
\/ F_1
\/ G_1
...
\/ G_k
\/ F_2
...
\/ F_n
Similar to scopes in Python, the TLA+ parser groups the expressions according
to the number of spaces in front of \/
. The formula in the above example
is equivalent to:
F_1 \/ (G_1 \/ ... \/ G_k) \/ F_2 \/ ... \/ F_n
The indented form allows you to combine conjunctions and disjunctions:
\/ /\ F
/\ G
\/ \/ H
\/ J
The above formula is equivalent to:
(F /\ G) \/ (H \/ J)
Negation
Notation: ~F
or \neg F
or \lnot F
LaTeX notation:
Arguments: One argument that should evaluate to a Boolean value.
Apalache type: Bool => Bool
Effect:
The value of ~F
is computed as follows:
 if
F
is evaluated toFALSE
, then~F
is evaluated toTRUE
,  if
F
is evaluated toTRUE
, then~F
is evaluated toFALSE
.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined, if the argument evaluates to a nonBoolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
~TRUE \* FALSE
~FALSE \* TRUE
~(1) \* error in TLC, type error in Apalache
Example in Python:
>>> not True
False
>>> not False
True
Implication
Notation: F => G
LaTeX notation:
Arguments: Two arguments. Although they can be arbitrary expressions, the result is only defined when both arguments are evaluated to Boolean values.
Apalache type: (Bool, Bool) => Bool
. Note that the =>
operator at the type level expresses the relation of inputs types to output types for operators, and as opposed to the =>
expressing the implication relation at the value level.
Effect: F => G
evaluates to:

TRUE
, ifF
evaluates toFALSE
, orF
evaluates toTRUE
andG
evaluates toTRUE
. 
FALSE
, ifF
evaluates toTRUE
andG
evaluates toFALSE
.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined, if one of the arguments evaluates to a nonBoolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
FALSE => TRUE \* TRUE
TRUE => TRUE \* TRUE
FALSE => FALSE \* TRUE
TRUE => FALSE \* FALSE
FALSE => 1 \* TRUE in TLC, type error in Apalache
TRUE => 1 \* runtime error in TLC, type error in Apalache
1 => TRUE \* runtime error in TLC, type error in Apalache
Example in Python:
Recall that A => B
is equivalent to ~A \/ B
.
>>> (not False) or True
True
>>> (not True) or True
True
>>> (not False) or False
True
>>> (not True) or False
False
Equivalence
Notation: F <=> G
or F \equiv G
LaTeX notation: or
Arguments: Two arguments. Although they can be arbitrary expressions, the result is only defined when both arguments are evaluated to Boolean values.
Apalache type: (Bool, Bool) => Bool
Effect: F <=> G
evaluates to:

TRUE
, if bothF
andG
evaluate toTRUE
, or bothF
andG
evaluate toFALSE
. 
FALSE
, if one of the arguments evaluates toTRUE
, while the other argument evaluates toFALSE
.
How is F <=> G
different from F = G
? Actually, F <=> G
is equality
that is defined only for Boolean values. In other words, if F
and G
are
evaluated to Boolean values, then F <=> G
and F = G
are evaluated to the
same Boolean value. We prefer F <=> G
to F = G
, as F <=> G
clearly
indicates the intended types of F
and G
and thus makes the logical
structure more obvious.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined, if one of the arguments evaluates to a nonBoolean value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
FALSE <=> TRUE \* FALSE
TRUE <=> TRUE \* TRUE
FALSE <=> FALSE \* TRUE
TRUE <=> FALSE \* TRUE
FALSE <=> 1 \* runtime error in TLC, type error in Apalache
1 <=> TRUE \* runtime error in TLC, type error in Apalache
Example in Python:
Assume that both expressions are Boolean. Then, in TLA+, F <=> G
is
equivalent to F = G
. In Python, we express Boolean equality using ==
.
>>> False == True
False
>>> True == True
True
>>> False == False
True
>>> True == False
False
Control Flow and Nondeterminism in TLA+
Author: Igor Konnov
Peer review: Shon Feder, Jure Kukovec
Nondeterminism is one of the TLA+ features that makes it different from
mainstream programming languages. However, it is very easy to overlook it: There is no
special syntax for expressing nondeterminism. In pure TLA+, whether your
specification is deterministic or not depends on the evaluation of the initial
predicate and of the transition predicate. These are usually called Init
and
Next
, respectively. In the following, we first intuitively explain what nondeterminism
means in the mathematical framework of TLA+, and then proceed with the
explanation that is friendly to computers and software engineers.
Explaining nondeterminism to humans
States, transitions, actions, computations. Every TLA+ specification comes
with a set of state variables. For instance, the following specification
declares two state variables x
and y
:
 MODULE coord 
VARIABLES x, y
Init == x = 0 /\ y = 0
Next == x' = x + 1 /\ y' = y + 1
================================
A state is a mapping from state variables to TLA+ values. We do not go into the mathematical depths of precisely defining TLA+ values. Due to the background theory of ZFC, this set is welldefined and is not subject to logical paradoxes. Basically, the values are Booleans, integers, strings, sets, functions, etc.
In the above example, the operator Init
evaluates to TRUE
on exactly one
state, which we can conveniently write using the record
constructor as follows: [x > 0, y > 0]
.
The operator Next
contains primes ('
) and thus represents pairs of states,
which we call transitions. An operator over unprimed and primed variables
is called an action in TLA+. Intuitively, the operator Next
in our example
evaluates to TRUE
on infinitely many pairs of states. For instance, Next
evaluates to TRUE
on the following pairs:
<<[x > 0, y > 0], [x > 1, y > 1]>>
<<[x > 1, y > 1], [x > 2, y > 2]>>
<<[x > 2, y > 2], [x > 3, y > 3]>>
...
In our example, the second state of every transition matches the first state of the next transition in the list. This is because the above sequence of transitions describes the following sequence of states:
[x > 0, y > 0]
[x > 1, y > 1]
[x > 2, y > 2]
[x > 3, y > 3]
...
Actually, we have just written a computation of our specification.
A finite computation is a finite sequence of states s_0, s_1, ..., s_k
that satisfies the following properties:
 The operator
Init
evaluates toTRUE
on states_0
, and  The operator
Next
evaluates toTRUE
on every pair of states<<s_i, s_j>>
for0 <= i < k
andj = i + 1
.
We can also define an infinite computation by considering an infinite
sequence of states that are connected via Init
and Next
as above, but
without stopping at any index k
.
Below we plot the values of x
and y
in the first 16 states with red dots.
Not surprisingly, we just get a line.
Note: In the above examples, we only showed transitions that could be
produced by computations, which (by our definition) originate from the initial
states. These transitions contain reachable states. In principle, Next
may
also describe transitions that contain unreachable states. For instance, the
operator Next
from our example evaluates to TRUE
on the following pairs as
well:
<<[x > 100, y > 100], [x > 99, y > 99]>>
<<[x > 100, y > 100], [x > 99, y > 101]>>
<<[x > 100, y > 100], [x > 101, y > 99]>>
...
There is no reason to restrict transitions only to the reachable states (and it would be hard to do, technically). This feature is often used to reason about inductive invariants.
Determinism and nondeterminism. Our specification is quite boring: It describes exactly one initial state, and there is no variation in computing the next states. We can make it a bit more interesting:
 MODULE coord2 
VARIABLES x, y
Init == x = 0 /\ (y = 0 \/ y = 1 \/ y = 2)
Next == x' = x + 1 /\ y' = y + 1
==========================================
Now our plot has a bit more variation. It presents three computations
that are starting in three different initial states: [x > 0, y > 0]
,
[x > 0, y > 1]
, and [x > 0, y > 2]
.
However, there is still not much variation in Next
. For every state s
,
we can precisely say which state follows s
according to Next
. We can
define Next
as follows (note that Init
is defined as in coord
):
 MODULE coord3 
VARIABLES x, y
Init == x = 0 /\ y = 0
Next == x' = x + 1 /\ (y' = x \/ y' = x + 1)
============================================
The following plot shows the states that are visited by the computations
of the specification coord3
:
Notice that specification coord
describes one infinite computation (and
infinitely many finite computations that are prefixes of the infinite
computation). Specification coord2
describes three infinite computations.
Specification coord3
describes infinitely many infinite computations: At
every step, Next
may choose between y' = x
or y' = x + 1
.
Why are these specifications so different? The answer lies in nondeterminism.
Specification coord
is completely deterministic: There is just one state that
evaluates Init
to TRUE
, and every state is the first component of exactly
one transition, as specified by Next
. Specification coord2
has
nondeterminism in the operator Init
. Specification coord3
has
nondeterminism in the operator Next
.
Discussion. So far we have been talking about the intuition. If you would like to know more about the logic behind TLA+ and the semantics of TLA+, check Chapter 16 of Specifying Systems and The Specification Language TLA+.
When we look at the operators like Init
and Next
in our examples, we can
guess the states and transitions. If we could ask our logician friend to guess
the states and transitions for us every time we read a TLA+ specification, that
would be great. But this approach does not scale well.
Can we explain nondeterminism to a computer? It turns out that we can. In fact, many model checkers support nondeterminism in their input languages. For instance, see Boogie and Spin. Of course, this comes with constraints on the structure of the specifications. After all, people are much better at solving certain logical puzzles than computers, though people get bored much faster than computers.
To understand how TLC enumerates states, check Chapter 14 of Specifying Systems. In the rest of this document, we focus on treatment of nondeterminism that is close to the approach in Apalache.
Explaining nondeterminism to computers
To see how a program could evaluate a TLA+ expression, we need two more ingredients: bindings and oracles.
Bindings. We generalize states to bindings: Given a set of names N
, a
binding maps every name from N
to a value. When N
is the set of all
state variables, a binding describes a state. However, a binding does not have
to assign values to all state variables. Moreover, a binding may assign values
to names that are not the names of state variables. In the following, we are
using bindings over subsets of names that contain: (1) names of the state
variables, and (2) names of the primed state variables.
To graphically distinguish bindings from states, we use parentheses and arrows
to define bindings. For instance, (x > 1, x' > 3)
is a binding that maps
x
to 1 and x'
to 3. (This is our notation, not a common TLA+ notation.)
Evaluating deterministic expressions. Consider the specification coord
,
which was given above. By starting with the empty binding ()
, we can see how
to automatically evaluate the body of the operator Init
:
x = 0 /\ y = 0
By following semantics of conjunction, we see that /\
is
evaluated from lefttoright. The lefthand side equality x = 0
is treated as
an assignment to x
, since x
is not assigned a value in the empty binding
()
, which it is evaluated against. Hence, the expression x = 0
produces
the binding (x > 0)
. When applied to this binding, the righthand side
equality y = 0
is also treated as an assignment to y
. Hence, the expression
y = 0
results in the binding (x > 0, y > 0)
. This binding is defined over
all state variables, so it gives us the only initial state [x > 0, y > 0]
.
Let's see how to evaluate the body of the operator Next
:
x' = x + 1 /\ y' = y + 1
As we have seen, Next
describes pairs of states. Thus, we will produce
bindings over nonprimed and primed variables, that is, over x, x', y, y'
.
Nonprimed variables represent the state before a transition fires, whereas
primed variables represent the state after the transition has been fired.
Consider evaluation of Next
in the state [x > 3, y > 3]
, that is, the
evaluation starts with the binding (x > 3, y > 3)
. Similar to the
conjunction in Init
, the conjunction in Next
first produces the binding (x > 3, y > 3, x' > 4)
and then the binding (x > 3, y > 3, x' > 4, y' > 4)
. Moreover, Next
evaluates to TRUE
when it is evaluated against the
binding (x > 3, y > 3)
. Hence, the state [x > 3, y > 3]
has the only
successor [x > 4, y > 4]
, when following the transition predicate Next
.
In contrast, if we evaluate Next
when starting with the binding (x > 3, y > 3, x' > 1, y' > 1)
, the result will be FALSE
, as the lefthand side of
the conjunction x' = x + 1
evaluates to FALSE
. Indeed, x'
has value 1
,
whereas x
has value 3
, so x' = x + 1
is evaluated as 1 = 3 + 1
against
the binding (x > 3, y > 3, x' > 1, y' > 1)
, which gives us FALSE
.
Hence, the pair of states [x > 3, y > 3]
and [x > 1, y > 1]
is not
a valid transition as represented by Next
.
So far, we only considered unconditional operators. Let's have a look at the
operator A
:
A ==
y > x /\ y' = x /\ x' = x
Evaluation of A
against the binding (x > 3, y > 10)
produces the binding
(x > 3, y > 10, x' > 3, y' > 3)
and the result TRUE
. However, in the
evaluation of A
against the binding (x > 10, y > 3)
, the leftmost
condition y > x
evaluates to FALSE
, so A
evaluates to FALSE
against the
binding (x > 10, y > 3)
. Hence, no next state can be produced from the
the state [x > 3, y > 10]
by using operator A
.
Until this moment, we have been considering only deterministic examples, that is, there was no "branching" in our reasoning. Such examples can be easily put into a program. What about the operators, where we can choose from multiple options that are simultaneously enabled? We introduce an oracle to resolve this issue.
Oracles. For multiple choices, we introduce an external device that we call
an oracle. More formally, we assume that there is a device called GUESS
that
has the following properties:
 For a nonempty set
S
, a callGUESS S
returns some valuev \in S
.  A call
GUESS {}
halts the evaluation.  There are no assumptions about fairness of
GUESS
. It is free to return elements in any order, produce duplicates and ignore some elements.
Why do we call it a device? We cannot call it a function, as functions are
deterministic by definition. For the same reason, it is not a TLA+
operator. In logic, we would say that GUESS
is simply a binary relation on
sets and their elements, which would be no different from the membership
relation \in
.
Why do we need GUESS S
and cannot use CHOOSE x \in S: TRUE
instead?
Actually, CHOOSE x \in S: TRUE
is deterministic. It is guaranteed to return
the same value, when it is called on two equals sets: if S = T
, then
(CHOOSE x \in S: TRUE) = (CHOOSE x \in T: TRUE)
. Our GUESS S
does not have
this guarantee. It is free to return an arbitrary element of S
each time
we call it.
How to implement GUESS S
? There is no general answer to this question.
However, we know of multiple sources of nondeterminism in computer science. So
we can think of GUESS S
as being one of the following implementations:

GUESS S
can be a remote procedure call in a distributed system. Unless we have centralized control over the distributed system, the returned value of RPC may be nondeterministic. 
GUESS S
can be simply the user input. In this case, the user resolves nondeterminism. 
GUESS S
can be controlled by an adversary, who is trying to break the system. 
GUESS S
can pick an element by calling a pseudorandom number generator. However, note that RNG is a very special way of resolving nondeterminism: It assumes probabilistic distribution of elements (usually, it is close to the uniform distribution). Thus, the probability of producing an unfair choice of elements with RNG will be approaching 0.
As you see, there are multiple sources of nondeterminism. With GUESS S
we can
model all of them. As TLA+ does not introduce special primitives for different
kinds of nondeterminism, neither do we fix any implementation of GUESS S
.
Halting. Note that GUESS {}
halts the evaluation. What does it mean? The
evaluation cannot continue. It does not imply that we have found a deadlock in
our TLA+ specification. It simply means that we made wrong choices on the way.
If we would like to enumerate all possible state successors, like TLC does, we
have to backtrack (though that needs fairness of GUESS
). In general, the
course of action depends on the program analysis that you implement. For
instance, a random simulator could simply backtrack and randomly choose another
value.
Nondeterminism in \E x \in S: P
We only have to consider the following case: \E x \in S: P
is evaluated against
a binding s
, and there is a primed state variable y'
that satisfies two
conditions:
 The predicate
P
refers toy'
, that is,P
has to assign a value toy'
.  The value of
y'
is not defined yet, that is, bindings
does not have a value for the namey'
.
If the above assumptions do not hold true, the expression \E x \in S: P
does
not have nondeterminism and it can be evaluated by following the standard
deterministic semantics of exists, see Logic.
Note: We do not consider action operators like UNCHANGED y
. They can be
translated into an equivalent form, e.g., UNCHANGED x
is equivalent to x' = x
.
Now it is very easy to evaluate \E x \in S: P
. We simply evaluate the
following expression:
LET x == GUESS S IN P
It is the job of GUESS S
to tell us what value of x
should be
evaluated. There are three possible outcomes:
 Predicate
P
evaluates toTRUE
when using the provided value ofx
. In this case,P
assigns the value of an expressione
toy'
as soon as the evaluator meets the expressiony' = e
. The evaluation may continue.  Predicate
P
evaluates toFALSE
when using the provided value ofx
. Well, that was a wrong guess. According to our semantics, the evaluation halts. See the above discussion on "halting".  The set
S
is empty, andGUESS S
halts. See the above discussion on "halting".
Example. Consider the following specification:
VARIABLE x
Init == x = 0
Next ==
\E i \in Int:
i > x /\ x' = i
It is easy to evaluate Init
: It does not contain nondeterminism and it
produces the binding (x > 0)
and the state [x > 0]
, respectively. When
evaluating Next
against the binding (x > 0)
, we have plenty of choices.
Actually, we have infinitely many choices, as the set Int
is infinite. TLC
would immediately fail here. But there is no reason for our evaluation to fail.
Simply ask the oracle. Below we give three examples of how the evaluation
works:
1. (GUESS Int) returns 10. (LET i == 10 IN i > x /\ x' = i) is TRUE, x' is assigned 10.
2. (GUESS Int) returns 0. (LET i == 0 IN i > x /\ x' = i) is FALSE. Halt.
3. (GUESS Int) returns 20. (LET i == 20 IN i > x /\ x' = i) is FALSE. Halt.
Nondeterminism in disjunctions
Consider a disjunction that comprises n
clauses:
\/ P_1
\/ P_2
...
\/ P_n
Assume that we evaluate the disjunction against a binding s
. Further,
let us say that Unassigned(s)
is the set of variables that are not
defined in s
. For every P_i
we construct the set of state variables
Use_i
that contains every variable x'
that is mentioned in P_i
.
There are three cases to consider:
 All sets
Use_i
agree on which variables are to be assigned. Formally,Use_i \intersect Unassigned(s) = Use_j \intersect Unassigned(s) /= {}
fori, j \in 1..n
. This is the case that we consider below.  Two clauses disagree on the set of variables to be assigned.
Formally, there is a pair
i, j \in 1..n
that satisfy the inequality:Use_i \intersect Unassigned(s) /= Use_j \intersect Unassigned(s)
. In this case, the specification is illstructured. TLC would raise an error when it found a binding like this. Apalache would detect this problem when preprocessing the specification.  The clauses do not assign values to the primed variables.
Formally,
Use_i \intersect Unassigned(s) = {}
fori \in 1..n
. This is the deterministic case. It can be evaluated by using the deterministic semantics of Boolean operators.
We introduce a fresh variable to contain the choice of the clause. Here we
call it choice
. In a real implementation of an evaluator, we would have to
give it a unique name. Now we evaluate the following conjunction:
LET choice == GUESS 1..n IN
/\ (choice = 1) => P_1
/\ (choice = 2) => P_2
...
/\ (choice = n) => P_n
Importantly, at most one clause in the conjunction will be actually evaluated. As a result, we cannot produce conflicting assignments to the primed variables.
Example: Consider the following specification:
VARIABLES x, y
Init == x == 0 /\ y == 0
Next ==
\/ x >= 0 /\ y' = x /\ x' = x + 1
\/ x <= 0 /\ y' = x /\ x' = (x + 1)
As you can see, the operator Next
is nondeterministic since both clauses may
be activated when x = 0
.
First, let's evaluate Next
against the binding (x > 3, y > 3)
:
1. (GUESS 1..2) returns 1. (LET i == 1 IN Next) is TRUE, x' is assigned 4, y' is assigned 3.
2. (GUESS 1..2) returns 2. (LET i == 2 IN Next) is FALSE. Halt.
Second, evaluate Next
against the binding (x > 3, y > 3)
:
1. (GUESS 1..2) returns 1. (LET i == 1 IN Next) is FALSE. Halt.
2. (GUESS 1..2) returns 2. (LET i == 2 IN Next) is TRUE, x' is assigned 4, y' is assigned 3.
Third, evaluate Next
against the binding (x > 0, y > 0)
:
1. (GUESS 1..2) returns 1. (LET i == 1 IN Next) is TRUE. x' is assigned 1, y' is assigned 0.
2. (GUESS 1..2) returns 2. (LET i == 2 IN Next) is TRUE, x' is assigned 1, y' is assigned 0.
Important note. In contrast to shortcircuiting of disjunction in the deterministic case, we have nondeterministic choice here. Hence, shortcircuiting does not apply to nondeterministic disjunctions.
Nondeterminism in Boolean IFTHENELSE
For the deterministic use of IFTHENELSE
, see Deterministic
conditionals.
Consider an IFTHENELSE
expression to be evaluated in a partial state s
:
IF A THEN B ELSE C
In Apalache, this operator has the polymorphic type (Bool, a, a) => a
,
where a
can be replaced with a concrete type. Here, we consider the case
(Bool, Bool, Bool) => Bool
.
Here we assume that both B
and C
produce Boolean results and B
and C
refer to at least one primed variable y'
that is undefined in s
. Otherwise, the
expression can be evaluated as a deterministic
conditional.
In this case, IFTHENELSE
can be evaluated as the equivalent expression:
\/ A /\ B
\/ ~A /\ C
We do not recommend you to use IFTHENELSE with nondeterminism. The structure of the disjunction provides a clear indication that the expression may assign to variables as a side effect. IFTHENELSE has two thinking steps: what is the expected result, and what are the possible side effects.
Warning: While it is technically possible to write x' = e
inside the
condition, the effect of x' = e
is not obvious when x'
is not assigned a
value.
Nondeterminism in Boolean CASE
For the deterministic use of CASE
,
see Deterministic conditionals.
CASE without OTHER.
Consider a CASE
expression:
CASE P_1 > e_1
[] P_2 > e_2
...
[] P_n > e_n
Here, we assume that e_1, ..., e_n
produce Boolean results. Or, in terms of
Apalache types, this expression has the type: (Bool, Bool, ..., Bool, Bool) => Bool
. Otherwise, see Deterministic conditionals.
This operator is equivalent to the following disjunction:
\/ P_1 /\ e_1
\/ P_2 /\ e_2
...
\/ P_n /\ e_n
Similar to IFTHENELSE, we do not recommend using CASE for expressing nondeterminism. When you are using disjunction, the Boolean result and possible side effects are expected.
CASE with OTHER. The more general form of CASE
is like follows:
CASE P_1 > e_1
[] P_2 > e_2
...
[] P_n > e_n
[] OTHER > e_other
This operator is equivalent to the following disjunction:
\/ P_1 /\ e_1
\/ P_2 /\ e_2
...
\/ P_n /\ e_n
\/ ~P_1 /\ ... /\ ~P_n /\ e_other
The use of CASE with OTHER together with nondeterminism is quite rare. It is not clear why would one need a fallback option in the Boolean formula. We recommend you to use the disjunctive form instead.
Deterministic conditionals
In this section, we consider the instances of IFTHENELSE
and CASE
that
may not update primed variables. For the case, when the operators inside
IFTHENELSE
or CASE
can be used to do nondeterministic assignments, see
Control Flow and Nondeterminism.
Warning: Because frequent use of IFTHENELSE
is very common in most
programming languages, TLA+ specification authors with programming experience
often default to writing expressions such as IF A THEN B ELSE C
. We
encourage those authors to use this construct more sparingly. In our
experience, the use of IFTHENELSE
is rarely required. Many things can be
done with Boolean operators, which provide more structure in
TLA+ code than in programming languages. We recommend using IFTHENELSE
to
compute predicatedependent values, not to structure code.
Warning 2: CASE
is considered deterministic in this
section, as it is defined with the CHOOSE
operator in
Specifying Systems, Section 16.1.4.
For this reason, CASE
should only be used when all of its guards are mutually exclusive.
Given all the intricacies of CASE
,
we recommend using nested IFTHENELSE
instead.
Deterministic IFTHENELSE
Use it when choosing between two values, not to structure your code.
Notation: IF A THEN B ELSE C
LaTeX notation: the same
Arguments: a Boolean expression A
and two expressions B
and C
Apalache type: (Bool, a, a) => a
. Note that a
can be replaced with
Bool
. If a
is Bool
, and only in that case, the expression IF A THEN B ELSE C
is equivalent to (A => B) /\ (~A => C)
.
Effect: IF A THEN B ELSE C
evaluates to:
 The value of
B
, ifA
evaluates toTRUE
.  The value of
C
, ifA
evaluates toFALSE
.
Determinism: This is a deterministic version. For the nondeterministic version, see Control Flow and Nondeterminism.
Errors: If A
evaluates to a nonBoolean value, the result is undefined.
TLC raises an error during model checking. Apalache raises a type error when
preprocessing. Additionally, if B
and C
may evaluate to values of different
types, Apalache raises a type error.
Example in TLA+: Consider the following TLA+ expression:
IF x THEN 100 ELSE 0
As you most likely expected, this expression evaluates to 100
, when x
evaluates to TRUE
; and it evaluates to 0
, when x
evaluates to FALSE
.
Example in Python:
100 if x else 0
Note that we are using the expression syntax for ifelse
in python.
This is because we write an expression, not a series of statements that assign
values to variables!
Deterministic CASE
Read the description and never use this operator
Notation:
CASE p_1 > e_1
[] p_2 > e_2
...
[] p_n > e_n
LaTeX notation:
Arguments: Boolean expressions p_1, ..., p_n
and expressions e_1, ..., e_n
.
Apalache type: (Bool, a, Bool, a, ..., Bool, a) => a
, for some type a
.
If a
is Bool
, then the case operator can be a part of a Boolean formula.
Effect: Given a state s
, define the set I \subseteq 1..n
as follows:
The set I
includes the index j \in 1..n
if
and only if p_j
evaluates to TRUE
in the state s
.
Then the above CASE
expression evaluates to:
 the value of the expression
e_i
for somei \in I
, ifI
is not empty; or  an undefined value, if the set
I
is empty.
As you can see, when several predicates {p_i: i \in I}
are evaluated
to TRUE
in the state s
, then the result of CASE
is equal to one of the
elements in the set {e_i: i \in I}
. Although the result should be stable,
the exact implementation is unknown.
Whenever I
is a singleton set, the result is easy to define: Just take the
only element of I
. Hence, when p_1, ..., p_n
are mutually exclusive,
the result is deterministic and implementationagnostic.
Owing to the flexible semantics of simultaneously enabled predicates,
TLC interprets the above CASE
operator as a chain of IFTHENELSE
expressions:
IF p_1 THEN e_1
ELSE IF p_2 THEN e_2
...
ELSE IF p_n THEN e_n
ELSE TLC!Assert(FALSE)
As TLC fixes the evaluation order, TLC may miss a bug in an arm that is never activated in this order!
Note that the last arm of the ITEseries ends with Assert(FALSE)
, as the
result is undefined, when no predicate evaluates to TRUE
. As the type
of this expression cannot be precisely defined, Apalache does not support CASE
expressions, but only supports CASEOTHER
expressions (see below), which
it treats as a chain of IFTHENELSE
expressions.
Determinism. The result of CASE
is deterministic, if there are no primes
inside. For the nondeterministic version, see [Control Flow and
Nondeterminism]. When the predicates are
mutually exclusive, the evaluation result is clearly specified. When the predicates are
not mutually exclusive, the operator is still deterministic, but only one of
the simultaneously enabled branches is evaluated.
Which branch is evaluated depends on the CHOOSE
operator, see [Logic].
Errors: If one of p_1, ..., p_n
evaluates to a nonBoolean value, the
result is undefined. TLC raises an error during model checking. Apalache
raises a type error when preprocessing. Additionally, if e_1
, ..., e_n
may evaluate to values of different types, Apalache raises a type error.
Example in TLA+: The following expression classifies an integer variable
n
with one of the three strings: "negative", "zero", or "positive".
CASE n < 0 > "negative"
[] n = 0 > "zero"
[] n > 0 > "positive"
Importantly, the predicates n < 0
, n = 0
, and n > 0
are mutually
exclusive.
The following expression contains nonexclusive predicates:
CASE n % 2 = 0 > "even"
[] (\A k \in 2..(1 + n \div 2): n % k /= 0) > "prime"
[] n % 2 = 1 > "odd"
Note that by looking at the specification, we cannot tell, whether this
expression returns "odd" or "prime", when n = 17
. We only know that the
case expression should consistently return the same value, whenever it is
evaluated with n = 17
.
Example in Python: Consider our first example in TLA+. Similar to TLC, we give executable semantics for the fixed evaluation order of the predicates.
def case_example(n):
if n < 0:
return "negative"
elif n == 0:
return "zero"
elif n > 0:
return "positive"
Deterministic CASEOTHER
Better use IFTHENELSE.
Notation:
CASE p_1 > e_1
[] p_2 > e_2
...
[] p_n > e_n
[] OTHER > e_0
LaTeX notation:
Arguments: Boolean expressions p_1, ..., p_n
and expressions e_0, e_1, ..., e_n
.
Apalache type: (Bool, a, Bool, a, ..., Bool, a, a) => a
, for some type a
.
If a
is Bool
, then the case operator can be a part of a Boolean formula.
Effect: This operator is equivalent to the following version of CASE
:
CASE p_1 > e_1
[] p_2 > e_2
...
[] p_n > e_n
[] ~(p_1 \/ p_2 \/ ... \/ p_n) > e_0
Both TLC and Apalache interpret this CASE
operator as a chain of
IFTHENELSE
expressions:
IF p_1 THEN e_1
ELSE IF p_2 THEN e_2
...
ELSE IF p_n THEN e_n
ELSE e_0
All the idiosyncrasies of CASE
apply to CASEOTHER
. Hence, we recommend
using IFTHENELSE
instead of CASEOTHER
. Although IFTHENELSE
is a bit more verbose, its semantics are precisely defined.
Determinism. The result of CASEOTHER
is deterministic, if e_0
, e_1
,
..., e_n
may not update primed variables. For the nondeterministic version,
see [Control Flow and Nondeterminism]. When
the predicates are mutually exclusive, the semantics is clearly specified. When
the predicates are not mutually exclusive, the operator is still deterministic,
but only one of the simultaneously enabled branches is evaluated. The choice of
the branch is implemented with the operator CHOOSE
, see
[Logic].
Errors: If one of p_1, ..., p_n
evaluates to a nonBoolean value, the
result is undefined. TLC raises an error during model checking. Apalache
raises a type error when preprocessing. Additionally, if e_0
, e_1
, ...,
e_n
may evaluate to values of different types, Apalache raises a type error.
Integers
The integer literals belong to the core language. They are written by
using the standard syntax: 0, 1, 1, 2, 2, 3, 3, ... Importantly, TLA+
integers are unbounded. They do not have any fixed bit width, and they cannot
overflow. In Apalache, these literals have the type Int
.
The integer operators are defined in the standard module Integers
. To use
it, write the EXTENDS
clause in the first lines of your module. Like this:
 MODULE MyArithmetics 
EXTENDS Integers
...
==============================
Integers in Apalache and SMT
Although you can write arbitrary expressions over integers in TLA+, Apalache
translates these expressions as constraints in
SMT. Some
expressions are easier to solve than the others. For instance, the expression
2 * x > 5
belongs to linear integer arithmetic, which can be solved more
efficiently than general arithmetic. For state variables x
and y
, the
expression x * y > 5
belongs to nonlinear integer arithmetic, which is
harder to solve than linear arithmetic.
When your specification is using only integer literals, e.g., 1
, 2
, 42
,
but none of the operators from the Integers
module, the integers can
be avoided altogether. For instance, you can replace the integer constants
with string constants, e.g., "1"
, "2"
, "42"
. The string constants are
translated as constants in the SMT constraints. This simple trick may bring
your specification into a much simpler theory. Sometimes, this trick allows z3
to use parallel algorithms.
Constants
The module Integers
defines two constant sets (technically, they are
operators without arguments):
 The set
Int
that consists of all integers. This set is infinite. In Apalache, the setInt
has the typeSet(Int)
. A bit confusing, right? 😎  The set
Nat
that consists of all natural numbers, that is,Nat
contains every integerx
that has the propertyx >= 0
. This set is infinite. In Apalache, the setNat
has the type...Set(Int)
.
Operators
Integer range
Notation: a..b
LaTeX notation: a..b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Set(Int)
.
Effect: a..b
evaluates to the finite set {i \in Int: a <= i /\ i <= b}
,
that is, the set of all integers in the range from a
to b
, including a
and b
. If a > b
, then a..b
is the empty set {}
.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
0..10 \* { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
5..3 \* { 5, 4, 3, 2, 1, 0, 1, 2, 3 }
10..0 \* { }
"a".."z" \* runtime error in TLC, type error in Apalache
{1}..{3} \* runtime error in TLC, type error in Apalache
Example in Python: a..b
can be written as set(range(a, b + 1))
in
python.
>>> set(range(0, 10 + 1))
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
>>> set(range(10, 2))
set()
Unary integer negation
Notation: i
LaTeX notation: i
Arguments: One argument. The result is only defined when the argument evaluates to an integer.
Apalache type: Int => Int
.
Effect: i
evaluates to the negation of i
.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined if the argument evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
(5) \* 5, note that '5' is just a literal, not operator application
(5) \* 5
x \* negated value of x
Example in Python:
>>> (5)
5
>>> (5)
5
Integer addition
Notation: a + b
LaTeX notation: a + b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Int
.
Effect: a + b
evaluates to the sum of a
and b
.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
5 + 3 \* 8
(5) + 3 \* 2
Example in Python:
>>> 5 + 3
8
>>> (5) + 3
2
Integer subtraction
Notation: a  b
LaTeX notation: a  b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Int
.
Effect: a  b
evaluates to the difference of a
and b
.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
5  3 \* 2
(5)  3 \* 8
(5)  (3) \* 2
Example in Python:
>>> 5  3
2
>>> (5)  3
8
>>> (5)  (3)
2
Integer multiplication
Notation: a * b
LaTeX notation: a * b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Int
.
Effect: a * b
evaluates to the product of a
and b
.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
5 * 3 \* 15
(5) * 3 \* 15
Example in Python:
>>> 5 * 3
15
>>> (5) * 3
15
Integer division
Notation: a \div b
LaTeX notation:
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values, and the second argument is different from 0.
Apalache type: (Int, Int) => Int
.
Effect: a \div b
is defined as follows:
 When
a >= 0
andb > 0
, then the result ofa \div b
is the integerc
that has the property:a = b * c + d
for somed
in0..(b1)
.  When
a < 0
andb > 0
, then the result ofa \div b
is the integerc
that has the property:a = b * c + d
for somed
in0..(b1)
.  When
a >= 0
andb < 0
, then the result ofa \div b
is the integerc
that has the property:a = b * c + d
for somed
in0..(b1)
.  When
a < 0
andb < 0
, then the result ofa \div b
is the integerc
that has the property:a = b * c + d
for somed
in0..(b1)
.
When a < 0
or b < 0
, the result of the integer division a \div b
according to the TLA+ definition is different from the integer division a / b
in the programming languages (C, Java, Scala, Rust). See the
table below.
C (clang 12)  Scala 2.13  Rust  Python 3.8.6  TLA+ (TLC)  SMT (z3 4.8.8) 

100 / 3 == 33  100 / 3 == 33  100 / 3 == 33  100 // 3 == 33  (100 \div 3) = 33  (assert (= 33 (div 100 3))) 
100 / 3 == 33  100 / 3 == 33  100 / 3 == 33  100 // 3 == 34  ((100) \div 3) = 34  (assert (= ( 0 34) (div ( 0 100) 3))) 
100 / (3) == 33  100 / (3) == 33  100 / (3) == 33  100 // (3) == 34  (100 \div (3)) = 34  (assert (= ( 0 33) (div 100 ( 0 3)))) 
100 / (3) == 33  100 / (3) == 33  100 / (3) == 33  100 // (3) == 33  ((100) \div (3)) = 33  (assert (= 34 (div ( 0 100) ( 0 3)))) 
Unfortunately, Specifying Systems only gives us the definition for the case
b > 0
(that is, cases 12 in our description). The implementation in SMT and
TLC produce incompatible results for b < 0
. See issue #331 in
Apalache.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if
one of the arguments evaluates to a noninteger value. In this case, Apalache
statically reports a type error, whereas TLC reports a runtime error. The value
of a \div b
is undefined for b = 0
.
Example in TLA+: Here are the examples for the four combinations of signs (according to TLC):
100 \div 3 \* 33
(100) \div 3 \* 34
100 \div (3) \* 34 in TLC
(100) \div (3) \* 33 in TLC
Example in Python: Here are the examples for the four combinations of signs to produce the same results as in TLA+:
>>> 100 // 3
33
>>> 100 // 3
34
>>> 100 // (3)
34
>>> (100) // (3)
33
Integer remainder
Notation: a % b
LaTeX notation: a % b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values, and the second argument is different from 0.
Apalache type: (Int, Int) => Int
.
Effect: a % b
is the number c
that has the property:
a = b * (a \div b) + c
.
Note that when a < 0
or b < 0
, the result of the integer remainder a % b
according to the TLA+ definition is different from the integer remainder a % b
in the programming languages (C, Python, Java, Scala, Rust). See the
examples below.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if
one of the arguments evaluates to a noninteger value. In this case, Apalache
statically reports a type error, whereas TLC reports a runtime error. The value
of a % b
is undefined for b = 0
.
Example in TLA+: Here are the examples for the four combinations of signs:
100 % 3 \* 1
100 % (3) \* 2
100 % (3) \* 1
100 % 3 \* 2
Example in Python: Here are the examples for the four combinations of signs to produce the same results as in TLA+:
>>> 100 % 3
1
>>> 100 % (3) + 3
2
>>> 100 % (3) + 3
1
>>> 100 % 3
2
Integer exponentiation
Notation: a^b
LaTeX notation:
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values, and these values fall into one of the several cases:
b > 0
,b = 0
anda /= 0
.
Apalache type: (Int, Int) => Int
.
Effect: a^b
evaluates to a
raised to the b
th power:
 If
b = 1
, thena^b
is defined asa
.  If
a = 0
andb > 0
, thena^b
is defined as0
.  If
a /= 0
andb > 1
, thena^b
is defined asa * a^(b1)
.  In all other cases,
a^b
is undefined.
In TLA+, a^b
extends to reals, see Chapter 18 in Specifying Systems.
For instance, 3^(5)
is defined on reals. However, reals are supported
neither by TLC, nor by Apalache.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
5^3 \* 125
(5)^3 \* 125
0^3 \* 0
1^5 \* 1
(1)^5 \* 1
0^0 \* undefined on integers, TLC reports a runtime error
5^(3) \* undefined on integers, TLC reports a runtime error
Example in Python:
>>> 5 ** 3
125
>>> (5) ** 3
125
>>> 0 ** 3
0
>>> 1 ** 5
1
>>> (1) ** 5
1
>>> 0 ** 0
1
>>> 5 ** (3)
0.008
Integer lessthan
Notation: a < b
LaTeX notation: a < b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Bool
.
Effect: a < b
evaluates to:
TRUE
, ifa
is less thanb
,FALSE
, otherwise.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
1 < 5 \* TRUE
5 < 5 \* FALSE
5 < 1 \* FALSE
Example in Python:
>>> 1 < 5
True
>>> 5 < 5
False
>>> 5 < 1
False
Integer lessthanorequal
Notation: a <= b
or a =< b
or a \leq b
LaTeX notation:
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Bool
.
Effect: a <= b
evaluates to:
TRUE
, ifa < b
ora = b
.FALSE
, otherwise.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
1 <= 5 \* TRUE
5 <= 5 \* TRUE
5 <= 1 \* FALSE
Example in Python:
>>> 1 <= 5
True
>>> 5 <= 5
True
>>> 5 <= 1
False
Integer greaterthan
Notation: a > b
LaTeX notation: a > b
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Bool
.
Effect: a > b
evaluates to:
TRUE
, ifa
is greater thanb
,FALSE
, otherwise.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
1 > 5 \* FALSE
5 < 5 \* FALSE
5 > 1 \* TRUE
Example in Python:
>>> 1 > 5
False
>>> 5 > 5
False
>>> 5 > 1
True
Integer greaterthanorequal
Notation: a >= b
or a \geq b
LaTeX notation:
Arguments: Two arguments. The result is only defined when both arguments are evaluated to integer values.
Apalache type: (Int, Int) => Bool
.
Effect: a >= b
evaluates to:
TRUE
, ifa > b
ora = b
.FALSE
, otherwise.
Determinism: Deterministic.
Errors: No overflow is possible. In pure TLA+, the result is undefined if one of the arguments evaluates to a noninteger value. In this case, Apalache statically reports a type error, whereas TLC reports a runtime error.
Example in TLA+:
1 >= 5 \* FALSE
5 >= 5 \* TRUE
5 >= 1 \* TRUE
Example in Python:
>>> 1 >= 5
False
>>> 5 >= 5
True
>>> 5 >= 1
True
Equality and inequality
The operators a = b
and a /= b
are core operators of TLA+ and thus they are
not defined in the module Integers
, see Logic.
Sets
Sets are the foundational data structure in TLA+. (Similar to what lists are in Lisp and Python). The other TLA+ data structures can be all expressed with sets: functions, records, tuples, sequences. In theory, even Booleans and integers can be expressed with sets. In practice, TLA+ tools treat Booleans and integers as special values that are different from sets. It is important to understand TLA+ sets well. In contrast to programming languages, there is no performance penalty for using sets instead of sequences: TLA+ does not have a compiler, the efficiency is measured in the time it takes the human brain to understand the specification.
Immutability. In TLA+, a set is an immutable data structure that stores its elements in no particular order. All elements of a set are unique. In fact, those two sentences do not make a lot of sense in TLA+. We have written them to build the bridge from a programming language to TLA+, as TLA+ does not have a memory model. 😉
Sets may be constructed by enumerating values in some order, allowing for duplicates:
{ 1, 2, 3, 2, 4, 3 }
Note that the above set is equal to the sets { 1, 2, 3, 4 }
and { 4, 3, 2, 1 }
. They are actually the same set, though they are constructed by passing
various number of arguments in different orders.
The most basic set operation is the set membership that checks, whether a set contains a value:
3 \in S
TLA+ sets are similar to
frozenset
in
Python and immutable Set[Object]
in Java. In contrast to programming
languages, set elements do not need hashes, as implementation efficiency is not
an issue in TLA+.
Types. In pure TLA+, sets may contain any kinds of elements. For instance, a set may mix integers, Booleans, and other sets:
{ 2020, { "is" }, TRUE, "fail" }
TLC restricts set elements to comparable values. See Section 14.7.2 of Specifying Systems. In a nutshell, if you do not mix the following five kinds of values in a single set, TLC would not complain about your sets:
 Booleans,
 integers,
 strings,
 sets,
 functions, tuples, records, sequences.
Apalache requires set elements to have the same type, that is, Set(a)
for
some type a
. This is enforced by the type checker. (Records are an exception
to this rule, as some records can be unified to a common type.)
Operators
Set constructor by enumeration
Notation: {e_1, ..., e_n}
LaTeX notation: {e_1, ..., e_n}
Arguments: Any number of arguments, n >= 0
.
Apalache type: (a, ..., a) => Set(a)
, for some type a
.
Effect: Produce the set that contains the values of the expressions e_1, ..., e_n
, in no particular order, and only these values. If n = 0
, the
empty set is constructed.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the set elements. They can be any combination of TLA+ values: Booleans, integers, strings, sets, functions, etc.
TLC only allows a user to construct sets out of elements that are comparable. For instance, two integers are comparable, but an integer and a set are not comparable. See Section 14.7.2 of Specifying Systems.
Apalache goes further and requires that all set elements have the same type. If this is not the case, the type checker flags an error.
Example in TLA+:
{ 1, 2, 3 } \* a flat set of integers
{ { 1, 2 }, { 2, 3 } } \* a set of sets of integers
{ FALSE, 1 } \* a set of mixed elements.
\* Model checking error in TLC, type error in Apalache
Example in Python:
>>> {1, 2, 3}
{1, 2, 3}
>>> {frozenset({2, 3}), frozenset({1, 2})}
{frozenset({2, 3}), frozenset({1, 2})}
>>> {False, 1}
{False, 1}
Set membership
Notation: e \in S
LaTeX notation:
Arguments: Two arguments. If the second argument is not a set, the result is undefined.
Apalache type: (a, Set(a)) => Bool
, for some type a
.
Effect: This operator evaluates to:
TRUE
, ifS
is a set that contains an element that is equal to the value ofe
; andFALSE
, ifS
is a set and all of its elements are not equal to the value ofe
.
Warning: If you are using the special form x' \in S
, this operator may
assign a value to x'
as a side effect. See Control Flow and Nondeterminism.
Determinism: Deterministic, unless you are using the special form x' \in S
to assign a value to x'
, see Control Flow and Nondeterminism.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that e
cannot be compared to the
elements of S
. Apalache produces a static type error, if the type of e
is
incompatible with the type of elements of S
, or if S
is not a set.
Example in TLA+:
1 \in { 1, 2, 3 } \* TRUE
10 \in { 1, 2, 3 } \* FALSE
{} \in { {1}, {2} } \* FALSE
1 \in { "a", "b" } \* model checking error in TLC,
\* static type error in Apalache
Example in Python: Python conveniently offers us in
:
>>> 1 in {1, 2, 3}
True
>>> 10 in {1, 2, 3}
False
>>> 1 in {"a", "b"}
False
Set nonmembership
Notation: e \notin S
LaTeX notation:
Arguments: Two arguments. If the second argument is not a set, the result is undefined.
Apalache type: (a, Set(a)) => Bool
, for some type a
.
Effect: This operator evaluates to:
FALSE
, ifS
is a set that contains an element that is equal to the value ofe
; andTRUE
, ifS
is a set and all of its elements are not equal to the value ofe
.
Warning: In contrast to x' \in S
, the expression x' \notin T
,
which is equivalent to ~(x' \in T)
is never
treated as an assignment in Apalache and TLC.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that e
cannot be compared to the
elements of S
. Apalache produces a static type error, if the type of e
is
incompatible with the type of elements of S
, or if S
is not a set.
Example in TLA+:
1 \notin { 1, 2, 3 } \* FALSE
10 \notin { 1, 2, 3 } \* TRUE
{} \notin { {1}, {2} } \* TRUE
1 \notin { "a", "b" } \* model checking error in TLC,
\* static type error in Apalache
Example in Python: Python conveniently offers us not in
:
>>> 1 not in {1, 2, 3}
False
>>> 10 not in {1, 2, 3}
True
>>> 1 not in {"a", "b"}
True
Equality and inequality
The operators a = b
and a /= b
are core operators of TLA+,
see Logic.
Set inclusion
Notation: S \subseteq T
LaTeX notation:
Arguments: Two arguments. If both arguments are not sets, the result is undefined.
Apalache type: (Set(a), Set(a)) => Bool
, for some type a
.
Effect: This operator evaluates to:
TRUE
, ifS
andT
are sets, and every element ofS
is a member ofT
;FALSE
, ifS
andT
are sets, and there is an element ofS
that is not a member ofT
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that elements of S
cannot be compared
to the elements of T
. Apalache produces a static type error, if S
and T
are either not sets, or sets of incompatible types.
Example in TLA+:
{ 1, 2 } \subseteq { 1, 2, 3 } \* TRUE
{ 1, 2, 3 } \subseteq { 1, 2, 3 } \* TRUE
{ 1, 2, 3 } \subseteq { 1, 2 } \* FALSE
{ {1} } \subseteq { 1, 2, 3 } \* FALSE, model checking error in TLC
\* static type error in Apalache
Example in Python: Python conveniently offers us <=
:
>>> {1, 2} <= {1, 2, 3}
True
>>> {1, 2, 3} <= {1, 2, 3}
True
>>> {1, 2, 3} <= {1, 2}
False
>>> {frozenset({1})} <= {1, 2, 3}
False
Binary set union
Notation: S \union T
or S \cup T
LaTeX notation:
Arguments: Two arguments. If both arguments are not sets, the result is undefined.
Apalache type: (Set(a), Set(a)) => Set(a)
, for some type a
.
Effect: This operator evaluates to the set that contains the elements
of S
as well as the elements of T
, and no other values.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that elements of S
cannot be compared
to the elements of T
. Apalache produces a static type error, if S
and T
are either not sets, or sets of incompatible types.
Example in TLA+:
{ 0, 1, 2 } \union { 1, 2, 3 } \* { 0, 1, 2, 3 }
{ } \union { 1, 2, 3 } \* { 1, 2, 3 }
{ 1, 2, 3 } \union { } \* { 1, 2, 3 }
{ {1} } \union { 1, 2 } \* { {1}, 1, 2 }, model checking error in TLC
\* static type error in Apalache
Example in Python: Python conveniently offers us union
that can be written as 
:
>>> {0, 1, 2}  {1, 2, 3}
{0, 1, 2, 3}
>>> set()  {1, 2, 3}
{1, 2, 3}
>>> {1, 2, 3}  set()
{1, 2, 3}
>>> {frozenset({1})}  {1, 2}
{1, frozenset({1}), 2}
Set intersection
Notation: S \intersect T
or S \cap T
LaTeX notation:
Arguments: Two arguments. If both arguments are not sets, the result is undefined.
Apalache type: (Set(a), Set(a)) => Set(a)
, for some type a
.
Effect: This operator evaluates to the set that contains only those elements
of S
that also belong to T
, and no other values.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that elements of S
cannot be compared
to the elements of T
. Apalache produces a static type error, if S
and T
are either not sets, or sets of incompatible types.
Example in TLA+:
{ 0, 1, 2 } \intersect { 1, 2, 3 } \* { 1, 2 }
{ } \intersect { 1, 2, 3 } \* { }
{ 1, 2, 3 } \intersect { } \* { }
{ {1} } \intersect { 1, 2 } \* { }, model checking error in TLC
\* static type error in Apalache
Example in Python: Python conveniently offers us intersection
, which
can be also written as &
:
>>> {0, 1, 2} & {1, 2, 3}
{1, 2}
>>> set() & {1, 2, 3}
set()
>>> {1, 2, 3} & set()
set()
>>> {frozenset({1})} & {1, 2}
set()
Set difference
Notation: S \ T
LaTeX notation:
Arguments: Two arguments. If both arguments are not sets, the result is undefined.
Apalache type: (Set(a), Set(a)) => Set(a)
, for some type a
.
Effect: This operator evaluates to the set that contains only those elements
of S
that do not belong to T
, and no other values.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, when it discovers that elements of S
cannot be compared
to the elements of T
. Apalache produces a static type error, if S
and T
are either not sets, or sets of incompatible types.
Example in TLA+:
{ 0, 1, 2 } \ { 1, 2, 3 } \* { 0 }
{ } \ { 1, 2, 3 } \* { }
{ 1, 2, 3 } \ { } \* { 1, 2, 3 }
{ {1} } \ { 1, 2 } \* { {1} }, model checking error in TLC
\* static type error in Apalache
Example in Python: Python conveniently offers us difference
, which
can be also written as 
:
>>> {0, 1, 2}  {1, 2, 3}
{0}
>>> set()  {1, 2, 3}
set()
>>> {1, 2, 3}  set()
{1, 2, 3}
>>> {frozenset({1})}  {1, 2}
{frozenset({1})}
Set filter
Notation: { x \in S: P }
LaTeX notation:
Arguments: Three arguments: a variable name (or a tuple of names, see Advanced syntax), a set, and an expression.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
{ x \in S: P }
has the typeSet(a)
.
Effect: This operator constructs a new set F
as follows. For every
element e
of S
, do the following (we give a sequence of steps to ease
the understanding):
 Bind the element
e
to variablex
,  Evaluate the predicate
P
,  If
P
evaluates toTRUE
under the binding[x > e]
, then insert the element ofe
into setF
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible with the type of x
as expected in P
.
Advanced syntax: Instead of a single variable x
, one can use a tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write { <<x, y>> \in S: P }
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The filter expression constructs the set of elements (tuples) that make
P
evaluate to TRUE
.
Example in TLA+:
{ x \in {1, 2, 3, 4}: x > 2 } \* { 3, 4 }
{ x \in {1, 2, 3, 4}: x > 10 } \* { }
\* check the section on tuples to understand the following syntax
{ <<x, y>> \in (1..4) \X (1..4): y = 3 } \* {<<1, 3>>, <<2, 3>>, <<3, 3>>, <<4, 3>>}
Example in Python: Python conveniently offers us the set comprehension syntax:
>>> S = {1, 2, 3, 4}
>>> { x for x in S if x > 2 }
{3, 4}
>>> { x for x in S if x > 10 }
set()
>>> S2 = {(x, y) for x in S for y in S}
>>> {(x, y) for (x, y) in S2 if y == 3}
{(2, 3), (3, 3), (1, 3), (4, 3)}
Set map
Notation: { e: x \in S }
or { e: x \in S, y \in T }
, or more arguments
LaTeX notation:
Arguments: At least three arguments: a mapping expression, a variable name (or a tuple of names, see Advanced syntax), a set. Additional arguments are variables names and sets, interleaved.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description for the oneargument case:
x
has the typea
, for some typea
,S
has the typeSet(a)
,e
has the typeb
, for some typeb
, the expression
{ e: x \in S }
has the typeSet(b)
.
Effect: We give the semantics for two arguments.
We write it as a sequence of steps to ease understanding.
This operator constructs a new set M
as follows.
For every element e_1
of S
and every element e_2
of T
:
 Bind the element
e_1
to variablex
,  Bind the element
e_2
to variabley
,  Compute the value of
e
under the binding[x > e_1, y > e_2]
,  Insert the value
e
into the setM
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible in the context of e
when an element of S
is bound to x
.
Advanced syntax: Instead of a single variable x
, one can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write { x + y: <<x, y>> \in S }
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The map expression constructs the set of expressions that are computed
under this binding.
Example in TLA+:
{ 2 * x: x \in { 1, 2, 3, 4 } } \* { 2, 4, 6, 8 }
{ x + y: x \in 1..2, y \in 1..2 } \* { 2, 3, 4 }
\* check the section on tuples to understand the following syntax
{ x + y: <<x, y>> \in (1..2) \X (1..2) } \* { 2, 3, 4 }
Example in Python: Python conveniently offers us the set comprehension syntax:
>>> S = frozenset({1, 2, 3, 4})
>>> {2 * x for x in S}
{8, 2, 4, 6}
>>> T = {1, 2}
>>> {x + y for x in T for y in T}
{2, 3, 4}
>>> T2 = {(x, y) for x in T for y in T}
>>> T2
{(1, 1), (1, 2), (2, 1), (2, 2)}
>>> {x + y for (x, y) in T2}
{2, 3, 4}
Powerset
Notation: SUBSET S
LaTeX notation: SUBSET S
Arguments: One argument. If it is not a set, the result is undefined.
Apalache type: Set(a) => Set(Set(a))
, for some type a
.
Effect: This operator computes the set of all subsets of S
.
That is, the set T
the has the following properties:
 If
X \in T
, thenX \subseteq S
.  If
X \subseteq S
, thenX \in T
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator argument. TLC flags a
model checking error, when it discovers that S
is not a set. Apalache
produces a static type error, if the type of S
is not a set type.
Example in TLA+:
SUBSET { 0, 1, 2 }
\* { {}, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2} }
Example in Python: An implementation of SUBSET S
in Python is not trivial.
To appreciate the power of TLA+, see subset.py.
Set flattening
Notation: UNION S
LaTeX notation: UNION S
Warning: Do not confuse UNION S
with S \union T
. These are two
different operators, which unfortunately have similarlooking names.
Arguments: One argument. If it is not a set of sets, the result is undefined.
Apalache type: Set(Set(a)) => Set(a)
, for some type a
.
Effect: Given that S
is a set of sets, this operator computes the set
T
that contains all elements of elements of S
:
 If
X \in S
, thenX \subseteq T
.  If
y \in T
, then there is a setY \in S
that containsy
, that is,y \in Y
.
In particular, UNION
flattens the powerset that is produced by SUBSET
. That
is, (UNION (SUBSET S)) = S
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator argument. TLC flags a
model checking error, when it discovers that S
is not a set of sets.
Apalache produces a static type error, if the type of S
is different from a
set of sets.
Example in TLA+:
UNION { {0, 1}, {1, 2}, {3} }
\* { 0, 1, 2, 3 }
Example in Python: In contrast to SUBSET S
, an implementation of UNION S
in Python is quite simple:
>>> from functools import reduce
>>> s = { frozenset({0, 1}), frozenset({1, 2}), frozenset({3}) }
>>> reduce((lambda x, y: x  y), s, set())
{0, 1, 2, 3}
Set cardinality
Notation: Cardinality(S)
LaTeX notation: Cardinality(S)
Warning: Cardinality(S)
is defined in the module FiniteSets
.
Arguments: One argument. If S
is not a set, or S
is an infinite set,
the result is undefined.
Apalache type: Set(a) => Int
, for some type a
.
Effect: Cardinality(S)
evaluates to the number of (unique) elements in
S
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator argument. TLC flags a
model checking error, when it discovers that S
is not a set, or when it is an
infinite set. Apalache produces a static type error, if the type of S
is
different from a finite set.
Example in TLA+:
EXTENDS FiniteSets
...
Cardinality({ 1, 2, 3 })
\* 3
Example in Python: In Python, we just use the set size:
>>> S = { 1, 2, 3 }
>>> len(S)
3
Set finiteness
Notation: IsFinite(S)
LaTeX notation: IsFinite(S)
Warning: IsFinite(S)
is defined in the module FiniteSets
.
Arguments: One argument. If S
is not a set, the result is undefined.
Apalache type: Set(a) => Bool
, for some type a
.
Effect: IsFinite(S)
evaluates to:
TRUE
, whenS
is a finite set,FALSE
, whenS
is an infinite set.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator argument. TLC flags a
model checking error, when it discovers that S
is not a set. Apalache
produces a static type error, if the type of S
is different from a set.
Example in TLA+:
EXTENDS FiniteSets
...
IsFiniteSet({ 1, 2, 3 }) \* TRUE
IsFiniteSet(BOOLEAN) \* TRUE
IsFiniteSet(Nat) \* FALSE
IsFiniteSet(Int) \* FALSE
IsFiniteSet(STRING) \* FALSE
Example in Python: We can construct only finite sets in Python.
Logic
In this section, you find the operators that – together with Sets – form the foundation of TLA+. It is a bit strange that we call this section "Logic", as the whole language of TLA+ is a logic. However, the operators of this section are often seen in firstorder logic, as opposed to propositional logic (see Booleans).
Note that the special form \E y \in S: x' = y
is often used to express
nondeterminism in TLA+. See Control Flow and Nondeterminism. In this
section, we only consider the deterministic use of the existential quantifier.
Bounded universal quantifier
Notation: \A x \in S: P
LaTeX notation:
Arguments: At least three arguments: a variable name, a set, and an expression. As usual in TLA+, if the second argument is not a set, the result is undefined. You can also use multiple variables and tuples, see Advanced syntax.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
\A x \in S: P
has the typeBool
.
Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:
\A x \in S: P
evaluates toTRUE
, if for every elemente
ofS
, the expressionP
evaluates toTRUE
against the binding[x > e]
. Conversely,
\A x \in S: P
evaluates toFALSE
, if there exists an elemente
ofS
that makes the expressionP
evaluate toFALSE
against the binding[x > e]
.
Importantly, when S = {}
, the expression \A x \in S: P
evaluates to
TRUE
, independently of what is written in P
. Likewise, when {x \in S: P} = {}
, the expression \A x \in S: P
evaluates to TRUE
.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible with the type
of x
that is expected in the predicate P
.
Advanced syntax: Instead of a single variable x
, you can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write \A <<x, y>> \in S: P
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The predicate P
is evaluated against this binding.
Moreover, instead of introducing one variable, one can quantify over several
sets. For instance, you can write: \A x \in S, y \in T: P
. This form is
simply syntax sugar for the form with nested quantifiers: \A x \in S: \A y \in T: P
. Similarly, \A x, y \in S: P
is syntax sugar for
\A x \in S: \A y \in S: P
.
Example in TLA+:
\A x \in {1, 2, 3, 4}:
x > 0
\* TRUE
\A x \in {1, 2, 3, 4}:
x > 2
\* FALSE
\* check the section on tuples to understand the following syntax
\A <<x, y>> \in { 1, 2 } \X { 3, 4 }:
x < y
\* TRUE
Example in Python: Python conveniently offers us a concise syntax:
>>> S = {1, 2, 3, 4}
>>> all(x > 0 for x in S)
True
>>> all(x > 2 for x in S)
False
>>> T2 = {(x, y) for x in [1, 2] for y in [3, 4]}
>>> all(x < y for (x, y) in T2)
True
Bounded existential quantifier
Notation: \E x \in S: P
LaTeX notation:
Arguments: At least three arguments: a variable name, a set, and an expression. As usual in TLA+, if the second argument is not a set, the result is undefined.You can also use multiple variables and tuples, see Advanced syntax.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
\E x \in S: P
has the typeBool
.
Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:
\E x \in S: P
evaluates toTRUE
, if there is an elemente
ofS
that makes the expressionP
evaluate toTRUE
against the binding[x > e]
. Conversely,
\E x \in S: P
evaluates toFALSE
, if for all elementse
ofS
, the expressionP
evaluate toFALSE
against the binding[x > e]
.
Importantly, when S = {}
, the expression \E x \ in S: P
evaluates to
FALSE
, independently of what is written in P
. Likewise, when {x \in S: P} = {}
, the expression \E x \ in S: P
evaluates to FALSE
.
As you probably have noticed, \E x \in S: P
is equivalent to ~(\A x \in S: ~P)
, and \A x \in S: P
is equivalent to ~(\E x \in S: ~P)
. This is called
duality in logic. But take care! If \E x \in S: P
may act as a
nondeterministic assignment, duality does not work anymore! See Control
Flow and Nondeterminism.
Determinism: Deterministic when P
contains no action operators (including
the prime operator '
). For the nondeterministic case, see Control Flow and
Nondeterminism.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible in the context of P
when an element of S
is bound to x
.
Advanced syntax: Instead of a single variable x
, you can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write \E <<x, y>> \in S: P
. In this case, for every
element e
of S
, the variable x
is bound to e[1]
and y
is bound to
e[2]
. The predicate P
is evaluated against this binding.
Moreover, instead of introducing one variable, one can quantify over several
sets. For instance, you can write: \E x \in S, y \in T: P
. This form is
simply syntax sugar for the form with nested quantifiers: \E x \in S: \E y \in T: P
. Similarly, \E x, y \in S: P
is syntax sugar for \E x \in S: \E y \in S: P
.
Example in TLA+:
\E x \in {1, 2, 3, 4}:
x > 0
\* TRUE
\E x \in {1, 2, 3, 4}:
x > 2
\* TRUE
\* check the section on tuples to understand the following syntax
\E <<x, y>> \in { 1, 2 } \X { 3, 4 }:
x < y
\* TRUE
Example in Python: Python conveniently offers us a concise syntax:
>>> S = {1, 2, 3, 4}
>>> any(x > 0 for x in S)
True
>>> any(x > 2 for x in S)
True
>>> T2 = {(x, y) for x in [1, 2] for y in [3, 4]}
>>> any(x < y for (x, y) in T2)
True
Equality
A foundational operator in TLA+
Notation: e_1 = e_2
LaTeX notation:
Arguments: Two arguments.
Apalache type: (a, a) => Bool
, for some type a
.
Effect: This operator evaluates to a Boolean value. It tests the values
of e_1
and e_2
for structural equality. The exact effect depends on the
values of e_1
and e_2
. Let e_1
and e_2
evaluate to the values
v_1
and v_2
. Then e_1 = e_2
evaluates to:

If
v_1
andv_2
are Booleans, thene_1 = e_2
evaluates tov_1 <=> v_2
. 
If
v_1
andv_2
are integers, thene_1 = e_2
evaluates toTRUE
if and only ifv_1
andv_2
are exactly the same integers. 
If
v_1
andv_2
are strings, thene_1 = e_2
evaluates toTRUE
if and only ifv_1
andv_2
are exactly the same strings. 
If
v_1
andv_2
are sets, thene_1 = e_2
evaluates toTRUE
if and only if the following expression evaluates toTRUE
:v_1 \subseteq v_2 /\ v_2 \subseteq v_1

If
v_1
andv_2
are functions, tuples, records, or sequences, thene_1 = e_2
evaluates toTRUE
if and only if the following expression evaluates toTRUE
:DOMAIN v_1 = DOMAIN v_2 /\ \A x \in DOMAIN v_1: v_1[x] = v_2[x]

In other cases,
e_1 = e_2
evaluates toFALSE
if the values have comparable types. 
TLC and Apalache report an error, if the values have incomparable types.
Determinism: Deterministic, unless e_1
has the form x'
, which can be
interpreted as an assignment to the variable x'
. For the nondeterministic
case, see Control Flow and Nondeterminism.
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if e_1
and e_2
evaluate to incomparable values.
Apalache produces a static type error, if the types of e_1
and e_2
do not
match.
Example in TLA+:
FALSE = FALSE \* TRUE
FALSE = TRUE \* FALSE
10 = 20 \* FALSE
15 = 15 \* TRUE
"Hello" = "world" \* FALSE
"Hello" = "hello" \* FALSE
"Bob" = "Bob" \* TRUE
{ 1, 2 } = { 2, 3} \* FALSE
{ 1, 2 } = { 2, 1} \* TRUE
{ 1 } \ { 1 } = { "a" } \ { "a" } \* TRUE in pure TLA+ and TLC,
\* type error in Apalache
{ { 1, 2 } } = { { 1, 2, 2, 2 } } \* TRUE
<<1, "a">> = <<1, "a">> \* TRUE
<<1, "a">> = <<1, "b">> \* FALSE
<<1, FALSE>> = <<2>> \* FALSE in pure TLA+ and TLC,
\* type error in Apalache
<<1, 2>> = <<1, 2, 3>> \* FALSE in pure TLA+ and TLC,
\* FALSE in Apalache, when both values
\* are treated as sequences
[ a > 1, b > 3 ] = [ a > 1, b > 3 ] \* TRUE
[ a > 1, b > 3 ] = [ a > 1 ] \* FALSE
[ x \in 2..2 > x + x ] = [ x \in {2} > 2 * x ] \* TRUE
[ x \in 2..3 > x + x ] = [ x \in {2, 3} > 2 * x ] \* TRUE
Example in Python: The standard data structures also implement
structural equality in Python, though we have to be careful to
use ==
instead of =
:
>>> False == False
True
>>> False == True
False
>>> 10 == 20
False
>>> 15 == 15
True
>>> "Hello" == "world"
False
>>> "Hello" == "hello"
False
>>> "Bob" == "Bob"
True
>>> { 1, 2 } == { 2, 3 }
False
>>> { 1, 2 } == { 2, 1 }
True
>>> { 1 }  { 1 } == { "a" }  { "a" }
True
>>> { frozenset({ 1, 2 }) } == { frozenset({ 1, 2, 2, 2 }) }
True
>>> (1, "a") == (1, "a")
True
>>> (1, "a") == (1, "b")
False
>>> (1, False) == (2, )
False
>>> (1, 2) == (1, 2, 3)
False
>>> { "a": 1, "b": 3 } == { "a": 1, "b": 3 }
True
>>> { "a": 1, "b": 3 } == { "a": 1 }
False
>>> { x: (x + x) for x in { 2 } } == { x: (x * x) for x in { 2 } }
True
>>> { x: (x + x) for x in { 2, 3 } } == { x: 2 * x for x in { 2, 3 } }
True
Inequality
Notation: e_1 /= e_2
or e_1 # e_2
LaTeX notation:
Arguments: Two arguments.
Apalache type: (a, a) => Bool
, for some type a
.
Effect: This operator is syntax sugar for ~(e_1 = e_2)
. Full stop.
Bounded Choice
This operator causes a lot of confusion. Read carefully!
Notation: CHOOSE x \in S: P
LaTeX notation:
Arguments: Three arguments: a variable name, a set, and an expression.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,P
has the typeBool
, the expression
CHOOSE x \in S: P
has the typea
.
Effect: This operator implements a blackbox algorithm that somehow picks
one element from the set {x \in S: P}
. Is it an algorithm? Yes! CHOOSE x \in S: P
is deterministic. When you give it two equal sets and two equivalent
predicates, CHOOSE
produces the same value. Formally, the only known property
of CHOOSE
is as follows (which is slightly more general than what we wrote
above):
{x \in S: P} = {y \in T: Q} =>
(CHOOSE x \in S: P) = (CHOOSE y \in T: Q)
Importantly, when {x \in S: P} = {}
, the expression CHOOSE x \ in S: P
evaluates to an undefined value.
How does CHOOSE
actually work? TLA+ does not fix an algorithm for CHOOSE
by
design. Maybe it returns the first element of the set? Sets are not ordered, so
there is no first element.
Why should you use CHOOSE
? Actually, you should not. Unless you have no other
choice 🎀
There are two common use cases, where the use of CHOOSE
is well justified:

Use case 1: Retrieving the only element of a singleton set. If you know that
Cardinality({x \in S: P}) = 1
, thenCHOOSE x \in S: P
returns the only element of{x \in S: P}
. No magic. For instance, see: Min and Max in FiniteSetsExt. 
Use case 2: Enumerating set elements in a fixed but unknown order. For instance, see: ReduceSet in FiniteSetsExt.
In other cases, we believe that CHOOSE
is bound to do Program synthesis.
So TLC does some form of synthesis by brute force when it has to evaluate
CHOOSE
.
Determinism: Deterministic. Very much deterministic. Don't try to model
nondeterminism with CHOOSE
. For nondeterminism, see:
Control Flow and Nondeterminism.
Apalache picks a set element that satisfies the predicate P
, but it does not
guarantee the repeatability property of CHOOSE
. It does not guarantee
nondeterminism either. Interestingly, this behavior does not really make a
difference for the use cases 1 and 2. If you believe that this causes a problem
in your specification, open an issue...
Errors: Pure TLA+ does not restrict the operator arguments. TLC flags a
model checking error, if S
is infinite. Apalache produces a static type
error, if the type of elements of S
is not compatible with the type of x
as expected in P
.
Example in TLA+:
CHOOSE x \in 1..3: x >= 3
\* 3
CHOOSE x \in 1..3:
\A y \in 1..3: y >= x
\* 1, the minimum
CHOOSE f \in [ 1..10 > BOOLEAN ]:
\E x, y \in DOMAIN f:
f[x] /\ ~f[y]
\* some Boolean function from 1..10 that covers FALSE and TRUE
Example in Python: Python does not have anything similar to CHOOSE
.
The closest possible solution is to sort the filtered set by the string values
and pick the first one (or the last one). So we have introduced a particular
way of implementing CHOOSE, see choose.py:
# A fixed implementation of CHOOSE x \in S: TRUE
# that sorts the set by the string representation and picks the head
def choose(s):
lst = sorted([(str(e), e) for e in s], key=(lambda pair: pair[0]))
(_, e) = lst[0]
return e
if __name__ == "__main__":
s = { 1, 2, 3}
print("CHOOSE {} = {}".format(s, choose(s)))
s2 = { frozenset({1}), frozenset({2}), frozenset({3})}
print("CHOOSE {} = {}".format(s2, choose(s2)))
Unbounded universal quantifier
Notation: \A x: P
LaTeX notation:
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to a Boolean value. It evaluates to TRUE
,
when every element in the logical universe makes the expression P
evaluate to
TRUE
against the binding [x > e]
. More precisely, we have to consider
only the elements that produced a defined result when evaluating P
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe. It requires a firstorder logic solver. This operator may be useful when writing proofs with TLAPS.
Unbounded existential quantifier
Notation: \E x: P
LaTeX notation:
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to a Boolean value. It evaluates to TRUE
,
when at least one element in the logical universe makes the expression P
evaluate to TRUE
against the binding [x > e]
. More precisely, we have to
consider only the elements that produced a defined result when evaluating P
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe. It requires a firstorder logic solver. This operator may be useful when writing proofs with TLAPS.
Unbounded CHOOSE
Notation: CHOOSE x: P
LaTeX notation: CHOOSE x: P
Arguments: At least two arguments: a variable name and an expression.
Effect: This operator evaluates to some value v
in the logical universe
that evaluates P
to TRUE
against the binding [x > v]
.
Neither TLC, nor Apalache support this operator. It is impossible to give operational semantics for this operator, unless we explicitly introduce the universe and introduce a fixed rule for enumerating its elements.
Congratulations! You have reached the bottom of this page. If you want to learn
more about unbounded CHOOSE
, read Section 16.1.2 of Specifying Systems.
Functions
Contributors: @konnov, @shonfeder, @Kukovec, @AlexanderN
Functions are probably the second most used TLA+ data structure after sets. TLA+
functions are not like functions in programming languages. In programming
languages, functions contain code that calls other functions. Although it is
technically possible to use functions when constructing a function in TLA+,
functions are more often used like tables or dictionaries: they are simple maps from a set of inputs to a set of outputs. For instance, in
Twophase commit, the function rmState
stores the transaction state for
each process:
argument  rmState[argument] 

"process1"  "working" 
"process2"  "aborted" 
"process3"  "prepared" 
In the above table, the first column is the value of the function argument, while the second column is the function result. An important property of this table is that no value appears in the first column more than once, so every argument value is assigned at most one result value.
Importantly, every function is defined in terms of the set of arguments over which it is
defined. This set is called the function's domain. There is even a special
operator DOMAIN f
, which returns the domain of a function f
.
In contrast to TLA+ operators, TLA+ functions are proper values, so they can be used as values in more complex data structures.
Construction. Typically, the predicate Init
constructs a function that
maps all elements of its domain to a default value.
In the example below we map the set { "process1", "process2", "process3" }
to the value "working":
Init ==
rmState = [ p \in { "process1", "process2", "process3" } > "working" ]
In general, we can construct a function by giving an expression that shows us how to map every argument to the result:
[ fahrenheit \in Int > (fahrenheit  32) * 5 \div 9 ]
Note that this function effectively defines an infinite table, as the set Int
is infinite. Both TLC and Apalache would give up on a function with an infinite
domain. (Though in the above example, it is obvious that we could treat the
function symbolically, without enumerating all of its elements.)
Another way to construct a function is to nondeterministically pick one
from a set of functions by using the function set constructor, >
. E.g.:
Init ==
\E f \in [ { "process1", "process2", "process3" } >
{ "working", "prepared", "committed", "aborted" } ]:
rmState = f
In the above example we are not talking about one function that is somehow
initialized "by default". Rather, we say that rmState
can be set to an
arbitrary function that receives arguments from the set { "process1", "process2", "process3" }
and returns values that belong to the set { "working", "prepared", "committed", "aborted" }
. As a result, TLC has to
enumerate all possible functions that match this constraint. On the contrary,
Apalache introduces one instance of a function and restricts it with the
symbolic constraints. So it efficiently considers all possible functions
without enumerating them. However, this trick only works with existential
quantifiers. If you use a universal quantifier over a set of functions,
both TLC and Apalache unfold this set.
Immutability. As you can see, TLA+ functions are similar to dictionaries in Python and maps in Java rather than to normal functions in programming languages. However, TLA+ functions are immutable. Hence, they are even closer to immutable maps in Scala. As in the case of sets, you do not need to define hash or equality, in order to use functions.
If you want to update a function, you have to produce another function and
describe how it is different from the original function. Luckily, TLA+ provides
you with operators for describing these updates in a compact way: By using the
function constructor (above) along with EXCEPT
. For instance, to produce a
new function from rmState
, we write the following:
[ rmState EXCEPT !["process3"] = "committed" ]
This new function is like rmState
, except that it returns "committed"
on the argument "process3"
:
"process1", "working"
"process2", "aborted"
"process3", "committed"
Importantly, you cannot extend the function domain by using EXCEPT
.
For instance, the following expression produces the function that is
equivalent to rmState
:
[ rmState EXCEPT !["process10"] = "working" ]
Types. In pure TLA+, functions are free to mix values of different types in their domains. See the example below:
[ x \in { 0, "FALSE", FALSE, 1, "TRUE", TRUE } >
IF x \in { 1, "TRUE", TRUE}
THEN TRUE
ELSE FALSE
]
TLA+ functions are also free to return any kinds of values:
[ x \in { 0, "FALSE", FALSE, 1, "TRUE", TRUE } >
CASE x = 0 > 1
[] x = 1 > 0
[] x = "FALSE" > "TRUE"
[] x = "TRUE" > "FALSE"
[] x = FALSE > TRUE
OTHER > FALSE
]
As in the case of sets, TLC restricts function domains to comparable values. See Section 14.7.2 of Specifying Systems. So, TLC rejects the two examples that are given above.
However, functions in TLC are free to return different kinds of values:
[ x \in { 1, 2 } >
IF x = 1 THEN FALSE ELSE 3 ]
This is why, in pure TLA+ and TLC, records, tuples, and sequences are just functions over particular domains (finite sets of strings and finite sets of integers).
Apalache enforces stricter types. It has designated types for all four
data structures: general functions, records, tuples, and sequences.
Moreover, all elements of the function domain must have the same type.
The same is true for the codomain. That is, general functions have the
type a > b
for some types a
and b
. This is enforced
by the type checker.
In this sense, the type restrictions of Apalache are similar to those for the generic collections of Java and Scala. As a result, the type checker in Apalache rejects the three above examples.
TLA+ functions and Python dictionaries. As we mentioned before, TLA+
functions are similar to maps and dictionaries in programming languages. To
demonstrate this similarity, let us compare TLA+ functions with Python
dictionaries. Consider a TLA+ function price
that is defined as follows:
[ meal \in { "Schnitzel", "Gulash", "Cordon bleu" } >
CASE meal = "Schnitzel" > 18
[] meal = "Gulash" > 11
[] meal = "Cordon bleu" > 12
]
If we had to define a similar dictionary in Python, we would normally introduce a Python dictionary like follows:
py_price = { "Schnitzel": 18, "Gulash": 11, "Cordon bleu": 12 }
As long as we are using the variable py_price
to access the dictionary, our
approach works. For instance, we can type the following in the python shell:
# similar to DOMAIN price in TLA+
py_price.keys()
In the above example, we used py_price.keys()
, which produces a view of the
mutable dictionary's keys. In TLA+, DOMAIN
returns a set. If we want to
faithfully model the effect of DOMAIN
, then we have to produce an immutable
set. We use
frozenset
,
which is a less famous cousin of the python set
. A frozen set can be
inserted into another set, in contrast to the standard (mutable) set.
>>> py_price = { "Schnitzel": 18, "Gulash": 11, "Cordon bleu": 12 }
>>> frozenset(py_price.keys()) == frozenset({'Schnitzel', 'Gulash', 'Cordon bleu'})
True
We can also apply our python dictionary similar to the TLA+ function price
:
>>> # similar to price["Schnitzel"] in TLA+
>>> py_price["Schnitzel"]
18
However, there is a catch! What if you like to put the function price
in a
set? In TLA+, this is easy: Simply construct the singleton set that contains
the function price
.
# TLA+: wrapping a function with a set
{ price }
Unfortunately, this does not work as easy in Python:
>>> py_price = { "Schnitzel": 18, "Gulash": 11, "Cordon bleu": 12 }
>>> # python expects hashable and immutable data structures inside sets
>>> frozenset({py_price})
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: unhashable type: 'dict'
Of course, this is an implementation detail of Python and it has nothing to do with TLA+. This example probably demonstrates that the builtin primitives of TLA+ are more powerful than the standard primitives of many programming languages (see this discussion).
Alternatively, we could represent a TLA+ function in Python as a set
of pairs (key, value)
and implement TLA+ function operators over such a
set. Surely, this implementation would be inefficient, but this is not
an issue for a specification language such as TLA+. For instance:
>>> py_price = { "Schnitzel": 18, "Gulash": 11, "Cordon bleu": 12 }
>>> { tuple(py_price.items()) }
{(('Schnitzel', 18), ('Gulash', 11), ('Cordon bleu', 12))}
If we try to implement TLA+like operators over this data structure, things will get complicated very quickly. For this reason, we are just using mutable dictionaries in the Python examples in the rest of this text.
Operators
Function constructor
Notation: [ x \in S > e ]
or [ x \in S, y \in T > e ]
, or more
arguments
LaTeX notation:
Arguments: At least three arguments: a variable name (or a tuple of names, see Advanced syntax), a set, and a mapping expression. Instead of one variable and one set, you can use multiple variables and multiple sets.
Apalache type: The formal type of this operator is a bit complex. Hence, we give an informal description:
x
has the typea
, for some typea
,S
has the typeSet(a)
,e
has the typeb
, for some typeb
, the expression
[ x \in S > e ]
has the typea > b
.
Effect: We give the semantics for one argument. We write a sequence of
steps to ease the understanding. This operator constructs a function f
over
the domain S
as follows. For every element elem
of S
, do the following:
 Bind the element
elem
to variablex
,  Compute the value of
e
under the binding[x > elem]
and store it in a temporary variable calledresult
.  Set
f[elem]
toresult
.
Of course, the semantics of the function constructor in Specifying Systems does not require us to compute the function at all. We believe that our description helps you to see that there is a way to compute this data structure, though in a very straightforward and inefficient way.
If the function constructor introduces multiple variables, then the constructed function maps a tuple to a value. See Example.
Determinism: Deterministic.
Errors: Pure TLA+ does not restrict the function domain and the mapping expression. They can be any combination of TLA+ values: Booleans, integers, strings, sets, functions, etc.
TLC accepts function domains of comparable values. For instance, two integers are comparable, but an integer and a set are not comparable. See Section 14.7.2 of Specifying Systems.
Apalache goes further: It requires the function domain to be welltyped (as a
set), and it requires the mapping expression e
to be welltyped. If this
is not the case, the type checker flags an error.
Advanced syntax: Instead of a single variable x
, one can use the tuple
syntax to unpack variables from a Cartesian product, see Tuples.
For instance, one can write [ <<x, y>> \in S > x + y ]
. In this case, for
every element e
of S
, the variable x
is bound to e[1]
and y
is bound
to e[2]
. The function constructor maps the tuples from S
to the values
that are computed under such a binding.
Example in TLA+:
[ x \in 1..3 > 2 * x ] \* a function that maps 1, 2, 3 to 2, 4, 6
[ x, y \in 1..3 > x * y ]
\* a function that maps <<1, 1>>, <<1, 2>>, <<1, 3>>, ..., <<2, 3>>, <<3, 3>>
\* to 1, 2, 3, ..., 6, 9
[ <<x, y>> \in (1..3) \X (4..6) > x + y ]
\* a function that maps <<1, 4>>, <<1, 5>>, <<1, 6>>, ..., <<2, 6>>, <<3, 6>>
\* to 5, 6, 7, ..., 8, 9
[ n \in 1..3 >
[ i \in 1..n > n + i ]]
\* a function that maps a number n from 1 to 3
\* to a function from 1..n to n + i. Like an array of arrays.
Example in Python:
In the following code, we write range(m, n)
instead of frozenset(range(m, n))
to simplify the presentation and produce idiomatic Python code. In the
general case, we have to iterate over a set, as the type and structure of the
function domain is not known in advance.
>>> # TLA: [ x \in 1..3 > 2 * x ]
>>> {x: 2 * x for x in range(1, 4)}
{1: 2, 2: 4, 3: 6}
>>> # TLA: [ x, y \in 1..3 > x * y ]
>>> {(x, y): x * y for x in range(1, 4) for y in range(1, 4)}
{(1, 1): 1, (1, 2): 2, (1, 3): 3, (2, 1): 2, (2, 2): 4, (2, 3): 6, (3, 1): 3, (3, 2): 6, (3, 3): 9}
>>> # TLA: [ <<x, y>> \in (1..3) \X (4..6) > x + y ]
>>> xy = {(x, y) for x in range(1, 4) for y in range(4, 7)}
>>> {(x, y): x + y for (x, y) in xy}
{(2, 4): 6, (3, 4): 7, (1, 5): 6, (1, 4): 5, (2, 6): 8, (3, 6): 9, (1, 6): 7, (2, 5): 7, (3, 5): 8}
>>> # TLA: [ n \in 1..3 >
>>> # [ i \in 1..n > n + i ]]
>>> {
... n: {i: n + i for i in range(1, n + 1)}
... for n in range(1, 4)
... }
{1: {1: 2}, 2: {1: 3, 2: 4}, 3: {1: 4, 2: 5, 3: 6}}
Function set constructor
Notation: [ S > T ]
LaTeX notation:
Arguments: Two arguments. Both have to be sets. Otherwise, the result is undefined.
Apalache type: (Set(a), Set(b)) => Set(a > b)
, for some types a
and b
.
Effect: This operator constructs the set of all possible functions that
have S
as their domain, and for each argument x \in S
return a value y \in T
.
Note that if one of the sets is infinite, then the set [S > T]
is infinite
too. TLC flags an error, if S
or T
are infinite. Apalache flags an error,
if S
is infinite, but when it does not have to explicitly construct [S > T]
, it may accept infinite T
. For instance:
\E f \in [ 1..3 > 4..6]:
...
Determinism: Deterministic.
Errors: In pure TLA+, if S
and T
are not sets, then [S > T]
is undefined. If either S
or T
is not a set, TLC flags a model checking error.
Apalache flags a static type error.
Example in TLA+:
[ 1..3 > 1..100 ]
\* the set of functions that map 1, 2, 3 to values from 1 to 100
[ Int > BOOLEAN ]
\* The infinite set of functions that map every integer to a Boolean.
\* Error in TLC.
Example in Python: We do not give here the code that enumerates all
functions. It should be similar in spirit to subset.py,
but it should enumerate strings over the alphabet of 0..(Cardinality(T)  1)
values, rather than over the alphabet of 2 values.
Function application
Notation: f[e]
or f[e_1, ..., e_n]
LaTeX notation: f[e]
or f[e_1, ..., e_n]
Arguments: At least two arguments. The first one should be a function,
the other arguments are the arguments to the function. Several arguments
are treated as a tuple. For instance, f[e_1, ..., e_n]
is shorthand for
f[<<e_1, ..., e_n>>]
.
Apalache type: In the singleindex case, the type is
((a > b), a) => b
, for some types a
and b
. In the multiindex case,
the type is ((<<a_1, ..., a_n>> > b), a_1, ..., a_n) => b
.
Effect: This operator is evaluated as follows:
 If
e \in DOMAIN f
, thenf[e]
evaluates to the value that functionf
associates with the value ofe
.  If
e \notin DOMAIN f
, then the value is undefined.
Determinism: Deterministic.
Errors: When e \notin DOMAIN f
, TLC flags a model checking error.
When e
has a type incompatible with the type of DOMAIN f
, Apalache flags
a type error. When e \notin DOMAIN f
, Apalache assigns some typecompatible
value to f[e]
, but does not report any error. This is not a bug in Apalache,
but a feature of the SMT encoding. Usually, an illegal access surfaces
somewhere, when checking a specification. If you want to detect an access
outside of the function domain, instrument your code with an additional state
variable.
Example in TLA+:
[x \in 1..10 > x * x][5] \* 25
[x \in 1..3, y \in 1..3 > x * y][2, 2]
\* Result = 4. Accessing a twodimensional matrix by a pair
[ n \in 1..3 >
[ i \in 1..n > n + i ]][3][2]
\* The first access returns a function, the second access returns 5.
[x \in 1..10 > x * x][100] \* model checking error in TLC,
\* Apalache produces some value
Example in Python:
In the following code, we write range(m, n)
instead of frozenset(range(m, n))
to simplify the presentation and produce idiomatic Python code. In the
general case, we have to iterate over a set, as the type and structure of the
function domain is not known in advance.
>>> # TLA: [x \in 1..10 > x * x][5]
>>> {x: x * x for x in range(1, 11)}[5]
25
>>> # TLA: [x, y \in 1..3 > x * y][2, 2]
>>> {(x, y): x * y for x in range(1, 4) for y in range(1, 4)}[(2, 2)]
4
>>> # TLA: [ n \in 1..3 > [ i \in 1..n > n + i ]][3][2]
>>> {n: {i: n + i for i in range(1, n + 1)} for n in range(1, 4)}[3][2]
5
Function replacement
Notation: [f EXCEPT ![a_1] = e_1, ..., ![a_n] = e_n]
LaTeX notation: [f EXCEPT ![a_1] = e_1, ..., ![a_n] = e_n]
Arguments: At least three arguments. The first one should be a function, the other arguments are interleaved pairs of argument expressions and value expressions.
Apalache type: In the case of a singlepoint update, the type is simple:
(a > b, a, b) => (a > b)
, for some types a
and b
. In the general case,
the type is: (a > b, a, b, ..., a, b) => (a > b)
.
Effect: This operator evaluates to a new function g
that is constructed
as follows:
 Set the domain of
g
toDOMAIN f
.  For every element
b \in DOMAIN f
, do: If
b = a_i
for somei \in 1..n
, then setg[b]
toe_i
.  If
b \notin { a_1, ..., a_n }
, then setg[b]
tof[b]
.
 If
Importantly, g
is a new function: the function f
is not modified!
Determinism: Deterministic.
Errors: When a_i \notin DOMAIN f
for some i \in 1..n
,
TLC flags a model checking error.
When a_1, ..., a_n
are not typecompatible with the type of DOMAIN f
,
Apalache flags a type error. When a_i \notin DOMAIN f
, Apalache ignores this
argument. This is consistent with the semantics of TLA+ in Specifying Systems.
Advanced syntax: There are three extensions to the basic syntax.
Extension 1. If the domain elements of a function f
are tuples, then, similar to
function application, the expressions a_1, ..., a_n
can be written without
the tuple braces <<...>>
. For example:
[ f EXCEPT ![1, 2] = e ]
In the above example, the element f[<<1, 2>>]
is replaced with e
.
As you can see, this is just syntax sugar.
Extension 2. The operator EXCEPT
introduces an implicit alias @
that refers to the element f[a_i]
that is going to be replaced:
[ f EXCEPT ![1] = @ + 1, ![2] = @ + 3 ]
In the above example, the element f[1]
is replaced with f[1] + 1
, whereas
the element f[2]
is replaced with f[2] + 3
.
This is also syntax sugar.
Extension 3. The advanced syntax of EXCEPT
allows for chained replacements.
For example:
[ f EXCEPT ![a_1][a_2]...[a_n] = e ]
This is syntax sugar for:
[ f EXCEPT ![a_1] =
[ @ EXCEPT ![a_2] =
...
[ @ EXCEPT ![a_n] = e ]]]
Example in TLA+:
LET f1 == [ p \in 1..3 > "working" ] IN
[ f1 EXCEPT ![2] = "aborted" ]
\* a new function that maps: 1 to "working", 2 to "aborted", 3 to "working"
LET f2 == [x \in 1..3, y \in 1..3 > x * y] IN
[ f2 EXCEPT ![1, 1] = 0 ]
\* a new function that maps:
\* <<1, 1>> to 0, and <<x, y>> to x * y when `x /= 0` or `y /= 0`
LET f3 == [ n \in 1..3 > [ i \in 1..n > n + i ]] IN
[ f3 EXCEPT ![2][2] = 100 ]
\* a new function that maps:
\* 1 to the function that maps: 1 to 2
\* 2 to the function that maps: 1 to 3, 2 to 100
\* 3 to the function that maps: 1 to 4, 2 to 5, 3 to 6
Example in Python:
In the following code, we write range(m, n)
instead of frozenset(range(m, n))
to simplify the presentation and produce idiomatic Python code. In the
general case, we have to iterate over a set, as the type and structure of the
function domain is not known in advance. Additionally, given a Python
dictionary f
, we write f.items()
to quickly iterate over the pairs of keys
and values. Had we wanted to follow the TLA+ semantics more precisely, we would
have to enumerate over the keys in the function domain and apply the function to
each key, in order to obtain the value that is associated with the key. This
code would be less efficient than the idiomatic Python code.
>>> # TLA: LET f1 == [ p \in 1..3 > "working" ] IN
>>> f1 = {i: "working" for i in range(1, 4)}
>>> f1
{1: 'working', 2: 'working', 3: 'working'}
>>> # TLA: [ f1 EXCEPT ![2] = "aborted" ]
>>> g1 = {i: status if i != 2 else "aborted" for i, status in f1.items()}
>>> g1
{1: 'working', 2: 'aborted', 3: 'working'}
>>> # TLA: LET f2 == [x, y \in 1..3 > x * y] IN
>>> f2 = {(x, y): x * y for x in range(1, 4) for y in range(1, 4)}
>>> # TLA: [ f2 EXCEPT ![1, 1] = 0
>>> g2 = {k: v if k != (1, 1) else 0 for k, v in f2.items()}
>>> g2
{(1, 1): 0, (1, 2): 2, (1, 3): 3, (2, 1): 2, (2, 2): 4, (2, 3): 6, (3, 1): 3, (3, 2): 6, (3, 3): 9}
>>> # TLA: [ n \in 1..3 > [ i \in 1..n > n + i ]]
>>> f3 = {n: {i: n + i for i in range(1, n + 1)} for n in range(4)}
>>> # TLA: [ f3 EXCEPT ![2][2] = 100 ]
>>> g3 = f3.copy()
>>> g3[2][2] = 100
>>> g3
{0: {}, 1: {1: 2}, 2: {1: 3, 2: 100}, 3: {1: 4, 2: 5, 3: 6}}
Function domain
Notation: DOMAIN f
LaTeX notation: DOMAIN f
Arguments: One argument, which should be a function (respectively, a record, tuple, sequence).
Apalache type: (a > b) => Set(a)
.
Effect: DOMAIN f
returns the set of values, on which the function
has been defined, see: Function constructor and Function set constructor.
Determinism: Deterministic.
Errors: In pure TLA+, the result is undefined, if f
is not a function
(respectively, a record, tuple, or sequence). TLC flags a model checking error
if f
is a value that does not have a domain. Apalache flags a type checking
error.
Example in TLA+:
LET f == [ x \in 1..3 > 2 * x ] IN
DOMAIN f \* { 1, 2, 3 }
Example in Python:
In the following code, we write range(m, n)
instead of frozenset(range(m, n))
to simplify the presentation and produce idiomatic Python code. In the
general case, we have to iterate over a set, as the type and structure of the
function domain is not known in advance.
>>> f = {x: 2 * x for x in range(1, 4)}
>>> f.keys()
dict_keys([1, 2, 3])
In the above code, we write f.keys()
to obtain an iterable over the
dictionary keys, which can be used in a further python code. In a more
principled approach that follows the semantics of TLA+, we would have to
produce a set, that is to write:
frozenset(f.keys())
Records
Records in TLA+ are special kinds of functions that have the following properties:
 The domain of a record contains only strings.
 The domain of a record is finite.
That is it in pure TLA+. Essentially, TLA+ is following the ducktyping principle for records: Any function over strings can be also treated as a record, and vice versa, a record is also a function. So you can use all function operators on records too.
Construction. TLA+ provides you with a convenient syntax for constructing
records. For instance, the following example shows how to construct a record
that has two fields: Field "a"
is assigned value 2
, and field "b"
is
assigned value TRUE
.
[ a > 2, b > TRUE ]
Similar to the function set [S > T]
, there is a record set constructor:
[ name: { "Alice", "Bob" }, year_of_birth: 1900..2000 ]
The expression in the above example constructs a set of records that have: the
name
field set to either "Alice" or "Bob", and the year_of_birth
field set
to an integer from 1900 to 2000.
Application. TLA+ provides you with a shorthand operator for accessing a record field by following Cstyle structmember notation. For example:
r.myField
This is essentially syntax sugar for r["myField"]
.
Immutability. As records are special kinds of functions, records are immutable.
Types. In contrast to pure TLA+ and TLC, the Apalache model checker distinguishes between general functions and records. When Apalache processes a record constructor, it assigns the record type to the result. This record type carries the information about the names of the record fields and their types. Similarly, Apalache assigns the type of a set of records, when it processes a record set constructor. See the Apalache ADR002 on types.
Owing to the type information, records are translated into SMT more efficiently by Apalache than the general functions.
Every record is assigned a type in Apalache. For instance, the record
[name > "A", a > 3]
has the type { name: Str, a: Int }
. In contrast to
TLC, the type checker statically flags an error, if a spec is trying to access
a nonexistent field. Consider the following example:
 MODULE TestUnsafeRecord 
\* the record in R has the type { name: Str, a: Int}
R == [name > "A", a > 3]
\* the type checker will report a type error in UnsafeAccess
UnsafeAccess == R.b
===============================================================================
If we run the type checker, it will immediately find unsafe record access:
$ apalachemc typecheck TestUnsafeRecord.tla
...
[TestUnsafeRecord.tla:6:176:19]: Cannot apply R() to the argument "b" in R()["b"]
Sometimes, record types can get tricky, when operators in a spec only have
partial type information. For example, consider operator GetX
:
 MODULE TestGetX 
GetX(r) == r.x
===============================================================================
If we run the type checker, it will complain about not being able to infer
the type of r
:
$ apalachemc typecheck TestGetX.tla
...
[TestGetX.tla:2:122:14]: Cannot apply r to the argument "x" in r["x"].
The reason is simple: The type checker could not decide, whether r
was a
record or a function. Even if we knew that r
was a record, what type should
it have? Luckily, the Apalache type checker supports Row polymorphism.
Hence, we can specify the type of r
as follows:
 MODULE TestGetXWithRows 
\* @type: { x: a, b } => a;
GetX(r) == r.x
===============================================================================
In the type annotation, we are saying that r
is a record that has the field
x
of some type a
(which we don't know), and the rest of the record does not
matter. This matches our intuition about the behavior of GetX
. This time the
type checker does not complain:
$ apalachemc typecheck TestGetXWithRows.tla
...
Type checker [OK]
In untyped TLA+, it is common to mix records of different shapes into sets. For
instance, see how the variable msgs
is updated in Paxos. It is not
possible to do so with records in Apalache. To address this pattern, Apalache
supports Variants.
Operators
In the Python examples, we are using the package frozendict, to produce an immutable dictionary.
Record constructor
Notation: [ field_1 > e_1, ..., field_n > e_n]
LaTeX notation:
Arguments: An even number of arguments: field names and field values, interleaved. At least one field is expected. Note that field names are TLA+ identifiers, not strings.
Apalache type: (a_1, ..., a_n) => { field_1: a_1, ..., field_n: a_n }
, for
some types a_1, ..., a_n
.
Effect: The record constructor returns a function r
that is constructed
as follows:
 set
DOMAIN r
to{ field_1, ..., field_n }
,  set
r[field_i]
to the value ofe_i
fori \in 1..n
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
[ name > "Printer", port > 631 ]
\* A record that has two fields:
\* field "name" that is equal to "Printer", and field "port" that is equal to 631.
Example in Python:
>>> { "name": "Printer", "port": 631 }
{'name': 'Printer', 'port': 631}
Record set constructor
Notation: [ field_1: S_1, ..., field_n: S_n]
LaTeX notation:
Arguments: An even number of arguments: field names and field values, interleaved. At least one field is expected. Note that field names are TLA+ identifiers, not strings.
Apalache type: (Set(a_1), ..., Set(a_n)) => Set({ field_1: a_1, ..., field_n: a_n })
, for some types a_1, ..., a_n
.
Effect: The record set constructor [ field_1: S_1, ..., field_n: S_n]
is syntax sugar for the set comprehension:
{ [ field_1 > e_1, ..., field_n > e_n]: e_1 \in S_1, ..., e_n \in S_n }
Determinism: Deterministic.
Errors: The arguments S_1, ..., S_n
must be sets. If they are not sets,
the result is undefined in pure TLA+. TLC raises a model checking error. Apalache
flags a static type error.
TLC raises a model checking error, whenever one of the sets S_1, ..., S_n
is
infinite. Apalache can handle infinite records sets in some cases, when one record
is picked with \E r \in [ field_1: S_1, ..., field_n: S_n]
.
Example in TLA+:
[ name: { "A", "B", "C" }, port: 1..65535 ]
\* A set of records. Each has two fields:
\* field "name" that has the value from the set { "A", "B", "C" }, and
\* field "port" that has the value from the set 1..65535.
Example in Python: TLA+ functions are immutable, so we are using frozendict:
frozenset({ frozendict({ "name": n, "port": p })
for n in { "A", "B", "C" } for p in range(1, 65535 + 1) })
Access by field name
Notation: r.field_i
LaTeX notation: r.field_i
Arguments: Two arguments: a record and a field name (as an identifier).
Apalache type: { field_i: a, b } => b
, for some types a
and b
(technically, b
is a row that captures the other fields).
Note that r.field_i
is just a syntax sugar for r["field_i"]
in TLA+.
Hence, if the Apalache type checker cannot choose between r
being a record or
a function, the type checker fails with a type error. In this case, you have to
typeannotate the definition that contains r
.
Effect: As records are also functions, this operator works as r["field_i"]
.
Apalache treats records as values of a record type. In comparison to the
general function application r["field"]
, the operator r.field
is handled
much more efficiently in Apalache. Due to the use of types, Apalache can
extract the respective field when translating the access expression into SMT.
Determinism: Deterministic.
Example in TLA+:
LET r == [ name > "Printer", port > 631 ] IN
r.name \* "Printer"
Example in Python:
>>> r = { "name": "Printer", "port": 631 }
>>> r["name"]
'Printer'
Tuples
Tuples in TLA+ are special kinds of functions that satisfy one of the following properties:
 The domain is either empty, that is,
{}
, or  The domain is
1..n
for somen > 0
.
That is right. You can construct the empty tuple <<>>
in TLA+ as well as a
singleelement tuple, e.g., <<1>>
. You can also construct pairs, triples, an
so on, e.g., <<1, TRUE>>
, <<"Hello", "world", 2020>>
. If you think that
empty tuples do not make sense: In TLA+, there is no difference between tuples
and sequences. Again, it is duck typing: Any function with
the domain 1..n
can be also treated as a tuple (or a sequence!), and vice
versa, tuples and sequences are also functions. So you can use all function
operators on tuples.
Importantly, the domain of a nonempty tuple is 1..n
for some n > 0
. So tuples never
have a 0th element. For instance, <<1, 2>>[1]
gives us 1, whereas <<1, 2>>[2]
gives us 2.
Construction. TLA+ provides you with a convenient syntax for constructing
tuples. For instance, the following example shows how to construct a tuple
that has two fields: Field 1 is assigned value 2
, and field 2 is
assigned value TRUE
.
<<2, TRUE>>
There is a tuple set constructor, which is wellknown as Cartesian product:
{ "Alice", "Bob" } \X (1900..2000)
The expression in the above example constructs a set of tuples <<n, y>>
: the
first field n
is set to either "Alice" or "Bob", and the second field y
is set
to an integer from 1900 to 2000.
Application. Simply use function application, e.g., t[2]
.
Immutability. As tuples are special kinds of functions, tuples are immutable.
Types. In contrast to pure TLA+ and TLC, the Apalache model checker
distinguishes between general functions, tuples, and sequences. They all have
different types. Essentially, a function has the type A > B
that
restricts the arguments and results as follows: the arguments have the type
A
and the results have the type B
. A sequence has the type
Seq(C)
, which restricts the sequence elements to have the same type C
. In
contrast, tuples have more finegrained types in Apalache: <<T_1>>
, <<T_1, T_2>>
, <<T_1, T_2, T_3>>
and so on. As a result, different tuple fields are
allowed to carry elements of different types, whereas functions and sequences
are not allowed to do that. See the Apalache ADR002 on types for details.
As tuples are also sequences in TLA+, this poses a challenge for the Apalache
type checker. For instance, it can immediately figure out that <<1, "Foo">>
is a tuple, as Apalache does not allow sequences to carry elements of different
types. However, there is no way to say, whether <<1, 2, 3>>
should be treated
as a tuple or a sequence. Usually, this problem is resolved by annotating the
type of a variable or the type of a user operator. See HOWTO write type
annotations.
Owing to the type information, tuples are translated into SMT much more efficiently by Apalache than the general functions and sequences!
Operators
In the Python examples, we are using the package frozendict, to produce an immutable dictionary.
Tuple/Sequence constructor
Notation: <<e_1, ..., e_n>>
LaTeX notation:
Arguments: An arbitrary number of arguments.
Apalache type: This operator is overloaded. There are two potential types:
 A tuple constructor:
(a_1, ..., a_n) => <<a_1, ..., a_n>>
, for some typesa_1, ..., a_n
.  A sequence constructor:
(a, ..., a) => Seq(a)
, for some typea
.
That is why the Apalache type checker is sometimes asking you to add annotations, in order to resolve this ambiguity.
Effect: The tuple constructor returns a function t
that is constructed
as follows:
 set
DOMAIN t
to1..n
,  set
r[i]
to the value ofe_i
fori \in 1..n
.
In Apalache, this constructor may be used to construct either a tuple, or a sequence. To distinguish between them, you will sometimes need a [type annotation].
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
<<"Printer", 631>>
Example in Python: Python provides us with the syntax for constructing tuples, which are indexed with 0!. If we want to stick to the principle "tuples are functions", we have to use a dictionary.
>>> ("Printer", 631) # the pythonic way, introducing fields 0 and 1
('Printer', 631)
>>> { 1: "Printer", 2: 631 } # the "tuplesarefunctions" way
{1: 'Printer', 2: 631}
Cartesian product
Notation: S_1 \X ... \X S_n
(or S_1 \times ... \times S_n
)
LaTeX notation:
Arguments: At least two arguments. All of them should be sets.
Apalache type: (Set(a_1), ..., Set(a_n)) => Set(<<a_1, ..., a_n>>)
,
for some types a_1, ..., a_n
.
Effect: The Cartesian product S_1 \X ... \X S_n
is syntax sugar for the set comprehension:
{ << e_1, ..., e_n >> : e_1 \in S_1, ..., e_n \in S_n }
Determinism: Deterministic.
Errors: The arguments S_1, ..., S_n
must be sets. If they are not sets,
the result is undefined in pure TLA+. TLC raises a model checking error. Apalache
flags a static type error.
TLC raises a model checking error, whenever one of the sets S_1, ..., S_n
is
infinite. Apalache can handle infinite sets in some cases, e.g., when one tuple
is picked with \E t \in S_1 \X S_2
.
Example in TLA+:
{ "A", "B", "C" } \X (1..65535)
\* A set of tuples. Each tuple has two fields:
\*  field 1 has the value from the set { "A", "B", "C" }, and
\*  field 2 has the value from the set 1..65535.
Example in Python: TLA+ functions are immutable, so we are using frozendict:
# the pythonic way: a set of python tuples (indexed with 0, 1, ...)
frozenset({ (n, p)
for n in { "A", "B", "C" } for p in range(1, 65535 + 1) })
# the TLA+ way
frozenset({ frozendict({ 1: n, 2: p })
for n in { "A", "B", "C" } for p in range(1, 65535 + 1) })
Function application
As tuples are functions, you can access tuple elements by function
application, e.g., tup[2]
. However, in the case of a
tuple, the type of the function application will be: (<<a_1, ..., a_i, ..., a_n>>, Int) => a_i
, for some types a_1, ..., a_n
.
Sequences
On the surface, TLA+ sequences are very much like lists in your programming language of choice. If you are writing code in Java, Python, Lisp, C++, Scala, you will be tempted to use sequences in TLA+ too. This is simply due to the fact that arrays, vectors, and lists are the most efficient collections in programming languages (for many tasks, but not all of them). But TLA+ is not about efficient compilation of your data structures! Many algorithms can be expressed in a much nicer way with sets and functions. In general, use sequences when you really need them.
In pure TLA+, sequences are just tuples. As a tuple, a sequence is
a function of the domain 1..n
for some n >= 0
(the domain may be empty).
The ducktyping principle applies to sequences too: Any function with the domain 1..n
can also be
treated as a sequence (or a tuple), and vice versa, tuples and sequences are
also functions. So you can use all function and tuple operators on sequences.
Importantly, the domain of a sequence is 1..n
for some n >= 0
. So the
indices in a sequence start with 1, not 0. For instance, <<1, 2>>[1]
gives us
1, whereas <<1, 2>>[2]
gives us 2.
The operators on sequences are defined in the standard module Sequences
. To
use it, write the EXTENDS
clause in the first lines of your module. Like
this:
 MODULE MyLists ====
EXTENDS Sequences
...
==============================
Construction. Sequences are constructed exactly as tuples in TLA+:
<<2, 4, 8>>
Sometimes, you have to talk about all possible sequences. The operator
Seq(S)
constructs the set of all (finite) sequences that draw elements
from the set S
. For instance, <<1, 2, 2, 1>> \in Seq({1, 2, 3})
.
Note that Seq(S)
is an infinite set. To use it with TLC, you often have
to override this operator, see Specifying Systems, page 237.
Application. Simply use function application, e.g., s[2]
.
Immutability. As sequences are special kinds of functions, sequences are immutable.
Sequence operators. The module Sequences
provides you with convenient
operators on sequences:
 Add to end:
Append(s, e)
 First and rest:
Head(s)
andTail(s)
 Length:
Len(s)
 Concatenation:
s \o t
 Subsequence:
SubSeq(s, i, k)
 Sequence filter:
SelectSeq(s, Test)
See the detailed description in Operators.
Types. In contrast to pure TLA+ and TLC, the Apalache model checker
distinguishes between general functions, tuples, and sequences. They all have
different types. Essentially, a function has the type T_1 > T_2
that
restricts the arguments and results as follows: the arguments have the type
T_1
and the results have the type T_2
. A sequence has the type Seq(T_3)
,
which restricts the sequence elements to have the same type T_3
.
As sequences are also tuples in TLA+, this poses a challenge for the Apalache
type checker. For instance, it can immediately figure out that <<1, "Foo">>
is a tuple, as Apalache does not allow sequences to carry elements of different
types. However, there is no way to say, whether <<1, 2, 3>>
should be treated
as a tuple or a sequence. Usually, this problem is resolved by annotating the
type of a variable or the type of a user operator. See HOWTO write type
annotations.
The current SMT encoding of sequences in Apalache is not optimized, so operations on sequences are often significantly slower than operations on sets.
Operators
Tuple/Sequence constructor
Notation: <<e_1, ..., e_n>>
LaTeX notation:
Arguments: An arbitrary number of arguments.
Apalache type: This operator is overloaded. There are two potential types:
 A tuple constructor:
(a_1, ..., a_n) => <<a_1, ..., a_n>>
, for some typesa_1, ..., a_n
.  A sequence constructor:
(a, ..., a) => Seq(a)
, for some typea
.
That is why the Apalache type checker is sometimes asking you to add annotations, in order to resolve this ambiguity.
Effect: The tuple/sequence constructor returns a function t
that is
constructed as follows:
 set
DOMAIN t
to1..n
,  set
r[i]
to the value ofe_i
fori \in 1..n
.
In Apalache, this constructor may be used to construct either a tuple, or a sequence. To distinguish between them, you will sometimes need a [type annotation].
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
<<"Printer", 631>>
Example in Python: Python provides us with the syntax for constructing lists, which are indexed with 0!. If we want to stick to the principle "sequences are functions", we have to use a dictionary.
>>> ["Printer", 631] # the pythonic way, a twoelement list
['Printer', 631]
>>> { 1: "Printer", 2: 631 } # the "sequencesarefunctions" way
{1: 'Printer', 2: 631}
Sequence append
Notation: Append(seq, e)
LaTeX notation: Append(seq, e)
Arguments: Two arguments. The first argument should be a sequence, the second one is an arbitrary expression.
Apalache type: (Seq(a), a) => Seq(a)
, for some type a
.
Effect: The operator Append(seq, e)
constructs a new sequence newSeq
as follows:
 set
DOMAIN newSeq
to be(DOMAIN seq) \union { Len(seq) + 1 }
.  set
newSeq[i]
toseq[i]
fori \in 1..Len(seq)
.  set
newSeq[Len(seq) + 1]
toe
.
Determinism: Deterministic.
Errors: The argument seq
must be a sequence, that is, a function over
integers 1..n
for some n
. Otherwise, the result is undefined in pure TLA+.
TLC raises a model checking error. Apalache flags a static type error.
Apalache flags a static type error, when the type of e
is not compatible with
the type of the sequence elements.
Example in TLA+:
Append(<<1, 2>>, 5)
\* The sequence <<1, 2, 5>>
Example in Python:
>>> # the pythonic way: a list (indexed with 0, 1, ...)
>>> l = [ 1, 2 ]
>>> l.append(5)
>>> l
[1, 2, 5]
>>> # the TLA+ way
>>> l = { 1: 1, 2: 2 }
>>> { i: l[i] if i <= len(l) else 5
... for i in range(1, len(l) + 2) }
{1: 1, 2: 2, 3: 5}
Function application
As sequences are functions, you can access sequence elements with function
application, e.g., seq[2]
. However, in the case of a
sequence, the type of the function application is: (Seq(a), Int) => a
, for
some type a
.
Sequence head
Notation: Head(seq)
LaTeX notation: Head(seq)
Arguments: One argument. The argument should be a sequence (or a tuple).
Apalache type: Seq(a) => a
, for some type a
.
Effect: The operator Head(seq)
evaluates to seq[1]
.
If seq
is an empty sequence, the result is undefined.
Determinism: Deterministic.
Errors: The arguments seq
must be a sequence (or a tuple), that is, a
function over integers 1..n
for some n
. Otherwise, the result is undefined
in pure TLA+. TLC raises a model checking error. Apalache flags a static type
error.
Example in TLA+:
Head(<<3, 4>>)
\* 3
Example in Python:
>>> # the pythonic way: using the list
>>> l = [ 3, 4 ]
>>> l[0]
3
>>> # the TLA+ way
>>> l = { 1: 3, 2: 4 }
>>> l[1]
3
Sequence tail
Notation: Tail(seq)
LaTeX notation: Tail(seq)
Arguments: One argument. The argument should be a sequence (or a tuple).
Apalache type: Seq(a) => Seq(a)
, for some type a
.
Effect: The operator Tail(seq)
constructs a new sequence newSeq
as
follows:
 set
DOMAIN newSeq
to be(DOMAIN seq) \ { Len(seq) }
.  set
newSeq[i]
toseq[i + 1]
fori \in 1..(Len(seq)  1)
.
If seq
is an empty sequence, the result is undefined.
Apalache encodes a sequences as a triple <<fun, start, end>>
, where
start
and end
define a slice of the function fun
. As a result,
Tail
is a very simple operation that just increments start
.
Determinism: Deterministic.
Errors: The arguments seq
must be a sequence (or a tuple), that is, a
function over integers 1..n
for some n
. Otherwise, the result is undefined
in pure TLA+. TLC raises a model checking error. Apalache flags a static type
error.
Example in TLA+:
Tail(<<3, 4, 5>>)
\* <<4, 5>>
Example in Python:
>>> # the pythonic way: using the list
>>> l = [ 3, 4, 5 ]
>>> l[1:]
[4, 5]
>>> # the TLA+ way
>>> l = { 1: 3, 2: 4, 3: 5 }
>>> { i: l[i + 1] for i in range(1, len(l)) }
{1: 4, 2: 5}
Sequence length
Notation: Len(seq)
LaTeX notation: Len(seq)
Arguments: One argument. The argument should be a sequence (or a tuple).
Apalache type: Seq(a) => Int
, for some type a
.
Effect: The operator Len(seq)
is semantically equivalent to
Cardinality(DOMAIN seq)
.
Apalache encodes a sequences as a triple <<fun, start, end>>
, where
start
and end
define a slice of the function fun
. As a result,
Len
is simply computed as end  start
.
Determinism: Deterministic.
Errors: The argument seq
must be a sequence (or a tuple), that is, a
function over integers 1..n
for some n
. Otherwise, the result is undefined
in pure TLA+. TLC raises a model checking error. Apalache flags a static type
error.
Example in TLA+:
Len(<<3, 4, 5>>)
\* 3
Example in Python:
>>> # the pythonic way: using the list
>>> l = [ 3, 4, 5 ]
>>> len(l)
3
>>> # the TLA+ way
>>> l = { 1: 3, 2: 4, 3: 5 }
>>> len(l.keys())
3
Sequence concatenation
Notation: s \o t
(or s \circ t
)
LaTeX notation:
Arguments: Two arguments: both should be sequences (or tuples).
Apalache type: (Seq(a), Seq(a)) => Seq(a)
, for some type a
.
Effect: The operator s \o t
constructs a new sequence newSeq
as follows:
 set
DOMAIN newSeq
to be1..(Len(s) + Len(t))
.  set
newSeq[i]
tos[i]
fori \in 1..Len(s)
.  set
newSeq[Len(s) + i]
tot[i]
fori \in 1..Len(t)
.
Determinism: Deterministic.
Errors: The arguments s
and t
must be sequences, that is, functions
over integers 1..n
and 1..k
for some n
and k
. Otherwise, the result is
undefined in pure TLA+. TLC raises a model checking error. Apalache flags a
static type error.
Apalache flags a static type error, when the types of s
and t
are
incompatible.
Example in TLA+:
<<3, 5>> \o <<7, 9>>
\* The sequence <<3, 5, 7, 9>>
Example in Python:
>>> # the pythonic way: a list (indexed with 0, 1, ...)
>>> l1 = [ 3, 5 ]
>>> l2 = [ 7, 9 ]
>>> l1 + l2
[3, 5, 7, 9]
>>> # the TLA+ way
>>> l1 = { 1: 3, 2: 5 }
>>> l2 = { 1: 7, 2: 9 }
>>> { i: l1[i] if i <= len(l1) else l2[i  len(l1)]
... for i in range(1, len(l1) + len(l2) + 1) }
{1: 3, 2: 5, 3: 7, 4: 9}
Subsequence
Notation: SubSeq(seq, m, n)
LaTeX notation: SubSeq(seq, m, n)
Arguments: Three arguments: a sequence (tuple), and two integers.
Apalache type: (Seq(a), Int, Int) => Seq(a)
, for some type a
.
Effect: The operator SubSeq(seq, m, n)
constructs a new sequence newSeq
as follows:
 set
DOMAIN newSeq
to be1..(n  m)
.  set
newSeq[i]
tos[m + i  1]
fori \in 1..(n  m + 1)
.
If m > n
, then newSeq
is equal to the empty sequence << >>
.
If m < 1
or n > Len(seq)
, then the result is undefined.
Determinism: Deterministic.
Errors: The argument seq
must be a sequence, that is, a function over
integers 1..k
for some k
. The arguments m
and n
must be integers.
Otherwise, the result is undefined in pure TLA+. TLC raises a model checking
error. Apalache flags a static type error.
Example in TLA+:
SubSeq(<<3, 5, 9, 10>>, 2, 3)
\* The sequence <<5, 9>>
Example in Python:
>>> # the pythonic way: a list (indexed with 0, 1, ...)
>>> l = [ 3, 5, 9, 10 ]
>>> l[1:3]
[5, 9]
>>> # the TLA+ way
>>> l = { 1: 3, 2: 5, 3: 9, 4: 10 }
>>> m = 2
>>> n = 3
>>> { i: l[i + m  1]
... for i in range(1, n  m + 2) }
{1: 5, 2: 9}
Sequence filter
Notation: SelectSeq(seq, Test)
LaTeX notation: SelectSeq(seq, Test)
Arguments: Two arguments: a sequence (a tuple) and a oneargument
operator that evaluates to TRUE
or FALSE
when called with
an element of seq
as its argument.
Apalache type: (Seq(a), (a => Bool)) => Seq(a)
, for some type a
.
Effect: The operator SelectSeq(seq, Test)
constructs a new sequence
newSeq
that contains every element e
of seq
on which Test(e)
evaluates
to TRUE
.
It is much easier to describe the effect of SelectSeq
in words than to
give a precise sequence of steps. See Examples.
Determinism: Deterministic.
Errors: If the arguments are not as described in Arguments, then the result is undefined in pure TLA+. TLC raises a model checking error.
Example in TLA+:
LET Test(x) ==
x % 2 = 0
IN
SelectSeq(<<3, 4, 9, 10, 11>>, Test)
\* The sequence <<4, 10>>
Example in Python:
>>> # the pythonic way: a list (indexed with 0, 1, ...)
>>> def test(x):
... return x % 2 == 0
>>>
>>> l = [ 3, 4, 9, 10, 11 ]
>>> [ x for x in l if test(x) ]
[4, 10]
>>> # the TLA+ way
>>> l = { 1: 3, 2: 4, 3: 9, 4: 10, 5: 11 }
>>> as_list = sorted(list(l.items()))
>>> filtered = [ x for (_, x) in as_list if test(x) ]
>>> { i: x
... for (i, x) in zip(range(1, len(filtered) + 1), filtered)
... }
{1: 4, 2: 10}
All sequences
Notation: Seq(S)
LaTeX notation: Seq(S)
Arguments: One argument that should be a set.
Apalache type: Set(a) => Set(Seq(a))
, for some type a
.
Effect: The operator Seq(S)
constructs the set of all (finite) sequences
that contain elements from S
. This set is infinite.
It is easy to give a recursive definition of all sequences whose length
is bounded by some n >= 0
:
RECURSIVE BSeq(_, _)
BSeq(S, n) ==
IF n = 0
THEN {<< >>} \* the set that contains the empty sequence
ELSE LET Shorter == BSeq(S, n  1) IN
Shorter \union { Append(seq, x): seq \in Shorter, x \in S }
Then we can define Seq(S)
to be UNION { BSeq(S, n): n \in Nat }
.
Determinism: Deterministic.
Errors: The argument S
must be a set.
Apalache flags a static type error, if S
is not a set.
TLC raises a model checking error, when it meets Seq(S)
, as Seq(S)
is
infinite. You can override Seq(S)
with its bounded version BSeq(S, n)
for some n
. See: Overriding Seq in TLC.
Apalache does not support Seq(S)
yet. As a workaround, you can manually
replace Seq(S)
with BSeq(S, n)
for some constant n
. See the progress in
Issue 314.
Example in TLA+:
Seq({1, 2, 3})
\* The infinite set
{ <<>>,
<<1>>, <<2>>, <<3>>,
<<1, 1>>, <<1, 2>>, <<1, 3>>,
<<2, 1>>, <<2, 2>>, <<2, 3>>, <<3, 1>>, <<3, 2>>, <<3, 3>>
...
}
Example in Python: We cannot construct an infinite set in Python. However,
we could write an iterator that enumerates the sequences in Seq(S)
till the end of the universe.
Apalache extensions
Apalache provides the user with several TLA+ modules. These modules introduce TLA+ operators to allow for more efficient model checking with Apalache. Since our users may run Apalache and TLC interchangeably, the modules contain default definitions in TLA+ that are compatible with TLC. Apalache overrides these definitions internally for more efficient treatment compared to the default TLA+ definitions.
Currently supported modules:
Apalache operators
In addition to the standard TLA+ operators described in the previous section, Apalache defines a number of operators, which do not belong to the core language of TLA+, but which Apalache uses to provide clarity, efficiency, or special functionality. These operators belong to the module Apalache
, and can be used in any specification by declaring EXTENDS Apalache
.
Assignment
Notation: v' := e
LaTeX notation:
Arguments: Two arguments. The first is a primed variable name, the second is arbitrary.
Apalache type: (a, a) => Bool
, for some type a
Effect: The expression v' := e
evaluates to v' = e
. At the level of Apalache static analysis, such expressions indicate parts of an action, where the value of a statevariable in a successor state is determined. See here for more details about assignments in Apalache.
Determinism: Deterministic.
Errors: If the first argument is not a primed variable name, or if the assignment operator is used where assignments are prohibited, Apalache statically reports an error.
Example in TLA+:
x' := 1 \* x' = 1
x' := (y = z) \* x' = (y = z)
x' := (y' := z) \* x' = (y' = z) in TLC, assignment error in Apalache
x' := 1 \/ x' := 2 \* x' = 1 \/ x' = 2
x' := 1 /\ x' := 2 \* FALSE in TLC, assignment error in Apalache
x' := 1 \/ x' := "a" \* Type error in Apalache
(x' + 1) := 1 \* (x' + 1) = 1 in TLC, assignment error in Apalache
IF x' := 1 THEN 1 ELSE 0 \* Assignment error in Apalache
Example in Python:
>> a = 1 # a' := 1
>> a == 1 # a' = 1
True
>> a = b = "c" # b' := "c" /\ a' := b'
>> a = (b == "c") # a' := (b = "c")
Nondeterministically guess a value
Notation: Guess(S)
LaTeX notation: Guess(S)
Arguments: One argument: a finite set S
, possibly empty.
Apalache type: Set(a) => a
, for some type a
.
Effect: Nondeterministically pick a value out of the set S
, if S
is
nonempty. If S
is empty, return some value of the proper type.
Determinism: Nondeterministic if S
is nonempty, that is, two subsequent
calls to Guess(S)
may return x, y \in S
that can differ (x /= y
) or may
be equal (x = y
). Moreover, Apalache considers all possible combinations of
elements of S
in the model checking mode. If S
is empty, Guess(S)
produces the same value of proper type.
Errors:
If S
is not a set, Apalache reports an error.
Example in TLA+:
/\ 1 = Guess({ 1, 2, 3 }) \* TRUE or FALSE
/\ 2 = Guess({ 1, 2, 3 }) \* TRUE or FALSE
/\ 3 = Guess({ 1, 2, 3 }) \* TRUE or FALSE
/\ 4 /= Guess({ 1, 2, 3 }) \* TRUE
/\ Guess({ 1, 2, 3 }) \in Int \* TRUE
Value generators
Notation: Gen(bound)
LaTeX notation: Gen(bound)
Arguments: One argument: an integer literal or a constant expression (of the integer type).
Apalache type: Int => a
, for some type a
.
Effect: A generator of a data structure. Given a positive integer bound
,
and assuming that the type of the operator application is known, we recursively
generate a TLA+ data structure as a tree, whose width is bound by the number
bound
.
Determinism: The generated data structure is unrestricted. It is effectively implementing data nondeterminism.
Errors:
If the type of Gen
cannot be inferred from its application context,
or if bound
is not an integer, Apalache reports an error.
Example in TLA+:
\* produce an unrestricted integer
LET \* @type: Int;
oneInt == Gen(1)
IN
\* produce a set of integers up to 10 elements
LET \* @type: Set(Int);
setOfInts == Gen(10)
IN
\* produce a sequence of up to 10 elements
\* that are integers up to 10 elements each
LET \* @type: Seq(Set(Int));
sequenceOfInts == Gen(10)
IN
...
Folding
The operators ApaFoldSet
and ApaFoldSeqLeft
are explained in more detail in a dedicated section here.
Convert a set of pairs to a function
Notation: SetAsFun(S)
LaTeX notation: SetAsFun(S)
Arguments: One argument: A set of pairs S
, which may be empty.
Apalache type: Set(<<a, b>>) => (a > b)
, for some types a
and b
.
Effect: Convert a set of pairs S
to a function F
, with the property that F(x) = y => <<x,y>> \in S
. Note that if S
contains at least two pairs <<x, y>>
and <<x, z>>
, such that y /= z
, then F
is not uniquely defined.
We use CHOOSE
to resolve this ambiguity. The operator SetAsFun
can be defined as follows:
SetAsFun(S) ==
LET Dom == { x: <<x, y>> \in S }
Rng == { y: <<x, y>> \in S }
IN
[ x \in Dom > CHOOSE y \in Rng: <<x, y>> \in S ]
Apalache implements a more efficient encoding of this operator than the default one.
Determinism: Deterministic.
Errors:
If S
is illtyped, Apalache reports an error.
Example in TLA+:
SetAsFun({ <<1, 2>>, <<3, 4>> }) = [x \in { 1, 3 } > x + 1] \* TRUE
SetAsFun({}) = [x \in {} > x] \* TRUE
LET F == SetAsFun({ <<1, 2>>, <<1, 3>>, <<1, 4>> }) IN
\* this is all we can guarantee, when the relation is nondeterministic
\/ F = [x \in { 1 } > 2]
\/ F = [x \in { 1 } > 3]
\/ F = [x \in { 1 } > 4]
Construct a sequence
Notation: MkSeq(n, F)
LaTeX notation: MkSeq(n, F)
Arguments: Two arguments: sequence length n
(a constant integer
expression), and element constructor F(i)
.
Apalache type: (Int, (Int => a)) => Seq(a)
, for some type a
.
Effect: Produce the sequence of n
elements <<F(1), .., F(n)>>
.
Determinism: Deterministic.
Errors:
If n
is not a constant, or is negative, Apalache reports an error.
Example in TLA+:
LET Double(i) == 2 * i IN
MkSeq(3, Double) = <<2, 4, 6>> \* TRUE
Interpret a function as a sequence
Notation: FunAsSeq(fn, len, maxLen)
LaTeX notation: FunAsSeq(fn, len, maxLen)
Arguments: Three arguments:
 A function
fn
that should be interpreted as a sequence.  An integer
len
, denoting the length of the sequence, with the property1..len \subseteq DOMAIN fn
. Apalache does not check this requirement. It is up to the user to ensure that it holds. This expression is not necessarily constant.  An integer constant
maxLen
, which is an upper bound onlen
, that is,len <= maxLen
.
Apalache type: (Int > a, Int, Int) => Seq(a)
, for some type a
Effect: The expression FunAsSeq(fn, len, maxLen)
evaluates to the
sequence << fn[1], ..., fn[Min(len, maxLen)] >>
.
Determinism: Deterministic.
Errors: If the types of fn
, len
or maxLen
do not match the expected types, Apalache statically reports a
type error. Additionally, if it is not the case that 1..len \subseteq DOMAIN fn
, the result is undefined.
Example in TLA+:
Head([ x \in 1..5 > x * x ]) \* 1 in TLC, type error in Apalache
FunAsSeq([ x \in 1..5 > x * x ], 3, 3) \* <<1,4,9>>
Head(FunAsSeq([ x \in 1..5 > x * x ], 3, 3)) \* 1
FunAsSeq(<<1,2,3>>, 3, 3) \* <<1,2,3>> in TLC, type error in Apalache
FunAsSeq([ x \in {0,42} > x * x ], 3, 3) \* UNDEFINED
Example in Python:
# define a TLA+like dictionary via a python function
def boundedFn(f, dom):
return { x: f(x) for x in dom }
# this is how we could define funAsSeq in python
def funAsSeq(f, length, maxLen):
return [ f.get(i) for i in range(1, min(length, maxLen) + 1) ]
# TLA+: [ x \in 1..5 > x * x ]
f = boundedFn(lambda x: x * x, range(1,6))
# TLA+: [ x \in {0, 42} > x * x ]
g = boundedFn(lambda x: x * x, {0, 42})
>>> f[1]
1
>>> funAsSeq(f, 3, 3)
[1, 4, 9]
>>> funAsSeq(f, 3, 3)[1]
1
>>> funAsSeq(g, 3, 3)
[None, None, None]
Skolemization Hint
Notation: Skolem(e)
LaTeX notation: Skolem(e)
Arguments: One argument. Must be an expression of the form \E x \in S: P
.
Apalache type: (Bool) => Bool
Effect: The expression Skolem(\E x \in S: P)
provides a hint to Apalache, that the existential quantification may be skolemized. It evaluates to the same value as \E x \in S: P
.
Determinism: Deterministic.
Errors:
If e
is not a Boolean expression, throws a type error. If it is Boolean, but not an existentially quantified expression, throws a StaticAnalysisException
.
Note:
This is an operator produced internally by Apalache. You may see instances of this operator, when reading the .tla
sideoutputs of various passes. Manual use of this operator is discouraged and, in many cases, not supported.
Example in TLA+:
Skolem( \E x \in {1,2}: x = 1 ) \* TRUE
Skolem( 1 ) \* 1 in TLC, type error in Apalache
Skolem( TRUE ) \* TRUE in TLC, error in Apalache
Set expansion
Notation: Expand(S)
LaTeX notation: Expand(S)
Arguments: One argument. Must be either SUBSET SS
or [T1 > T2]
.
Apalache type: (Set(a)) => Set(a)
, for some a
.
Effect: The expression Expand(S)
provides instructions to Apalache, that the large set S
(powerset or set of functions) should be explicitly constructed as a finite set, overriding Apalache's optimizations for dealing with such collections. It evaluates to the same value as S
.
Determinism: Deterministic.
Errors:
If e
is not a set, throws a type error. If the expression is a set, but is not of the form SUBSET SS
or [T1 > T2]
, throws a StaticAnalysisException
.
Note:
This is an operator produced internally by Apalache. You may see instances of this operator, when reading the .tla
sideoutputs of various passes. Manual use of this operator is discouraged and, in many cases, not supported.
Example in TLA+:
Expand( SUBSET {1,2} ) \* {{},{1},{2},{1,2}}
Expand( {1,2} ) \* {1,2} in TLC, error in Apalache
Expand( 1 ) \* 1 in TLC, type error in Apalache
Cardinality Hint
Notation: ConstCardinality(e)
LaTeX notation: ConstCardinality(e)
Arguments: One argument. Must be an expression of the form Cardinality(S) >= k
.
Apalache type: (Bool) => Bool
Effect: The expression ConstCardinality(Cardinality(S) >= k)
provides a hint to Apalache, that Cardinality(S)
is a constant, allowing Apalache to encode the constraint e
without attempting to dynamically encode Cardinality(S). It evaluates to the same value as
e`.
Determinism: Deterministic.
Errors:
If S
is not a Boolean expression, throws a type error. If it is Boolean, but not an existentially quantified expression, throws a StaticAnalysisException
.
Note:
This is an operator produced internally by Apalache. You may see instances of this operator, when reading the .tla
sideoutputs of various passes. Manual use of this operator is discouraged and, in many cases, not supported.
Example in TLA+:
Skolem( \E x \in {1,2}: x = 1 ) \* TRUE
Skolem( 1 ) \* 1 in TLC, type error in Apalache
Skolem( TRUE ) \* TRUE in TLC, error in Apalache
Variants
Variants (also called tagged unions or sum types) are useful, when you want to combine values of different shapes in a single set or a sequence.
Idiomatic tagged unions in untyped TLA+. In untyped TLA+, one can construct sets, which contain records with different fields, where one filed is typically used as a disambiguation tag. For instance, we could create a set that contains two records of different shapes:
ApplesAndOranges == {
[ tag > "Apple", color > "red" ],
[ tag > "Orange", seedless > TRUE ]
}
We can dynamically reason about the elements of ApplesAndOranges
based on their tag:
\E e \in ApplesAndOranges:
/\ e.tag = "Apple"
/\ e.color /= "green"
This idiom is quite common in untyped TLA+. Tagged unions in Paxos is probably the most illuminating example of this idiom. Unfortunately, it is way too easy to make a typo in the tag name, since it is a string, or simply access a field, which records marked with the given tag do not have. For example:
\E e \in ApplesAndOranges:
/\ e.tag = "Apple"
/\ e.seedless
Variants module. Apalache formalizes the above idiom in the module Variants.tla. Apalache's type checker alerts users with a type error when they access a wrong value. Additionally, the default implementation raises an error in TLC when a variant is used incorrectly.
Immutability. All variants are immutable.
Construction. An instance of a variant can be constructed via the operator
Variant
:
Variant("Apple", "red")
If we just construct a variant like in the example above, it will be assigned a parametric variant type:
Apple(Str)  a
In this type, we know that whenever a value is tagged with "Apple" it should be of the string type. However, we know nothing about other options. Most of the time, we want to define variants that are sealed, that is, we know all available options. Suppose we wanted to reason about different kinds of fruit, but wanted to limit our model to only comparing apples and oranges. In Apalache, the type for a value that could be either an apple or an orange, but nothing else, would be as follows:
Apple(Str)  Orange(Bool)
To make it easier to represent the fruits, we can introduce variants together with userdefined constructors for each option::
\* @typeAlias: FRUIT = Apple(Str)  Orange(Bool);
\* @type: Str => FRUIT;
Apple(color) == Variant("Apple", color)
\* @type: Bool => FRUIT;
Orange(seedless) == Variant("Orange", seedless)
Now we can naturally construct apples and orange as follows:
Apple("red")
Orange(TRUE)
Variants can wrap records, for when we want to represent compound data with named fields:
\* @typeAlias: DRINK =
\* Water({ sparkling: Bool })
\*  Beer({ malt: Str, strength: Int });
\*
\* @type: Bool => DRINK;
Water(sparkling) == Variant("Water", [ sparkling > sparkling ])
\* @type: (Str, Int) => DRINK;
Beer(malt, strength) == Variant("Beer", [ malt > malt, strength > strength ])
Once a variant is constructed, it becomes opaque to the type checker, that is,
the type checker only knows that Water(TRUE)
and Beer("Dark", 5)
are both
of type DRINK
. This is exactly what we want, in order to combine these values
in a single set. However, we have lost the ability to access the fields of
these values. To deconstruct values of a variant type, we have to use other
operators, presented below.
Filtering by tag name. Following the idiomatic use of tagged unions in untyped TLA+, we can filter a set of variants:
LET Drinks == { Water(TRUE), Water(FALSE), Beer("Radler", 2) } IN
\E d \in VariantFilter("Beer", Drink):
d.strength < 3
We believe that VariantFilter
is the most commonly used way to partition a
set of variants. Note that VariantFilter
transforms a set of variants into a
set of values (that correspond to the associated tag name).
Typesafe get. Sometimes, we do have just a value that does not belong to a
set, so we cannot use VariantFilter
directly. In this case, we can use
VariantGetOrElse
:
LET water == Water(TRUE) IN
VariantGetOrElse("Beer", water,
[ malt > "Nonalcoholic", strength > 0])).strength
In the above example, we unpack water
by using the tag name "Beer"
. Since
water
is actually tagged with "Water"
, the operator falls back to the
default case and returns the record [ malt > "Nonalcoholic", strength > 0]
.
Typeunsafe get. Sometimes, using VariantFilter
and VariantGetOrElse
is a nuisance, when we know the exact value type. In this case, we can bypass
the type checker and get the value notwithstanding the tag:
LET drink == ... IN
LET nonFree ==
IF VariantTag(drink) = "Water"
THEN VariantGetUnsafe("Water", drink).sparkling
ELSE VariantGetUnsafe("Beer", drink).strength > 0
IN
...
In general, you should avoid using VariantGetUnsafe
, as it is type unsafe.
Consider the following example:
VariantGetUnsafe("Beer", Water(TRUE)).strength
In the above example, we treat water as beer. If you try this example with TLC,
it would complain about the missing field strength
, as it computes some form
of types dynamically. If you try this example with Apalache, it would compute
types statically and in the case of VariantGetUnsafe
it would simply produce
an arbitrary integer. Most likely, this arbitrary integer would propagate into
an invariant violation and will lead to a spurious counterexample.
Operators
Variant constructor
Notation: Variant(tagName, associatedValue)
LaTeX notation: same
Arguments: Two arguments: the tag name (a string literal) and a value (a TLA+ expression).
Apalache type: (Str, a) => tagName(a)  b
, for some types a
and b
.
Note that tagName
is an identifier in this notation. In this type, b
is a
type variable that captures other options in the variant type.
Effect: The variant constructor returns a new value of the variant type.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @typeAlias: DRINK =
\* Water({ sparkling: Bool })
\*  Beer({ malt: Str, strength: Int });
\*
\* @type: Bool => DRINK;
Water(sparkling) == Variant("Water", [ sparkling > sparkling ])
\* @type: (Str, Int) => DRINK;
Beer(malt, strength) == Variant("Beer", [ malt > malt, strength > strength ])
Variant tag
Notation: VariantTag(variant)
LaTeX notation: same
Arguments: One argument: a variant constructed via Variant
.
Apalache type: (tagName(a)  b) => Str
, for some types a
and b
. Note
that tagName
is an identifier in this notation. In this type, b
is a type
variable that captures other options in the variant type.
Effect: This operator simply returns the tag attached to the variant.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
VariantTag(Variant("Water", [ sparkling > sparkling ])) = "Water"
Variant filter
Notation: VariantFilter(tagName, set)
LaTeX notation: same
Arguments: Two arguments: the tag name (a string literal) and a set of variants (a TLA+ expression).
Apalache type: (Str, Set(tagName(a)  b)) => Set(a)
, for some types a
and b
. Note that tagName
is an identifier in this notation. In this type,
b
is a type variable that captures other options in the variant type.
Effect: The variant filter keeps the set elements that are tagged with
tagName
. It removes the tags from these elements and produces the set of
values that were packed with Variant
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @typeAlias: DRINK =
\* Water({ sparkling: Bool })
\*  Beer({ malt: Str, strength: Int });
\*
\* @type: Bool => DRINK;
Water(sparkling) == Variant("Water", [ sparkling > sparkling ])
\* @type: (Str, Int) => DRINK;
Beer(malt, strength) == Variant("Beer", [ malt > malt, strength > strength ])
LET Drinks == { Water(TRUE), Water(FALSE), Beer("Radler", 2) } IN
\E d \in VariantFilter("Beer", Drinks):
d.strength < 3
Unpacking a variant safely
Notation: VariantGetOrElse(tagName, variant, defaultValue)
LaTeX notation: same
Arguments: Three arguments: the tag name (a string literal), a variant
constructed via Variant
, a default value compatible with the value carried by
the variant.
Apalache type: (Str, tagName(a)  b, a) => a
, for some types a
and b
.
Note that tagName
is an identifier in this notation. In this type, b
is a
type variable that captures other options in the variant type.
Effect: The operator VariantGetOrElse
returns the value that was wrapped
via the Variant
constructor, if the variant is tagged with tagName
.
Otherwise, the operator returns defaultValue
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @typeAlias: DRINK =
\* Water({ sparkling: Bool })
\*  Beer({ malt: Str, strength: Int });
\*
\* @type: Bool => DRINK;
Water(sparkling) == Variant("Water", [ sparkling > sparkling ])
\* @type: (Str, Int) => DRINK;
Beer(malt, strength) == Variant("Beer", [ malt > malt, strength > strength ])
LET water == Water(TRUE) IN
VariantGetOrElse("Beer", water,
[ malt > "Nonalcoholic", strength > 0])).strength
Unpacking a variant unsafely
Notation: VariantGetUnsafe(tagName, variant)
LaTeX notation: same
Arguments: Two arguments: the tag name (a string literal) and a variant
constructed via Variant
.
Apalache type: (Str, tagName(a)  b) => a
, for some types a
and b
.
Note that tagName
is an identifier in this notation. In this type, b
is a
type variable that captures other options in the variant type.
Effect: The operator VariantGetUnsafe
unconditionally returns some value
that is compatible with the type of values tagged with tagName
. If variant
is tagged with tagName
, the returned value is the value that was wrapped via
the Variant
constructor. Otherwise, it is some arbitrary value of proper type. As such,
this operator does not guarantee that the retrieved value is always constructed
via Variant
, unless the operator is used with the right tag.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @typeAlias: DRINK =
\* Water({ sparkling: Bool })
\*  Beer({ malt: Str, strength: Int });
\*
\* @type: Bool => DRINK;
Water(sparkling) == Variant("Water", [ sparkling > sparkling ])
\* @type: (Str, Int) => DRINK;
Beer(malt, strength) == Variant("Beer", [ malt > malt, strength > strength ])
LET drink == Beer("Dunkles", 4) IN
LET nonFree ==
IF VariantTag(drink) = "Water"
THEN VariantGetUnsafe("Water", drink).sparkling
ELSE VariantGetUnsafe("Beer", drink).strength > 0
IN
...
Option Types
Option types are useful when you
want to internalize reasoning about partial functions. A simple motivating
example is division over integers, for which n/0
is undefined.
The basic idea is as follows: given a partial function f : A > B
, we form the
type Option(B)
by extending B
with an element representing a missing value,
None
, and lift each value b
in B
to Some(b)
, allowing us to represent
the partial function pf : A > Option(B)
, such that, for each a
in A
,
pf(a) = Some(f(a))
iff f(a)
is defined, and None
otherwise.
Apalache leverages its support for variants to define a polymorphic option type along with some common utility functions in the module Option.tla.
The module defines a type alias $option
as
\* @typeAlias: option = Some(a)  None(UNIT);
However, due to the current lack of support for polymorphic aliases, this alias
has limited utility, and parametric option types can only be properly expressed
by writing out the full variant type Some(a)  None(UNIT)
. Nonetheless, in this manual
page, we will sometimes write $option(a)
as a shorthand for the type Some(a)  None(UNIT)
.
In the context of TLA+, our encoding of option types is generalized over
"partial operators", meaning operators which return a value of type
$option(a)
. Support for partial functions is supplied by two operators,
OptionPartialFun and OptionFunApp.
Operators
Constructing present optional values
Notation: Some(v)
LaTeX notation: same
Apalache type: (a) => Some(a)  None(UNIT)
, for some type a
.
Arguments: The value v
of type a
to be lifted into the
option.
Effect: Produces a new value of the optional type.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @type: Some(Int)  None(UNIT);
SomeInt == Some(42)
Constructing absent optional values
Notation: None
LaTeX notation: same
Apalache type: Some(a)  None(UNIT)
, for some type a
.
Arguments: None
Effect: Produces a representation of an absent value.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @type: Some(Int)  None(UNIT);
NoInt == None
Checking for presence or absence of a value
Notation: IsSome(o)
or IsNone(o)
LaTeX notation: same
Apalache type: (Some(a)  None(UNIT)) => Bool
, for some type a
.
Arguments: One argument: a value of type $option(a)
for some type a
.
Effect: These operators are TRUE
or FALSE
depending on whether the
optional value is present or absent, in the expected way.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
TRUE = IsSome(Some(5)) /\ IsNone(None)
Case analysis and elimination of optional values
Notation: OptionCase(o, caseSome, caseNone)
LaTeX notation: same
Apalache type: (Some(a)  None(UNIT), a => b, UNIT => b) => b
, for some types a
and b
.
Arguments:
o
an optional valuecaseSome
is an operator to be applied to a present valuecaseNone
is an operator to be applied to theUNIT
if the value is absent
Effect: OptionCase(o, caseSome, caseNone)
is caseSome(v)
if o = Some(v)
, or else caseNone(UNIT)
. This is a way of eliminating a value of type
Option(a)
to produce a value of type b
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
/\ LET
\* @type: Int => Int;
caseSome(x) == x + 1
IN LET
\* @type: UNIT => Int;
caseNone(u) == 0
IN
OptionCase(Some(3), caseSome, caseNone) = 4
/\ LET
\* @type: Int => Str;
caseSome(x) == "Some Number"
IN LET
\* @type: UNIT => Str;
caseNone(u) == "None"
IN
OptionCase(None, caseSome, caseNone) = "None"
Sequencing application of partial operators
Notation: OptionFlatMap(f, o)
LaTeX notation: same
Apalache type: (a => Some(b)  None(UNIT), Some(a)  None(UNIT)) => Some(b)  None(UNIT)
, for some types a
and b
.
Arguments:
f
is a partial operatoro
an optional value
Effect: OptionFlatMap(f, o)
is f(v)
if o = Some(v)
, or else None
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
LET incr(n) == Some(n + 1) IN
LET fail(n) == None IN
LET q == OptionFlatMap(incr, Some(1)) IN
LET r == OptionFlatMap(incr, q) IN
LET s == OptionFlatMap(fail, r) IN
/\ r = Some(3)
/\ s = None
Unwrapping optional values
Notation: OptionGetOrElse(o, default)
LaTeX notation: same
Apalache type: (Some(a)  None(UNIT), a) => a
, for some type a
.
Arguments:
o
an optional valuedefault
is a default value to return
Effect: OptionGetOrElse(o, default)
is v
iff o = Some(v)
, or else default
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
\* @type: Set(Int) => Some(Int)  None(UNIT);
MaxSet(s) ==
LET max(oa, b) ==
IF OptionGetOrElse(oa, b) > b
THEN oa
ELSE Some(b)
IN
ApaFoldSet(max, None, s)
Converting optional values
Converting to sequences
Notation: OptionToSeq(o)
LaTeX notation: same
Apalache type: (Some(a)  None(UNIT)) => Seq(a)
, for some type a
.
Arguments:
o
an optional value
Effect: OptionToSeq(o)
is <<v>>
iff o = Some(v)
, or else <<>>
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
LET \* @type: Seq(Int);
empty == <<>>
IN
/\ OptionToSeq(None) = empty
/\ OptionToSeq(Some(1)) = <<1>>
Converting to sets
Notation: OptionToSet(o)
LaTeX notation: same
Apalache type: (Some(a)  None(UNIT)) => Seq(a)
, for some type a
.
Arguments:
o
an optional value
Effect: OptionToSet(o)
is like OptionToSeq
, but producing a set.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
LET \* @type: Set(Int);
empty == {}
IN
/\ OptionToSet(None) = empty
/\ OptionToSet(Some(1)) = {1}
Obtaining an optional value from a set
Notation: OptionGuess(s)
LaTeX notation: same
Apalache type: Set(a) => Some(a)  None(UNIT)
, for some type a
.
Arguments:
s
is a set
Effect: OptionGuess(s)
is None
if s = {}
, otherwise it is Some(x)
,
where x \in s
. x
is selected from s
nondeterministically.
Determinism: Nondeterministic.
Errors: No errors.
Example in TLA+:
LET
\* @type: Set(Int);
empty == {}
IN
/\ OptionGuess(empty) = None
/\ LET choices == {1,2,3,4} IN
LET choice == OptionGuess(choices) IN
VariantGetUnsafe("Some", choice) \in choices
Apply a function to a partial value
Notation: OptionFunApp(f, o)
LaTeX notation: same
Apalache type: (a > b, Some(a)  None(UNIT)) => Some(b)  None(UNIT)
Arguments:
f
is a functiono
is an optional value
Effect: OptionFunApp(f, o)
is Some(f[v])
if o = Some(v)
or else None
.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
LET f == [x \in 1..3 > x + 1] IN
/\ OptionFunApp(f, Some(1)) = Some(2)
/\ OptionFunApp(f, None) = None
Extend a total function into a partial function
Notation: OptionPartialFun(f, undef)
LaTeX notation: same
Apalache type: (a > b, Set(a)) => (a > Some(b)  None(UNIT))
Arguments:
f
is a total functionundef
is a set of values for which the new function is to be "undefined"
Effect: OptionPartialFun(f, undef)
is a function mapping each value in
undef
to None
, and each value x \in (DOMAIN f \ undef)
to Some(f[x])
.
This can be used to extend a total function into a "partial function" whose domain is
extended to include the values in 'undef'.
Determinism: Deterministic.
Errors: No errors.
Example in TLA+:
LET def == 1..3 IN
LET undef == 4..10 IN
LET f == [x \in def > x + 1] IN
LET pf == OptionPartialFun(f, undef) IN
/\ \A n \in def: pf[n] = Some(n + 1)
/\ \A n \in undef: pf[n] = None
Userdefined operators
Like macros, to do a lot of things in one system step...
Userdefined operators in TLA+ may be confusing. At first, they look like
functions in programming languages. (Recall that TLA+
functions are more like dictionaries or hash maps, not
functions in PL.) Then you realize that operators such as Init
and Next
are
used as logic predicates. However, large specifications often contain operators
that are not predicates, but in fact are similar to pure functions in
programming languages: They are computing values over the system state but pose
no constraints over the system states.
Recently, Leslie Lamport has extended the syntax of TLA+ operators in TLA+ version 2, which supports recursive operators and lambda operators. We explain why Apalache does not support those in Recursive operators and functions.
The operator syntax that is described in Specifying Systems describes TLA+ version 1. This page summarizes the syntax of userdefined operators in versions 1 and 2.
Short digression. The most important thing to understand about userdefined
operators is that they are normally used inside Init
and Next
. While the
operator Init
describes the initial states, the operator Next
describes a
single step of the system. That is, these two operators are describing the
initial states and the possible transitions of the system, respectively. They
do not describe the whole system computation. Most of the time, we are writing
canonical specifications, which are written in temporal logic as Init /\ [][Next]_vars
. Actually, you do not have to understand temporal logic, in
order to write canonical specifications. A canonical specification is saying:
(1) Initialize the system as Init
prescribes, and (2) compute system
transitions as Next
prescribes. It also allows for stuttering, but this
belongs to Advanced topics.
After the digression, you should now see that userdefined operators in TLA+ are (normally) describing a single step of the system. Hence, they should be terminating. That is why user operators are often understood as macros. The same applies to [Recursive operator definitions]. They have to terminate within a single system step.
Quirks of TLA+ operators. Below we summarize features of userdefined operators that you would probably find unexpected:

Some operators are used as predicates and some are used to compute values (à la pure).

Operators may accept other operators as parameters. Such operators are called Higherorder operator definitions.

Although operators may be passed as parameters, they are not firstclass citizens in TLA+. For instance, an operator cannot be returned as a result of another operator. Nor can an operator be assigned to a variable (only the result of its application may be assigned to a variable).

Operators do not support Currying. That is, you can only apply an operator by providing values for all of its expected arguments.

Operators can be nested. However, nested operators require a slightly different syntax. They are defined with LETIN definitions.
Details about operators. We go in detail about different kinds of operators and recursive functions below:
Toplevel operator definitions
Quick example
Here is a quick example of a toplevel user operator (which has to be defined in a module) and of its application:
 MODULE QuickTopOperator 
...
Abs(i) == IF i >= 0 THEN i ELSE i
...
B(k) == Abs(k)
===============================================================================
As you most probably guessed, the operator Abs
expects one argument i
.
Given an integer j
, then the result of computing Abs(j)
is the absolute
value of j
. The same applies, when j
is a natural number or a real number.
Syntax of operator definitions
In general, operators of n
arguments are
defined as follows:
\* an operator without arguments (nullary)
Opa0 == body_0
\* an operator of one argument (unary)
Opa1(param1) == body_1
\* an operator of two arguments (binary)
Opa2(param1, param2) == body_2
...
In this form, the operator arguments are not allowed to be operators. If you want to receive an operator as an argument, see the syntax of Higherorder operators.
Here are concrete examples of operator definitions:
 MODULE FandC 
EXTENDS Integers
...
ABSOLUTE_ZERO_IN_CELCIUS ==
273
Fahrenheit2Celcius(t) ==
(t  32) * 10 / 18
Max(s, t) ==
IF s >= t THEN s ELSE t
...
===============================================================================
What is their arity (number of arguments)?
If you are used to imperative languages such as Python or Java, then you are
probably surprised that operator definitions do not have any return
statement. The reason for that is simple: TLA+ is not executed on any hardware.
To understand how operators are evaluated, see the semantics below.
Syntax of operator applications
Having defined an operator, you can apply it inside another operator as follows (in a module):
 MODULE FandC 
EXTENDS Integers
VARIABLE fahrenheit, celcius
\* skipping the definitions of
\* ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius, and Max
...
UpdateCelcius(t) ==
celcius' = Max(ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius(t))
Next ==
/\ fahrenheit' \in 1000..1000
/\ UpdateCelcius(fahrenheit')
...
===============================================================================
In the above example, you see examples of four operator applications:

The nullary operator
ABSOLUTE_ZERO_IN_CELCIUS
is applied without any arguments, just by its name. Note how a nullary operator does not require parentheses()
. Yet another quirk of TLA+. 
The oneargument operator Fahrenheit2Celcius is applied to
t
, which is a parameter of the operatorUpdateCelcius
. 
The twoargument operator
Max
is applied toABSOLUTE_ZERO_IN_CELCIUS
andFahrenheit2Celcius(t)
. 
The oneargument operator
UpdateCelcius
is applied tofahrenheit'
, which is the value of state variablefahrenheit
in the next state of the state machine. TLA+ has no problem applying the operator tofahrenheit'
or tofahrenheit
.
Technically, there are more than four operator applications in our example. However, all other operators are the standard operators. We do not focus on them here.
Note on the operator order. As you can see, we are applying operators after they have been defined in a module. This is a general rule in TLA+: A name can be only referred to, if it has been defined in the code before. TLA+ is not the first language to impose that rule. For instance, Pascal had it too.
Note on shadowing. TLA+ does not allow you to use the same name as an operator parameter, if it has been defined in the context of the operator definition. For instance, the following is not allowed:
 MODULE NoShadowing 
VARIABLE x
\* the following operator definition produces a semantic error:
\* the parameter x is shadowing the state variable x
IsZero(x) == x = 0
===============================================================================
There are a few tricky cases, where shadowing can actually happen, e.g., see
the operator dir
in SlidingPuzzles. However, we recommend to keep things
simple and avoid shadowing at all.
Semantics of operator application
Precise treatment of operator application is given on page 320 of Specifying Systems. In a nutshell, operator application in TLA+ is a Call by macro expansion, though it is a bit smarter: It does not blindly mix names from the operator's body and its application context. For example, the following semantics by substitution is implemented in the Apalache model checker.
Here we give a simple explanation for nonrecursive operators. Consider the
definition of an n
ary operator A
and its application in the definition
of another operator B
:
A(p_1, ..., p_n) == body_of_A
...
B(p_1, ..., p_k) ==
...
A(e_1, ..., e_n)
...
The following three steps allow us to replace application of the operator A
in B
:
 Change the names in the definition of
A
in such a way such they do not clash with the names inB
(as well as with other names that may be used inB
). This is the wellknown technique of Alpha conversion in programming languages. This may also require renaming of the parametersp_1, ..., p_n
. Let the result of alpha conversion be the following operator:
uniq_A(uniq_p_1, ..., uniq_p_n) == body_of_uniq_A

Substitute the expression
A(e_1, ..., e_n)
in the definition ofB
withbody_of_uniq_A
. 
Substitute the names
uniq_p_1, ..., uniq_p_n
with the expressionse_1, ..., e_n
, respectively.
The above transformation is usually called Beta reduction.
Example. Let's go back to the module FandC
, which we considered above. By
applying the substitution approach several times, we transform Next
in
several steps as follows:
First, by substituting the body of UpdateCelsius
:
Next ==
/\ fahrenheit' \in 1000..1000
/\ celcius' = Max(ABSOLUTE_ZERO_IN_CELCIUS, Fahrenheit2Celcius(fahrenheit'))
Second, by substituting the body of Max
:
Next ==
/\ fahrenheit' \in 1000..1000
/\ celcius' =
IF ABSOLUTE_ZERO_IN_CELCIUS >= Fahrenheit2Celcius(fahrenheit')
THEN ABSOLUTE_ZERO_IN_CELCIUS
ELSE Fahrenheit2Celcius(fahrenheit')
Third, by substituting the body of Fahrenheit2Celcius
(twice):
Next ==
/\ fahrenheit' \in 1000..1000
/\ celcius' =
IF ABSOLUTE_ZERO_IN_CELCIUS >= (fahrenheit'  32) * 10 / 18
THEN ABSOLUTE_ZERO_IN_CELCIUS
ELSE (fahrenheit'  32) * 10 / 18
You could notice that we applied beta reduction syntactically from top to
bottom, like peeling an onion. We could do it in another direction: First
starting with the application of Fahrenheit2Celcius
. This actually does not
matter, as long as our goal is to produce a TLA+ expression that is free of
userdefined operators. For instance, Apalache applies Alpha conversion and
Beta reduction to remove userdefined operator and then translates the TLA+
expression to SMT.
LETIN operator definitions
work in progress...
Higherorder operator definitions
work in progress...
Anonymous operator definitions
work in progress...
Local operator definitions
work in progress...
Idiomatic TLA+
Authors: Shon Feder, Igor Konnov, Gabriela Moreira + (who likes to contribute?)
This document is under construction. If you like to contribute, open a pull request.
Introduction
In this document, we collect specification idioms that aid us in writing TLA+ specifications that are:
 understood by distributed system engineers,
 understood by verification engineers, and
 understood by automatic analysis tools such as the Apalache model checker.
If you believe, that the above points are contradictory when put together, it is to some extent true. TLA+ is an extremely general specification language. As a result, it is easy to write a short specification that leaves a human reader puzzled . It is even easier to write a (syntactically correct) specification that turns any program trying to reason about TLA+ to dust.
Nevertheless, we find TLA+ quite useful when writing concise specifications of distributed protocols at Informal Systems. Other specification languages  especially, those designed for software verification  would require us to introduce unnecessary bookkeeping details that would both obfuscate the protocols and make their verification harder. However, we do not always need "all the power of mathematics", so we find it useful to introduce additional structure in TLA+ specifications.
Below, we summarize the idioms that help us in maintaining that structure. As a bonus, these idioms usually aid the Apalache model checker in analyzing the specifications. Our idioms are quite likely different from the original ideas of Leslie Lamport (the author of TLA+). So it is useful to read Lamport's Specifying Systems. Importantly, these are idioms, not rules set in stone. If you believe that one of those idioms does not work for you in your specific setting, don't follow it.
If this is the first page where you encounter the word "TLA+", we do not recommend that you continue to read the material. It is better to start with The TLA+ Video Course by Leslie Lamport. Once you have understood the basics and tried the language, it makes sense to ask the question: "How do I write a specification that other people understand?". We believe that many TLA+ users reinvent rules that are similar to our idioms. By providing you with a bit of guidance, we hope to reduce your discomfort when learning more advanced TLA+.
The idioms
Idiom 0: Keep state variables to the minimum 🔋
Idiom 1: Update state variables with assignments 📅
Idiom 2: Apply primes only to state variables 📌
Idiom 3: Isolate updates to VARIABLES 👻
Idiom 4: Isolate nondeterminism in actions 🔮
Idiom 5: Introduce pure operators 🙈
Idiom 6: Introduce a naming convention for operator parameters 🛂
Idiom 7: Use Boolean operators in actions, not IFTHENELSE
🙅
Idiom 8: CHOOSE
smart, prefer \E
💂♂
Idiom 9: Do not overstructure 🔬
Idiom 10: Do not overmodularize 🦆
Idiom 11: Separate normal paths from error paths. ⚡
Idiom 12: Do you really need those nice recursive operators? 🌀
Idiom 13: Do you really need set cardinalities? 🍕
Idiom 14: Do you really need integers? 🔢
Idiom 15: Replace sets of mixed records with disjoint unions 🛄
Idiom 0: Keep state variables to the minimum
In imperative programming, it is common to use mutable variable assignments liberally, but to exercise caution whenever mutable variables have a global scope. In TLA+, mutable variables are always global, so it is important to use them carefully and in a way that accurately reflects the global state of the system you are specifying.
Description
A good TLA+ specification minimizes the computation state and makes it visible.
TLA+ does not have special syntax for variable assignment. For a good reason. The power of TLA+ is in writing constraints on variables rather than in writing detailed commands. If you have been writing in languages such as C, C++, Java, Python, your first reflex would be to define a variable to store the intermediate result of a complex computation.
In programming languages, we introduce temporary variables for several reasons:
 To avoid repetitive computations of the same expression,
 To break down a large expression into a series of smaller expressions,
 To make the code concise.
Point 1 is a nonissue in TLA+, as it is mostly executed in the reader's brain, and people are probably less efficient in caching expressions than computers. Points 2 and 3 can be nicely addressed with LETdefinitions in TLA+. Hence, there is no need for auxiliary variables.
Usually, we should minimize the specification state, that is, the scope of the data
structures that are declared with VARIABLES
. It does not mean that one variable
is always better than two. It means that what is stored in VARIABLES
should be
absolutely necessary to describe the computations or the observed properties.
Advantages
By avoiding auxiliary state variables, we localize the updates to the state. This improves specification readability. It also helps the tools, as large parts of the specification become deterministic.
Disadvantages
Sometimes, we have to expose the internals of the computation. For instance, if we want to closely monitor the values of the computed expressions, when using the specification for modelbased testing.
Sometimes, we have to break this idiom to make the specification more readable. Here is an example by Markus Kuppe. The specification of BlockingQueue that has one more variable is easier to read than the original specification with a minimal number of variables.
Example
Consider the following implementation of Bubble sort in Python:
my_list = [5, 4, 3, 8, 1]
finished = False
my_list_len = len(my_list) # cache the length
while not finished:
finished = True
if my_list_len > 0:
prev = my_list[0] # save the first element to use in the loop
for i in range(1, my_list_len):
current = my_list[i]
if prev <= current:
# save current for the next iteration
prev = current
else:
# swap the elements
my_list[i  1] = current
my_list[i] = prev
finished = False
Notice that we have introduced three local variables to optimize the code:
my_list_len
to cache the length of the list,prev
to cache the previously accessed element of the list, in order to minimize the number of list accesses,current
to cache the iterated element of the list.
In TLA+, one usually does not introduce local variables for the intermediate
results of the computation, but rather introduces variables to represent the
essential part of the algorithm state. (While we have spent some time on code
optimization, we might have missed the fact that our sorting algorithm is not
as good as Quicksort.) In the above example, the essential variables are
finished
and my_list
.
Compare the above code to (a slightly more abstract) bubble sort in TLA+:
EXTENDS Integers, Sequences
in_list == <<5, 4, 3, 8, 1>>
VARIABLES my_list, finished
Init ==
/\ my_list = in_list
/\ finished = FALSE
IsSorted(lst) ==
\A i \in DOMAIN lst \ {1}:
lst[i  1] <= lst[i]
WhenSorted ==
/\ IsSorted(my_list)
/\ finished' = TRUE
/\ UNCHANGED my_list
WhenUnsorted ==
/\ \E i \in DOMAIN my_list \ {1}:
/\ my_list[i  1] > my_list[i]
/\ my_list' = [my_list EXCEPT ![i  1] = my_list[i],
![i] = my_list[i  1]]
/\ finished' = FALSE
Next ==
IF finished
THEN UNCHANGED <<my_list, finished>>
ELSE WhenSorted \/ WhenUnsorted
Our TLA+ code contains only two state variables: my_list
and finished
.
Other variables are introduced by quantifiers (e.g., \E i \in ...
).
The state variables are not updated in the sense of programming languages.
Rather, one writes constraints over unprimed and primed versions, e.g.:
...
/\ my_list' = [my_list EXCEPT ![i  1] = my_list[i],
![i] = my_list[i  1]]
Of course, one can introduce aliases for intermediate expressions, for instance, by using letdefinitions:
...
LET prev == my_list[i  1]
current == my_list[i]
IN
/\ prev > current
/\ my_list' = [my_list EXCEPT ![i  1] = current, ![i] = prev]
However, the letdefinitions are not variables, they are just aliases for more complex expressions. Importantly, one cannot update the value of an expression that is defined with a letdefinition. In this sense, TLA+ is similar to functional languages, where side effects are carefully avoided and minimized.
In contrast to functional languages, the value of TLA+ is not in computing the result of a function application, but in producing sequences of states (called behaviors). Hence, some parts of a useful TLA+ specification should have side effects to record the states.
Idiom 1: Update state variables with assignments
Description
The idiom "Keep state variables to the minimum" tells us to store the minimum necessary state variables. By following this idiom, we develop the specification by writing constraints over the primed variables.
TLA+ comes with a great freedom of expressing constraints over variables.
While we love TLA+ for that freedom, we believe that constraints over primed
variables are sometimes confusing.
TLA+ uses the same glyph, =
for three separate purposes: assignment, asserting equality, and binding variables. But these are very different operations and have different semantics.
Issue 1
tl;dr: Use :=
(supplied by the Apalache.tla
module) instead of =
for assignment.
Consider the expression:
x' = x + 1
It is all clear here. The value of x
in the next states (there may be many)
is equal to val(x)+1
, where val(x)
is the value of x
in the current
state.
Wait. Is it clear? What if that expression was just the first line of the following expression:
x' = x + 1
=> x' = 3
This says, "if x'
is equal to x + 1
, then assign the value of 3
to x'
in the next state", which
implies that x'
may receive a value from the set:
{ 3 } \union { y \in Int: y /= val(x) + 1 }
But maybe the author of that specification just made a typo and never
meant to put the implication =>
in the first place. Actually, the intended
specification looks like follows:
x' = x + 1
\/ x' = 3
We believe that it is helpful to label the expressions that intend to denote the
values of the state variables in the next state. Apalache introduces the infix
operator :=
in the module Apalache.tla
for that purpose:
x' := x + 1
\/ x' := 3
Hence, it would be obvious in our motivating example that the author made a typo:
x' := x + 1
=> x' := 3
because the assignment x' := x + 1
does not express a boolean value
and so cannot be the antecedent of the conditional.
Issue 2
tl;dr: Use existential variables with the :=
operator for nondeterministic assignment.
Another common use of primed variables is to select the next value of a variable from a set:
x' \in { 1, 2, 3 }
This expression can be rewritten as an equivalent one:
\E y \in { 1, 2, 3 }:
x' = y
Which one to choose? The first one is more concise. The second one highlights
the important effect, namely, nondeterministic choice of the next value of x
.
When combined with the operator :=
, the effect of nondeterministic choice is
clearly visible:
\E y \in { 1, 2, 3 }:
x' := y
In fact, every constraint over primes can be translated into the existential form. For instance, consider the expression:
x' * x' = 4
It can be written as:
\E y \in Int:
/\ y * y = 4
/\ x' := y
Advantages

The reader clearly sees the writer's intention about the updates to the primed variables.

Nondeterminism is clearly isolated in existential choice:
\E y \in S: x' := y
. If there is no existential choice, the assignment is deterministic. 
When the existential form is used, the range of the values is clearly indicated. This is in contrast to the negated form such as:
~(x' = 10)
. 
TLC treats the expressions of the form
x' = e
andx' \in S
as assignments, as long asx'
is not bound to a value. 
Apalache uses assignments to decompose the specification into smaller pieces. Although Apalache tries to find assignments automatically, it often has to choose from several expressions, some of them may be more complex than the others. By using the
:=
operator, Apalache gets unambiguous instructions about when assignment is taking place
Disadvantages
 Replacing
x' \in S
with\E y \in S: x' := y
makes the specification a bit larger.
Example
The following example deliver.tla demonstrates how
one can clearly mark assignments using the :=
operator.
 MODULE deliver 
(*
* A simple specification of two processes in the network: sender and receiver.
* The sender sends messages in sequence. The receiver may receive the sent
* messages out of order, but delivers them to the client in order.
*
* Igor Konnov, 2020
*)
EXTENDS Integers, Apalache
VARIABLES
sentSeqNo, \* the sequence number of the next message to be sent
sent, \* the messages that are sent by the sender
received, \* the messages that are received by the receiver
deliveredSeqNo \* the sequence number of the last delivered message
(* We assign to the unprimed state variables to set their initial values. *)
Init ==
/\ sentSeqNo := 0
/\ sent := {}
/\ received := {}
/\ deliveredSeqNo := 1
(* Subsequent assignments are all to primed variables, designating changed values
after state transition. *)
Send ==
/\ sent' := sent \union {sentSeqNo}
/\ sentSeqNo' := sentSeqNo + 1
/\ UNCHANGED <<received, deliveredSeqNo>>
Receive ==
(* We make the nonderministic assignment explicit, by use of existential quantification *)
/\ \E msgs \in SUBSET (sent \ received):
received' := received \union msgs
/\ UNCHANGED <<sentSeqNo, sent, deliveredSeqNo>>
Deliver ==
/\ (deliveredSeqNo + 1) \in received
/\ deliveredSeqNo' := deliveredSeqNo + 1
\* deliver the message with the sequence number deliveredSeqNo'
/\ UNCHANGED <<sentSeqNo, sent, received>>
Next ==
\/ Send
\/ Receive
\/ Deliver
Inv ==
(deliveredSeqNo >= 0) => deliveredSeqNo \in sent
================================================================================
Idiom 2: Apply primes only to state variables
(Until you learn how prime actually works!)
Description
In many formal languages, the notation x'
denotes the value that a variable
x
has after the system has fired a transition. The reason for having both x
and x'
is that the transitions are often described as relations over unprimed
and primed variables, e.g., x' = x+1
. It is easy to extend this idea to
vectors of variables, but for simplicity we will use only one variable.
TLA+ goes further and declares prime ('
) as an operator! This operator distributes over
any state variables in the scope of its application. For example, assume that we
evaluate a TLA+ expression A
over x
and x'
, and v[i]
and v[i+1]
are
meant to be the values of x
in the ith state and i+1th state, respectively.
Then x
is evaluated to v[i]
and x'
is evaluated to v[i+1]
. Naturally,
x + 3
is evaluated to v[i] + 3
, whereas x' + 4
is evaluated to v[i+1] + 4
. We can go further and evaluate (x + 4)'
, which can be rewritten as x' + 4
.
Intuitively, there is nothing wrong with the operator "prime". However, you have to understand this operator well, in order to use it right. For starters, check the warning by Leslie Lamport in Specifying Systems on page 82. The following example illustrates the warning:
 MODULE clocks3 
(* Model a system of three processes, each one equipped with a logical clock *)
EXTENDS Integers, Apalache
VARIABLES clocks, turn
\* a shortcut to refer to the clock of the process that is taking the step
MyClock == clocks[turn]
\* a shortcut to refer to the processes that are not taking the step
Others == DOMAIN clocks \ {turn}
Init ==
/\ clocks := [p \in 1..3 > 0] \* initialize the clocks with 0
/\ turn := 1 \* process 1 makes the first step
Next ==
\* update the clocks of the processes (the section Example shows a better way)
/\ \E f \in [1..3 > Int]:
clocks' := f
\* increment the clock of the process that is taking the step
/\ MyClock' = MyClock + 1
\* all clocks of the other processes keep their clock values
/\ \A i \in Others:
clocks'[i] = clocks[i]
\* use roundrobin to decide who makes the next step
/\ turn' := 1 + (turn + 1) % 3
===============================================================================
Did you spot a problem in the above example? If not, check these lines again:
\* increment the clock of the process that is taking the step
/\ MyClock' = MyClock + 1
The code does not match the comment. By writing MyClock'
, we get
(clocks[turn])'
that is equivalent to clocks'[turn']
. So our constraint
says: Increment the clock of the process that is taking the next step. By
looking at the next constraint, we can see that Next
can never be evaluated
to true (a logician would say that Next
is "unsatisfiable"):
\* all clocks of the other processes keep their clock values
/\ \A i \in Others:
clocks'[i] = clocks[i]
Our intention was to make the specification easier to read, but instead we have introduced a deadlock in the system. In a larger specification, this bug would be much harder to find.
We recommend to follow this simple rule: Apply primes only to state variables
Can we remove the "prime" operator altogether and agree to use x
and x'
as
names of the variables? Not really. More advanced features of TLA+ require this
operator. In a nutshell, TLA+ is built around the idea of refinement, that is,
replacing an abstract specification with a more detailed one. Concretely, this
idea is implemented by module instances in TLA+. It often happens that
refinement requires us to replace a state variable of the abstract
specification with an operator of the detailed specification. Voilà. You have
to apply prime to an expression. For the details,
see Chapter 5 and pages 312313 of Specifying Systems.
Advantages

It is easy to see, whether the specification author intended to talk about the variables in the next state or about the variable in the current state.

It is harder to make an unexpected substitution mistake, as in the above example.
Disadvantages
 Sometimes, the operator "prime" helps us in avoiding code duplication.
For instance, you can write a state invariant
Inv
and later evaluate it against a next state by simply writingInv'
. However, you have to be careful about propagation of primes inInv
.
Example
A better version of the clocks
example applies prime only to state variables.
By doing so, we notice that the specification can be further simplified:
 MODULE clocks3_2 
(* Model a system of three processes, each one equipped with a digital clock *)
EXTENDS Integers, Apalache
VARIABLES clocks, turn
Init ==
/\ clocks := [p \in 1..3 > 0] \* initialize the clocks with 0
/\ turn := 1 \* process 1 makes the first step
Next ==
\* update the clocks of the processes
/\ clocks' :=
[p \in 1..3 >
IF p = turn THEN clocks[turn] + 1 ELSE clocks[p]]
\* use roundrobin to decide who makes the next step
/\ turn' := 1 + (turn + 1) % 3
===============================================================================
Idiom 7: Use Boolean operators in actions, not IFTHENELSE
author: Gabriela Moreira
Description
TLA+ provides an IFTHENELSE
operator, and it can be pretty tempting to use it for flow control, as it's done in procedural programming. However, TLA+ is about transitions over a state machine, and a transitiondefining action declared with IFTHENELSE
can be more complex than 2 actions declared without it. Considering that any expression of the form IF b THEN x ELSE y
, where x
and y
are Booleans, can be rewritten as (b /\ x) \/ (~b /\ y)
, there's a pattern we can apply to get rid of some potentially troublesome IFTHENELSE
definitions.
The IFTHENELSE
operator can be used either to define a value, or to branch some action as a sort of flow control. Defining values with IFTHENELSE
is common practice and is similar to the use of IF
expressions in declarative programming languages. However, flow control in TLA+ can be done naturally by behavior definition through actions, making the use of IFTHENELSE
for flow control unnecessary. This idiom aims to clarify different usages of IFTHENELSE
expressions, keeping in mind the TLA+ essence of declaring actions to define transitions.
When to use IFTHENELSE
When the result is not Boolean
When the IFTHENELSE
expression doesn't evaluate to a Boolean value, it cannot be rewritten using Boolean operators, so this idiom doesn't apply. For example:
SafeDiv(x, y) == IF y /= 0 THEN x/y ELSE 0
When the result is a state formula
State formulas are formulas that don't contain any action operator (e.g. primed variables, UNCHANGED
). Using IFTHENELSE
on this type of formula can make it easier to read in some situations, and don't come with any disadvantage. This example state formula uses IFTHENELSE
to return a Boolean value:
ValidIdentity(person) == IF Nationalized(person) THEN ValidId(person) ELSE ValidPassport(person)
Although it could be rewritten with Boolean operators, it doesn't read as nicely:
ValidIdentity(person) == \/ /\ Nationalized(person)
/\ ValidId(person)
\/ /\ ~Nationalized(person)
/\ ValidPassport(person)
When there are dependent conditions
Nesting IFTHENELSE
expressions can be useful when there is a dependency between the conditions where some conditions are only relevant if other conditions are met. This is an example where using an IFTHENELSE
expressions is clearer than the Boolean operator form. Consider the following:
IF c1
THEN a1
ELSE IF c2
THEN a2
ELSE IF
...
ELSE IF cn
THEN an
ELSE a
The Boolean operator version is quite verbose:
\/ c1 /\ a1
\/ ~c1 /\ c2 /\ a2
\/ ...
\/ ~c1 /\ ... /\ ~c_{n1} /\ cn /\ an
\/ ~c1 /\ ... /\ ~c_{n1} /\ ~cn /\ a
When (and how) not to use IFTHENELSE
Mixing IFTHENELSE
expressions with action operators introduces unnecessary branching to definitions that could be selfcontained and look more like a transition definition.
Withdraw(amount) == IF balance >= amount
THEN /\ balance' = balance  amount
/\ response' = "SUCCESS"
ELSE /\ UNCHANGED balance
/\ response' = "FAILURE"
We could separate the two branches into their own actions with clarifying names and explicit conditions, and use a disjunction between the two actions instead of the IFTHENELSE
block:
WithdrawSuccess(amount) == /\ balance >= amount
/\ balance' = balance  amount
/\ response' = "SUCCESS"
WithdrawFailure(amount) == /\ balance < amount
/\ response' = "FAILURE"
/\ UNCHANGED balance
Withdraw(amount) == WithdrawSuccess(amount) \/ WithdrawFailure(amount)
Advantages
 Each action declares fewer transitions, so it's easier to reason about it
 A disjunction of actions is closer to a union of transition relations than an
IFTHENELSE
expression is  Nested
IFTHENELSE
expressions are an extrapolation of these problems and can overconstrain some branches if not done carefully. Using different actions defining its conditions explicitly leaves less room for implicit wrong constraints that anELSE
branch allows. See the example below.
Assuming C1()
is a condition for A1()
and C2()
is a condition for A2()
:
Next == IF C1()
THEN A1()
ELSE
IF C2()
THEN A2()
ELSE A3()
What if C1() /\ C2()
is true? In this case, only A1()
would be enabled, which is incorrect.
Next == \/ /\ C1()
/\ A1()
\/ /\ C2()
/\ A2()
\/ A3()
This second definition can allow more behaviors than the first one (depending on whether C1()
and C2()
overlap), and these additional behaviors can be unintentionally left out when IFTHENELSE
is used without attention.
Disadvantages
A disjunction in TLA+ may or may not represent nondeterminism, while an IFTHENELSE
is incapable of introducing nondeterminism. If it's important that readers can easily differentiate deterministic and nondeterministic definitions, using IFTHENELSE
expressions can help to make determinism explicit.
Idiom 15: Replace sets of mixed records with disjoint unions
Message sets are canonically modeled as sets of records with mixed types. While the current type system supports this, in the future, Apalache is likely going to change support for these kinds of sets and implement stricter typechecking. See this issue for a discussion. This document aims to provide instructions for users to migrate their specs to maintain type compatibility in the future (and improve the performance and robustness of current specs in the present).
The common approach
Apalache allows mixed sets of records, by defining the type of the set to be Set(r)
, where r
is the record type which contains all of the fields, which are held by at least one set member. For example:
{ [x: Int], [y: Str] }
would have the type Set([x:Int,y:Str])
. The only constraints Apalache imposes are that, if two set elements declared the same field name, the types of the fields have to match. Consequently, given
A == { [x: Int, z: Bool], [y: Str, z: Bool] }
B == { [x: Int, z: Bool], [y: Str, z: Int] }
A
is considered well typed, and is assigned the type Set([x:Int, y:Str, z:Bool])
, whereas B
is rejected by the type checker.
The treatment of record types was implemented in this fashion, to maintain backwardcompatibility with specifications of messagebased algorithms, which typically encoded different message types as records of the shape [ type: Str, ... ]
, where all messages shared a disambiguation filed (commonly named type
), the value of which described the category of the message. Additional fields depended on the value of type
.
The bellow snippet from Paxos.tla demonstrates this convention:
\* The set of all possible messages
Message == [type : {"1a"}, bal : Ballot]
\union [type : {"1b"}, acc : Acceptor, bal : Ballot,
mbal : Ballot \union {1}, mval : Value \union {None}]
\union [type : {"2a"}, bal : Ballot, val : Value]
\union [type : {"2b"}, acc : Acceptor, bal : Ballot, val : Value]
Ultimately, this approach both disagrees with our interpretation of the purpose of a typesystem for TLA+, as well as introduces unsoundness, in the sense that it makes it impossible, at the typechecking level, to detect recordfield access violations. Consider the following:
\E m \in Message: m.type = "1a" /\ m.mbal = 1
As defined above, messages for which m.type = "1a"
do not define a field named mbal
, however, the type of Message
is Set([type: Str, ..., mbal: Int, ...])
, which means, that m
is assumed to have an mbal
field, typed Int
. Thus, this access error can only be caught much later in the modelchecking process, instead of at the level of static analysis provided by the typechecker.
The proposed changes
This section outlines a proposed migration strategy, to replace such sets in older specifications. The convention presented in this section works with both the current version of Apalache, as well as the next iteration of the typechecker, currently in development.
Suppose we use messages with types t1,...,tn
in the specification and a message set variable msgs
, like in the snippet below:
VARIABLE
\* @type: Set( [ type: Str, x1: a1, ..., xn: an, ... ] );
msgs
...
\* Assuming S1: Set(a1), ..., Sn: Set(an)
\* @type: Set( [ type: Str, x1: a1, ..., xn: an, ... ] );
Message == [type : {"t1"}, x1: S1, ...]
\union ...
\union [type : {"tn"}, xn: Sn, ...]
...
TypeOk: msgs \subseteq Message
We propose the following substitution: Instead of modeling the union of all messages as a single set, we model their disjoint union explicitly, with a record, in the following way:
\* @type: [ int: Set([x: Int]), str: Set([y: Str]) ];
Messages == [
t1: [x1: S1, ...],
...,
tn: [xn: Sn, ...]
]
This way, Messages.t1
represents the set of all messages m
, for which m.type
would have been equal to "t1" in the original implementation, that is, [type: {"t1"}, x1: S1, ...]
.
For example, assume the original specification included
Messages == [type: {"t1"}, x: {1,2,3}] \union [type: {"t2"}, y:{"a","b","c"}]
that is, defined two types of messages: "t1", with an integervalued field "x" and "t2" with a stringvalued field "y". The type of any m \in Messages
would have been [type: Str, x: Int, y: Str]
in the old approach.
The rewritten version would be:
Messages == [ t1: [x:{1,2,3}], t2: [y:{"a","b", "c"}] ]
If we took m: [ t1: Set([x: Int]), t2: Set([y: Str]) ]
, m
would be a record pointing to two disjoint sets of messages (of categories "t1" and "t2" respectively). Values in m.t1
would be records with the type [x: Int]
and values in m.t2
would be records with the type [y: Str]
.
Note, however, that this approach also requires a change in the way messages are added to, or read from, the "set" of all messages (m
is a record representing a set, but not a set itself, in the new approach).
Previously, a message m
would be added by writing:
msgs' = msgs \union {m}
regardless of whether m.type = "t1"
or m.type = "t2"
. In the new approach, one must always specify which type of message is being added. However, the type no longer needs to be a property of the message itself, i.e. the type
field is made redundant.
To add a message m
of the category ti
one should write
msgs' = [ msgs EXCEPT !.ti = msgs.ti \union {m} ]
Similarly, reading/processing a message, which used to be done in the following way:
\E m \in msgs:
/\ m.type = "ti"
/\ A(m)
is replaced by
\E m \in msgs.ti: A(m)
Example
Below, we demonstrate this process on a concrete specification: The old approach:
 MODULE MsgSetOld 
VARIABLE
\* @type: Set( [ type: Str, x: Int, y: Str ] );
msgs,
\* @type: Bool;
found3,
\* @type: Bool;
foundC
Ints == {1,2,3}
Strs == {"a","b","c"}
\* @type: () => Set([ type: Str, x: Int, y: Str ] );
Messages == [ type: {"int"}, x: Ints ] \union [ type: {"str"}, y: Strs ]
Init ==
/\ msgs = {}
/\ found3 = FALSE
/\ foundC = FALSE
Send(m) == msgs' = msgs \union {m}
Rm(m) == msgs' = msgs \ {m}
AddIntMsg ==
/\ \E v \in Ints:
/\ Send( [type > "int", x > v] )
/\ UNCHANGED <<found3, foundC>>
CheckIntMsg ==
/\ \E m \in msgs:
/\ m.type = "int"
/\ found3' = ( m.x = 3 )
/\ Rm(m)
/\ UNCHANGED foundC
AddStrMsg ==
/\ \E v \in Strs:
/\ Send( [type > "str", y > v] )
/\ UNCHANGED <<found3, foundC>>
CheckStrMsg ==
/\ \E m \in msgs:
/\ m.type = "str"
/\ foundC' = ( m.y = "c" )
/\ Rm(m)
/\ UNCHANGED found3
Next ==
\/ AddIntMsg
\/ CheckIntMsg
\/ AddStrMsg
\/ CheckStrMsg
TypeOk == msgs \subseteq Messages
===============================================================================
The new approach:
 MODULE MsgSetNew 
VARIABLE
\* @type: [ int: Set([x: Int]), str: Set([y: Str]) ];
msgs,
\* @type: Bool;
found3,
\* @type: Bool;
foundC
Ints == {1,2,3}
Strs == {"a","b","c"}
\* no annotation required
Messages == [
int > [x: Ints],
str > [y: Strs]
]
Init ==
/\ msgs = [ int > {}, str > {} ]
/\ found3 = FALSE
/\ foundC = FALSE
\* @type: ([x: Int]) => Bool;
SendInt(m) ==
msgs' = [msgs EXCEPT !.int = msgs.int \union {m}]
\* @type: ([x: Int]) => Bool;
RmInt(m) ==
msgs' = [msgs EXCEPT !.int = msgs.int \ {m}]
\* @type: ([y: Str]) => Bool;
SendStr(m) ==
msgs' = [msgs EXCEPT !.str = msgs.str \union {m}]
\* @type: ([x: Int]) => Bool;
RmStr(m) ==
msgs' = [msgs EXCEPT !.str = msgs.str \ {m}]
AddIntMsg ==
/\ \E v \in Ints:
/\ SendInt( [x > v] )
/\ UNCHANGED <<found3, foundC>>
CheckIntMsg ==
/\ \E m \in msgs.int:
/\ found3' = ( m.x = 3 )
/\ RmInt(m)
/\ UNCHANGED foundC
AddStrMsg ==
/\ \E v \in Strs:
/\ SendStr( [y > v] )
/\ UNCHANGED <<found3, foundC>>
CheckStrMsg ==
/\ \E m \in msgs.str:
/\ foundC' = ( m.y = "c" )
/\ RmStr(m)
/\ UNCHANGED found3
Next ==
\/ AddIntMsg
\/ CheckIntMsg
\/ AddStrMsg
\/ CheckStrMsg
TypeOk ==
/\ msgs.int \subseteq Messages.int
/\ msgs.str \subseteq Messages.str
===============================================================================
Note that the new approach, in addition to being sound w.r.t. record types, also typically results in a performance improvement, since typeunification for record sets is generally expensive for the solver.
RFC 001: types and type annotations
Contributors (in alphabetical order): Shon Feder @shonfeder, Igor Konnov @konnov, Jure Kukovec @Kukovec, Markus Kuppe @lemmy, Andrey Kupriyanov @andreykuprianov, Leslie Lamport
This is an RFC that reviews a number of possibilities. A concrete proposal can be found in ADR002.
It is good to have a number of different opinions here. We have three questions:
 How to write types in TLA+.
 How to write type annotations (as a user).
 How to display and use inferred types.
1. How to write types in TLA+
Everybody has a different opinion here. It would be great to use the native TLA+ constructs to express types.
1.1. TypeOK syntax
The only way to write types in the TypeOK
style is by set membership.
For instance:
x
is an integer:x \in Int
f
is a function from an integer to an integer:f \in [Int > Int]
f
is a function from a set of integers to a set of integers:f \in [SUBSET Int > SUBSET Int]
r
is a record that has the fieldsa
andb
, wherea
is an integer andb
is a string:r \in [a: Int, b: STRING]
f
is a set of functions from a tuple of integers to an integer:f \in SUBSET [Int \X Int > Int]
Foo
is an operator of anInt
andSTRING
that returns anInt
:\A a \in Int: \A b \in STRING: Foo(a, b) \in Int
Bar
is a higherorder operator that takes an operator that takes anInt
andSTRING
and returns anInt
, and returns aBOOLEAN
.
Here is an approach to higherorder operators suggested by Leslie Lamport, where he uses a theorem:
THEOREM BarType ==
ASSUME NEW G(_,_),
\A x \in Int, y \in STRING : G(x,y) \in Int
PROVE Bar(G) \in BOOLEAN
Similar to that, we can write a theorem about the type of Foo
:
THEOREM FooType ==
\A a \in Int: \A b \in STRING: Foo(a, b) \in Int
1.2. Types as terms
A classical way of writing types is by using logical terms (or algebraic datatypes).
To this end, we can define a special module Types.tla
:
 MODULE Types 
\* Types as terms. The righthand side of an operator does not play a role,
\* but we define it as the corresponding set of values.
\* Alternatively, we could just define tuples of strings in rhs.
\* a type annotation operator that erases the type
value <: type == value
\* the integer type
IntT == Int
\* the Boolean type
BoolT == BOOLEAN
\* the string type
StrT == STRING
\* a set type
SetT(elemT) == SUBSET elemT
\* a function type
FunT(fromT, toT) == [fromT > toT]
\* a sequence type
SeqT(elemT) == Seq(elemT)
\* tuple types
Tup0T == {}
Tup1T(t1) == t1
Tup2T(t1, t2) == t1 \X t2
Tup3T(t1, t2, t3) == t1 \X t2 \X t3
\* and so on, e.g., Scala has 26 tuples. how many do we like to have?
\* Record types. We assume that field names are alphabetically ordered.
\* We cannot use recordset notation here,
\* as the field names are parameters. So I gave up here on giving corresponding sets.
Rec1T(f1, t1) == <<"Rec1", f1, t1>>
Rec2T(f1, t1, f2, t2) == <<"Rec2", f1, t1, f2, t2>>
Rec3T(f1, t1, f2, t2, f3, t3) == <<"Rec3", f1, t1, f2, t2, f3, t3>>
\* and so on
\* Operator types. No clear set semantics.
\* Note that the arguments can be operators as well!
\* So this approach gives us higherorder operators for free.
Oper0T(resT) == <<"Oper0", resT>>
Oper1T(arg1T, resT) == <<"Oper1", arg1T, res1T>>
Oper2T(arg1T, arg2T, resT) == <<"Oper2", arg1T, arg2T, res1T>>
\* and so on
======================
Assuming that we have some syntax for writing down that x
has type T
,
e.g., by writing x <: T
, we can write the above examples as follows:

x
is an integer:x <: IntT

f
is a function from an integer to an integer:f <: FunT(IntT, IntT)

f
is a function from a set of integers to a set of integers:f <: FunT(SetT(IntT), SetT(IntT))

r
is a record that has the fieldsa
andb
, wherea
is an integer andb
is a string:r <: Rec2T("a", IntT, "b", StrT)

f
is a set of functions from a tuple of integers to an integer:f <: SetT(FunT(Tup2T(IntT, IntT), IntT))

Foo
is an operator of anInt
andSTRING
that returns anInt
:\A a: \A b: Foo(a, b) <: Oper2(IntT, StrT, IntT)
. Here it gets tricky, as the TLA+ syntax does not allow us to mention an operator by name without applying it.

Bar
is a higherorder operator that takes an operator that takes anInt
andSTRING
and returns anInt
, and returns aBOOLEAN
.\A a, b, c: Bar(LAMBDA a, b: c) <: Oper1(Oper2(IntT, StrT, IntT), BoolT)
. Here we have to pull lambda operators, but at least it is possible to write down a type annotation.
1.3. Types as strings
Let us introduce the following grammar for types:
T ::= var  Bool  Int  Str  T > T  Set(T)  Seq(T) 
<<T, ..., T>>  [h_1 > T, ..., h_k > T]  (T, ..., T) => T
In this grammar, var
stands for a type variable, which can be instantiated with
concrete variable names such as a
, b
, c
, etc., whereas h_1
,...,h_k
are
field names. The rule T > T
defines a function, while the rule
(T, ..., T) => T
defines an operator.
Assuming that we have some syntax for writing down that x
has type T
,
e.g., by writing isType("x", "T")
, we can write the above examples as follows:

x
is an integer:isType("x", "Int")
. 
f
is a function from an integer to an integer:isType("f", "Int > Int")
. 
f
is a function from a set of integers to a set of integers:isType("f", "Set(Int) > Set(Int))"
. 
r
is a record that has the fieldsa
andb
, wherea
is an integer andb
is a string:isType("r", "[a > Int, b > Str])"
. 
f
is a set of functions from a tuple of integers to an integer:isType("f", "Set(<<Int, Int>> > Int))"
. 
Foo
is an operator of anInt
andSTRING
that returns anInt
:isType("Foo", "(Int, Str) => Int")
. 
Bar
is a higherorder operator that takes an operator that takes anInt
andSTRING
and returns anInt
, and returns aBOOLEAN
:isType("Bar", "((Int, Str) => Int) => Bool")
.
Note: We have to pass names as strings, as it is impossible to pass operator
names, e.g., Foo
and Bar
in other operators, unless Foo
and Bar
are nullary operators and isType
is a higherorder operator.
2. How to write type annotations (as a user)
Note: This question is not a priority, as we do not expect the user to write type annotations. However, it would be good to have a solution, as sometimes users want to write types.
Again, we have plenty of options and opinions here:
 Write type annotations by calling a special operator like
<:
or=
.  Write type annotations as assumptions.
 Write type annotations in comments.
 Write type annotations as operator definitions.
2.1. Type annotations with a special operator
This is the current approach in Apalache. One has to define an operator, e.g., <:
:
value <: type == value
Then an expression may be annotated with a type as follows:
VARIABLE S
Init ==
S = {} <: {Int}
Pros:
 Intutive notation, similar to programming languages.
Cons:
 This approach works well for expressions. However, it is not clear how to extend it to operators.
 This notation is more like type clarification, rather than a type annotation. Normally types are specified for names, that is, constants, variables, functions, operators, etc.
 Same expression may be annotated in a Boolean formula. What shall we do, if the
user writes:
x <: BOOLEAN \/ x <: Int
?
Note: The current approach has an issue. If one declares the operator <:
in
a module M
and then uses an unnamed instance INSTANCE M
in a module M2
,
then M
and M2
will clash on the operator <:
. We should define the operator
once in a special module Types
or Apalache
.
2.2. Type annotations as assumptions
One can use TLA+ syntax to write assumptions and assertions about the types. We are talking only about type assumptions here. The similar approach can be used to write theorems about types. Consider the following specification:
EXTENDS Sequences
CONSTANTS Range
VARIABLES list
Mem(e, es) ==
\E i \in DOMAIN es:
e = es[i]
In this example, the operator Mem
is polymorphic, whereas the types of Range
and list
are parameterized. If the user wants to
restrict the types of constants, variables, and operators, they could write (using the
TypeOK syntax):
ASSUME(Range \in SUBSET Int)
ASSUME(list \in Seq(Int))
ASSUME(\A e \in Int, \A es \in Seq(Int): Mem(e, es) \in BOOLEAN)
SANY parser only accepts the first assumption in the above example. The two other assumptions are rejected by the parser, as they refer to nonconstant values.
Moreover, using the proof syntax of
TLA+ Version 2,
we can annotate the
types of variables introduced inside the operators. For instance, we could
label the name i
as follows:
Mem(e, es) ==
\E i \in DOMAIN es:
e = es[i_use :: i]
And then write:
ASSUME(\A e, es, i: Mem(e, es)!i_use(i) \in Int)
Pros:
 The assumptions syntax is quite appealing, when writing types of CONSTANTS, VARIABLES, and toplevel operators.
Cons:
 The syntax gets verbose and hard to write, when writing types of LETIN operators and bound variables.
 It is not clear how to extend this syntax to higherorder operators.
 One cannot write assumptions about state variables.
2.3. Type annotations in comments
This solution basically gives up on TLA+ syntax and introduces a special syntax à la javadoc for type annotations:
EXTENDS Sequences
CONSTANTS Range \*@ Range: Set(Int)
VARIABLES list \*@ list: Seq(Int)
Mem(e, es) ==
\*@ Mem: (Int, Seq(Int)) => Bool
\E i \in DOMAIN es:
\*@ i: Int
e = es[i]
We have not come up with a good syntax for these annotaions. The above example demonstrates one possible approach.
Pros:
 Nonverbose, simple syntax
 Type annotations do not stand in the way of the specification author
 Type annotations may be collapsed, removed, etc.
 If we have an annotation preprocessor, we can use it for other kinds of annotations
Cons:
 As we give up on the TLA+ syntax, TLA+ Toolbox will not help us (though it is not uncommon for IDEs to parse javadoc annotations, so there is some hope)
 The users have to learn new syntax for writing type annotations and types
 We have to write an annotation preprocessor
2.4. Type annotations as operator definitions
Operators definitions and LETIN definitions can be written almost anywhere in TLA+. Instead of writing incomment annotations, we can write annotations with operator definitions (assuming types as strings, but this is not necessary):
EXTENDS Sequences
CONSTANTS Range
Range_type == "set(z)"
VARIABLES list
list_type == "seq(z)"
Mem(e, es) ==
LET Mem_type == "<a, seq(a)> => Bool" IN
\E i \in DOMAIN es:
LET i_type == "Int" IN
e = es[i]
Init ==
LET Init_type == "<> => Bool" IN
list = <<>>
Next ==
LET Next_type == "<> => Bool" IN
\E e \in Range:
LET e_type == "set(z)" IN
list' = Append(list, e)
Pros:
 No need for a comment preprocessor, easy to extract annotations from the operator definitions
Cons:
 Fruitless operator definitions
 Looks like a hack
3. How to display and use inferred types
TBD
Basically, use Language Server Protocol and introduce THEOREMs in the spirit of types as TypeOK.
ADR002: types and type annotations
authors  revision  revision date 

Shon Feder, Igor Konnov, Jure Kukovec  8  July 22, 2022 
This is an architectural decision record. For user documentation, check the Snowcat tutorial and Snowcat HOWTO.
This is a follow up of RFC001, which discusses plenty of alternative solutions. In this ADR002, we fix one solution that seems to be most suitable. The interchange format for the type inference tools will be discussed in a separate ADR.
 How to write types in TLA+ (Type Systems 1 and 1.2).
 How to write type annotations (as a user).
This document assumes that one can write a simple type checker that computes the types of all expressions based on the annotations provided by the user. Such an implementation is provided by the type checker Snowcat. See the manual chapter on Snowcat.
System engineers often want to write type annotations and quickly check types when writing TLA+ specifications. This document is filling this gap.
1. How to write types in TLA+
1.1. Type grammar (Type System 1, or TS1)
Upgrade warning. This system is replaced with Type System 1.2.
In October of 2022, we will stop supporting Type System 1. For the transition
period, pass features=norows
to Apalache, to enable Type System 1.
We write types as strings that follow the type grammar:
T ::= // Booleans
 'Bool'
// integers
 'Int'
// immutable constant strings
 'Str'
// functions
 T '>' T
// sets
 'Set' '(' T ')'
// sequences
 'Seq' '(' T ')'
// tuples
 '<<' T ',' ...',' T '>>'
// operators
 '(' T ',' ...',' T ')' '=>' T
// constant types (uninterpreted types)
 typeConst
// type variables
 typeVar
// parentheses, e.g., to change associativity of functions
 '(' T ')'
// imprecise records of Type System 1, removed in Type System 1.2
 '[' field ':' T ',' ...',' field ':' T ']'
field ::= <an identifier that matches [azAZ_][azAZ09_]*>
typeConst ::= <an identifier that matches [AZ_][AZ09_]*>
typeVar ::= <a single letter from [az]>
The type rules have the following meaning:
 The rules
Bool
,Int
,Str
produce primitive types: the Boolean type, the integer type, and the string type, respectively.  The rule
T > T
produces a function.  The rule
Set(T)
produces a set type over elements of typeT
.  The rule
Seq(T)
produces a sequence type over elements of typeT
.  The rule
<<T, ..., T>>
produces a tuple type over types that are produced byT
. Types at different positions may differ.  The rule
[field: T, ..., field: T]
produces a record type over types that are produced byT
. Types at different positions may differ. This syntax will change in Type System 1.2.  The rule
(T, ..., T) => T
defines an operator whose result type and parameter types are produced byT
.  The rule
typeConst
defines an uninterpreted type (or a reference to a type alias), look for an explanation below.  The rule
typeVar
defines a type variable, look for an explanation below.
Importantly, a multiargument function always receives a tuple, e.g., <<Int, Bool>> > Int
, whereas a singleargument function receives the type of its
argument, e.g., Int > Int
. The arrow >
is rightassociative, e.g., A > B > C
is understood as A > (B > C)
, which is consistent with programming
languages. If you like to change the priority of >
, use parentheses, as
usual. For example, you may write (A > B) > C
.
An operator always has the types of its arguments inside (...)
, e.g., (Int, Bool) => Int
and () => Bool
. If a
type T
contains a type variable, e.g.,
a
, then T
is a polymorphic type, in which a
can be instantiated with a monotype (a variablefree term). Type
variables are useful for describing the types of polymorphic operators. Although the grammar accepts an operator type
that returns an operator, e.g., Int => (Int => Int)
, such a type does not have a meaningful interpretation in TLA+.
Indeed, TLA+ does not allow operators to return other operators.
A type constant should be understood as a type we don't know and we don't want to know, that is, an uninterpreted type. Type constants are useful for fixing the types of CONSTANTS and using them later in a specification. Two different type constants correspond to two different  yet uninterpreted  types. If you know Microsoft Z3, a type constant can be understood as an uninterpreted sort in SMT. Essentially, values of an uninterpreted type can be only checked for equality.
Another use for a type constant is referring to a type alias, see Section 1.2. This is purely a convenience feature to make type annotations more concise and easier to maintain. We expect that only users will write type aliases: tools should always exchange data with types in the aliasfree form.
Examples.
x
is an integer. Its type isInt
.f
is a function from an integer to an integer. Its type isInt > Int
.f
is a function from a set of integers to a set of integers. Its type isSet(Int) > Set(Int)
.r
is a record that has the fieldsa
andb
, wherea
is an integer andb
is a string. Its type is[a: Int, b: Str]
. This is the old syntax for record types, see Type System 1.2.F
is a set of functions from a pair of integers to an integer. Its type isSet(<<Int, Int>> > Int)
.Foo
is an operator of an integer and of a string that returns an integer. Its type is(Int, Str) => Int
.Bar
is a higherorder operator that takes an operator that takes an integer and a string and returns an integer, and returns a Boolean. Its type is((Int, Str) => Int) => Bool
.Baz
is a polymorphic operator that takes two arguments of the same type and returns a value of the type equal to the types of its arguments. Its type is(a, a) => a
.Proc
andFaulty
are sets of the same type. Their type isSet(PID)
.
1.2. Type aliases
New syntax for type aliases
We introduce a special syntax for introducing type alises, which is defined by the following singlerule grammar:
A ::= aliasName "=" T
// an identifer in camel case, starting with a lowercase letter
aliasName ::= [az]+(?:[AZ][az]*)*
Typically, a type alias is defined via an annotation such as:
\* @typeAlias: setOfIntegers = Set(Int);
module_typedefs == TRUE
To refer to a type alias, we extend the grammar T
with one more option:
T ::= // all rules as above
 '$' aliasName
Whenever the type checker meets a reference like $aliasName
, it tries to
substitute $aliasName
with the type that was earlier defined with the type
alias. If no such alias is found, the type checker emits a type error.
Old syntax for type aliases
This is the old syntax. We will drop its support in September, 2022.
Similar to the old syntax, type aliases are defined via a onegrammar rule:
A_old ::= typeConst "=" T
In contrast to the new syntax, the rule A_old
uses the same syntax for
aliases as for type constants. This rule binds a type (produced by T
) to a
name (produced by typeConst
). As you can see from the definition of
typeConst
, the name should be an identifier in the upper case. The type
checker should use the bound type instead of the constant type.
In retrospect, this syntax confused the users and introduced usability issues. For instance, when the users forgot to include a type alias, the type alias was interpreted as a type constant, and the type checker showed incomprehensible error messages.
1.3. Type System 1.2, including precise records, variants, and rows
As discussed in ADR014, many users expressed the need for precise type checking for records in Snowcat. Records in untyped TLA+ are used in two capacities: as plain records and as variants. While the technical proposal is given in ADR014, we discuss the extension of the type grammar in this ADR002. If you do not know about row typing, it may be useful to check the Wikipedia page on Row polymorphism. We extend the grammar with new records, variants, and rows as follows:
// Type System 1.2
T12 ::=
// all types of Type System 1 except records
T
// A new record type with a fully defined structure.
// The set of fields may be empty. If typeVar is present,
// the record type is parameterized (typeVar must be of the 'row' kind).
 '{' field ':' T12 ',' ...',' field ':' T12 [',' typeVar] '}'
// A variant that contains several options,
// optionally parameterized (typeVar must be of the 'row' kind).
 variantOption '' ... '' variantOption '' [typeVar]
// A purely parameterized variant (typeVar must be of the 'row' kind).
 'Variant' '(' typeVar ')'
variantOption ::=
// A variant option with a fully defined structure,
// tagged with a name that is defined with 'identifier'
identifier '(' T12 ')'
// Special syntax for the rows, which is internal to the type checker.
row ::=
// A row with a fully defined structure
// (having at least one field).
 '(' field ':' T12 '' ...'' field ':' T12 ')'
// A row with a partially defined structure
// (having at least one field and ending with a variable of the 'row' kind).
 '(' field ':' T12 '' ...'' field ':' T12 '' typeVar ')'
Examples.

r1
is a record that has the fieldsa
andb
, wherea
is an integer andb
is a string. Its type is{ a: Int, b: Str }
. 
r2
is a record that has the fieldsa
of typeInt
andb
of typeStr
and other fields, whose precise structure is captured with a type variablec
. The type ofr2
is{ a: Int, b: Str, c }
. More precisely, the variablec
must be a row. For instance,c
can be equal to the row( f: Bool  g: Set(Int) )
; in this case,r2
would be a record of type{ a: Int, b: Str, f: Bool, g: Set(Int) }
. 
v1
is a variant that has one of the two possible shapes: Either it carries the tag
A
and an associated value of typeInt
, or  It carries the tag
B
and an associated value of typeBool
.  The type of
v1
isA(Int)  B(Bool)
.
 Either it carries the tag

v2
is a variant whose structure is entirely defined by the type variableb
. The type ofv2
isVariant(b)
. Note thatb
must be a row. For instance, it could be equal to( A: Int  B: Str )
.
Note that this syntax encapsulates rows in records and variants. We introduce the syntax for row types for completeness. Most likely, the users will never see messages that mention rows explicitly, without referring to records or variants.
1.4. Comments inside types
When you introduce records that have dozens of fields, it is useful to explain those fields right in the type annotations. For that reason, the type lexer supports oneline comments right in the type definitions. The following text presents a type definition that contains comments:
// packets are stored in a set
Set({
// unique sequence number
seqno: Int,
// payload hash
payloadHash: Str
})
The parser only supports oneline comments that starts with //
. Since type
annotations are currently written inside TLA+ comments, we feel that more
complex comments would complicate the matters.
1.5. Discussion
Our type grammar presents a minimal type system that, in our understanding,
captures all interesting cases that occur in practice. Obviously, this type
system considers illtyped some perfectly legal TLA+ values. For instance, we
cannot assign a reasonable type to {1, TRUE}
.
Legacy: Sets of tagged records in Type System 1. We can assign a reasonable type to the set:
{[type > "1a", bal > 1], [type > "2a", bal > 2, val > 3]}
This pattern often occurs in practice, e.g., see Paxos. The type of that
set will be Set([type: Str, bal: Int, val: Int])
, which is probably not what
you expected, but it is the best type we can actually compute without having
algebraic datatypes in TLA+. It also reminds the user that one must test the
field type
carefully.
In retrospect, we have found that almost every user of Apalache made typos in their record types (including the Apalache developers!). Hence, we are migrating to Type System 1.2.
Default: Sets of tagged records (variants) in Type System 1.2. Apalache provides the user with the module Variants.tla that implements operators over variant types.
Using variants, we can write the above set of messages as follows:
{
Variant("M1a", [bal > 1]),
Variant("M2a", [bal > 2, val > 3])
}
In Type System 1.2 (Section 1.3), this set has the type of a set over a variant type:
Set(
M1a({ bal: Int })
 M2a({ bal: Int, val: Int })
 a
)
Note that the variant type is openended (parameterized with a
) in the above
example, as we have not restricted its type. If we want to restrict the type to
exactly two options, we have to do that explicitly:
\* @typeAlias: MESSAGE = M1a({ bal: Int })  M2a({ bal: Int, val: Int });
LET \* @type: Int => MESSAGE;
M1a(bal) == Variant("M1a", [bal > bal])
IN
LET \* @type: (Int, Int) => MESSAGE;
M2a(bal, val) == Variant("M2a", [bal > bal, val > val])
IN
{ M1a(1), M2a(2, 3) }
Many programming languages would automatically declare constructors such as
M2a
and M1a
from the type declaration. Since we are extending TLA+ with
types, we have to introduce some idiomatic boilerplate code. This could be
handled better in a surface syntax that is designed with types in mind.
Other type systems. Type System 1 is also very much in line with the type system by Stephan Merz and Hernan Vanzetto, which is used internally by TLAPS when translating proof obligations in SMT. We introduce types for userdefined operators, on top of their types for TLA+ expressions that do not contain userdefined operators.
We expect that this type system will evolve in the future. That is why we call it Type System 1. Section 1.3 presents its extension to Type System 1.2. Feel free to suggest Type System 2.0 :)
2. How to write type annotations (as a user)
In the following, we discuss how to annotate different TLA+ declarations.
In the previous version of this document, we defined two operators:
AssumeType(_, _)
and _ ## _
. They are no longer needed as we have introduced Code annotations.
2.1. Annotating CONSTANTS and VARIABLES
Simply write an annotation @type: <your type>;
in a comment that precedes the declaration of a constant declaration or
a variable. See the following example:
CONSTANT
\* @type: Int;
N,
\* @type: Set(ID);
Base
VARIABLE
\* @type: ID;
x,
\* @type: Set(ID);
S
Why don't we use THEOREMs? It is tempting to declare the types of variables as theorems. For example:
THEOREM N <: "Int"
However, this theorem must be proven. A type inference engine would be able
to infer the type of N
and thus state such a theorem. However, with type
assumptions, the user merely states the variable types and the type checker
has a simple job of checking type consistency and finding the types of the
expressions.
2.2. Annotating operators
Again, write a type annotation @type: <your type>;
in a comment that precedes the operator declaration. For example:
\* @type: (a, Seq(a)) => Bool;
Mem(e, es) ==
(e \in {es[i]: i \in DOMAIN es})
Higherorder operators are also easy to annotate:
\* @type: ((a) => Bool, Seq(a)) => Int;
Find(Pred(_), es) ==
IF \E i \in DOMAIN es: Pred(es[i])
THEN CHOOSE i \in DOMAIN es: Pred(es[i])
ELSE 1
The following definition declares a (global) function, not an
operator. However, the annotation syntax is quite similar to that of the
operators (note though that we are using >
instead of =>
):
\* @type: (a > b) > Int;
CardDomain[f \in T] ==
LET \* @type: Set(a);
\* we could also write: "() => Set(a)" instead of just "Set(a)"
D == DOMAIN f
IN LET \* @type: (Int, Int) => Int;
PlusOne(p,q) == p + 1
IN FoldSet(PlusOne, 0, D)
In the definition of CardDomain
, we annotated the letdefinition D
with its type, though any type checker should be
able to compute the type of
D
from its context. So the type of D
is there for clarification. According to our type grammar, the type of D
should be () => Set(a)
, as D
is an operator. It is not obvious from the syntax: TLA+ blends in nullary operators with other names. We have found that LETdefinitions without arguments are so common, so it is more convenient to write the shorter type annotation, that is, just Set(a)
.
2.3. Dealing with bound variables
A number of TLA+ operators are defining bound variables. Following TLA+ Summary, we list these operators here (we omit the unbounded quantifiers and temporal quantifiers):
\A x \in S: P
\E x \in S: P
CHOOSE x: P
{x \in S: P}
{e: x \in S}
[x \in S > e}
We do not introduce any special annotation to support these operators. Indeed, they are all introducing bound variables that range over sets. In most cases, the type checker should be able to extract the element type from a set expression.
However, there are a few pathological cases arising from empty collections. For example:
/\ \E x \in {}: x > 1
/\ f = [x \in {} > 2]
/\ z \in DOMAIN << >>
Similar typing issues occur in programming languages, e.g., Scala and Java. In these rare cases, you can write an auxiliary LETdefinition to specify the type of the empty collection:
/\ LET \* @type: Set(Int);
EmptyInts == {}
IN
\E x \in EmptyInts: x > 1
/\ LET \* @type: Set(Str);
EmptyStrings == {}
IN
f = [x \in EmptyStrings > 2]
/\ LET \* @type: Seq(Int);
EmptyIntSeq == {}
IN
z \in DOMAIN EmptyIntSeq
The type checker uses the type annotation to refine the type of an empty set (or, of an empty sequence).
2.4. Introducing and using type aliases
A type alias is introduced with the annotation @typeAlias: <ALIAS> = <Type>;
.
Since it is convenient to group type aliases of a module MyModule
in one place, we usually use the following idiom:
\* @typeAlias: id = Int;
\* @typeAlias: entry = { a: $id, b: Bool };
MyModule_typedefs == TRUE
VARIABLE
\* @type: Set($entry);
msgs
\* @type: (Set($entry), $entry) => $entry;
Foo(ms, m) ==
msgs' = ms \union {m}
The use of the dummy operator is a convention followed to simplify reasoning about where type aliases belong, and to ensure all aliases are located in one place. The prefix such as the module name protects against name clashes when the module is extended or instantiated.
The actual rules around the placement of the @typeAlias
annotation allows more
flexibility:

You can define a type alias with
@typeAlias
anywhere you can define a@type
. 
The names of type aliases must be unique in a module.

There is no scoping for aliases within a module. Even if an alias is defined deep in a tree of LETIN definitions, it can be referenced at any level in the module.
3. Example
As an example that contains nontrivial type information, we chose the specification of Cigarette Smokers by @mryndzionek from TLA+ Examples. In this document, we focus on the type information and give a shorter version of the specification. For detailed comments, check the original specification.
 MODULE CigaretteSmokersTyped 
(***************************************************************************)
(* A specification of the cigarette smokers problem, originally *)
(* described in 1971 by Suhas Patil. *)
(* https://en.wikipedia.org/wiki/Cigarette_smokers_problem *)
(* *)
(* This specification has been extended with type annotations for the *)
(* demonstration purposes. Some parts of the original specification are *)
(* omitted for brevity. *)
(* *)
(* The original specification by @mryndzionek can be found here: *)
(* https://github.com/tlaplus/Examples/blob/master/specifications/CigaretteSmokers/CigaretteSmokers.tla *)
(***************************************************************************)
EXTENDS Integers, FiniteSets
CONSTANT
\* @type: Set(INGREDIENT);
Ingredients,
\* @type: Set(Set(INGREDIENT));
Offers
VARIABLE
\* @type: INGREDIENT > { smoking: Bool };
smokers,
\* @type: Set(INGREDIENT);
dealer
(* try to guess the types in the code below *)
ASSUME /\ Offers \subseteq (SUBSET Ingredients)
/\ \A n \in Offers : Cardinality(n) = Cardinality(Ingredients)  1
vars == <<smokers, dealer>>
(***************************************************************************)
(* 'smokers' is a function from the ingredient the smoker has *)
(* infinite supply of, to a BOOLEAN flag signifying smoker's state *)
(* (smoking/not smoking) *)
(* 'dealer' is an element of 'Offers', or an empty set *)
(***************************************************************************)
TypeOK == /\ smokers \in [Ingredients > [smoking: BOOLEAN]]
/\ dealer \in Offers \/ dealer = {}
\* @type: (Set(INGREDIENT), (INGREDIENT) => Bool) => INGREDIENT;
ChooseOne(S, P(_)) ==
(CHOOSE x \in S : P(x) /\ \A y \in S : P(y) => y = x)
Init ==
/\ smokers = [r \in Ingredients > [smoking > FALSE]]
/\ dealer \in Offers
startSmoking ==
/\ dealer /= {}
/\ smokers' = [r \in Ingredients >
[smoking > {r} \union dealer = Ingredients]]
/\ dealer' = {}
stopSmoking ==
/\ dealer = {}
(* the type of LAMBDA should be inferred from the types
of ChooseOne and Ingredients *)
/\ LET r == ChooseOne(Ingredients, LAMBDA x : smokers[x].smoking)
IN smokers' = [smokers EXCEPT ![r].smoking = FALSE]
/\ dealer' \in Offers
Next ==
startSmoking \/ stopSmoking
Spec ==
Init /\ [][Next]_vars
FairSpec ==
Spec /\ WF_vars(Next)
AtMostOne ==
Cardinality({r \in Ingredients : smokers[r].smoking}) <= 1
=============================================================================
ADR003: transition executor (TRex)
author  revision 

Igor Konnov  1 
Transition executor is a new abstraction layer between the model checker and the translator of TLA+ expressions to SMT. The goal of this layer is to do the following:
 encapsulate the interaction with:
 the translator to SMT (called
SymbStateRewriter
)  the SMT solver (accessed via
Z3SolverContext
)  the type checker (accessed via
TypeFinder
)
 the translator to SMT (called
 provide the model checker with an API for symbolic execution:
 independent of the assumptions about how satisfiability of TLA+ formulas is checked
 constraints can be added and removed incrementally, even if the background SMT solver is nonincremental (this is important as some constraints are better solved by incremental solvers and some constraints are better solved by offline solvers)
 the state of the symbolic execution (context) can be saved and restored on another machine (essential for a multicore or distributed model checker)
TRex can be thought of as an API for a satisfiability solver on top of TLA+ (in the fragment of KerA+). We can even say that TRex is a solver for TLA+, in contrast to an SMT solver, which is a solver for theories in firstorder logic. As TLA+ is built around the concepts of a state and a transition, the TRex API abstracts symbolic execution in terms of symbolic states and symbolic transitions.
Classes
The figure below shows the class diagram of the essential classes
in TRex. TransitionExecutor
provides the higher level (a model checker) with
an API for symbolic execution. TransitionExecutorImpl
is the implementation
of TransitionExecutor
. It maintains ExecutionContext
that interacts with
the lower layer: the translator to SMT, the SMT solver, and the type checker.
Importantly, there are two implementations of ExecutionContext
: an
incremental one (IncrementalExecutionContext
) and an offline one
(OfflineExecutionContext
). In contrast to the standard stack API of SMT
solvers (push/pop), ExecutionContext
operates in terms of differential
snapshots. The implementation decides on how to translate differential
snapshots into interactions with the SMT solver.
IncrementalExecutionContext
simply maintains the SMT context stack by calling
push
and pop
. When a snapshot must be taken, it simply returns the depth of
the context stack. Recovery from a snapshot is done by a sequence of calls to
pop. (IncrementalExecutionContext
is not able to recover to an arbitrary
snapshot that is not subsumed by its current stack.) Thus,
IncrementalExecutionContext
can be used for efficient interaction with an
incremental SMT solver on a single machine (even in a single thread, as Z3
contexts are not multithreaded).
OfflineExecutionContext
records calls to SMT with the wrapper
RecordingZ3SolverContext
. A snapshot produces an hierarchical log of calls to
SMT that can be replayed in another OfflineExecutionContext
, even on another
machine.
Interaction with TransitionExecutor
We demonstrate a typical interaction with TransitionExecutor
for the
following TLA+ specification, which has been preprocessed by the passes
preceding the model checker pass:
 MODULE Test 
EXTENDS Integers
CONSTANT N
VARIABLES x
ConstInit ==
N > 0
Init$0 ==
x = 10
Next$0 ==
x < 0 /\ x' = x + N
Next$1 ==
x >= 0 /\ x' = x  N
Inv ==
x >= 0
=======================================
The sequence diagram below shows how the sequential model checker translates
ConstInit
to SMT and then translates Init$0
.
The sequence diagram below shows how the sequential model checker translates
Next$0
and Next$1
to SMT. It first finds that Next$0
is disabled and
then it finds that Next$1
is enabled. The enabled transition is picked.
The sequence diagram below shows how the sequential model checker translates
~Inv
to SMT and checks, whether there is a concrete state that witnesses
the negation of the invariant.
As you can see, TransitionExecutor
is still offering a lot flexibility to the
model checker layer, while it is completely hiding the lower layer. We do not
explain how the parallel checker is working. This is a subject to another ADR.
To sum up, this layer is offering you a nice abstraction to write different model checking strategies.
ADR004: Syntax for Javalike annotations in TLA+ comments
author  revision 

Igor Konnov  2 
This ADR documents our decision on using Javalike annotations in comments. Our main motivation to have annotations is to simplify type annotations, as presented in ADR002. Hence, in the following text, we are using examples for type annotations. However, the annotations framework is not restricted to types. Similar to Java and Scala, we can use annotations to decorate operators with hints, which may aid the model checker.
1. What can be annotated
Annotations should be written in comments that are written in front of a declaration. The following declarations are supported:
 Constant declarations, e.g.,
CONSTANT N
.  Variable declarations, e.g.,
VARIABLE x
.  Operator declarations, including:
 Toplevel operator declarations, e.g.,
Foo(x) == e
.  Operators defined via LETIN, e.g.,
Foo(x) == LET Inner(y) == e IN f
.  Recursive operators, e.g.,
RECURSIVE Fact(_) Fact(n) == ...
 Recursive and nonrecursive functions including:
 Toplevel functions, e.g.,
foo[i \in Int] == e
.  Functions defined via LETIN, e.g.,
Foo == LET foo[i \in Int] == e IN f
For an example, see Section 3.
2. Annotations syntax
An annotation is a string that follows the grammar (question mark denotes optional rules):
Annotation ::= '@' tlaIdentifier ( '(' ArgList? ')'  ':' inlineArg ';' )?
ArgList ::= (Arg) ( ',' Arg )*
Arg ::= (string  integer  boolean  tlaIdentifier)
string ::= '"' <char sequence> '"'
integer ::= ''? [09]+
boolean ::= ('false'  'true')
inlineArg ::= <char sequence excluding ';' and '@'>
The sequence <char sequence>
is a sequence of characters admitted by the TLA+ parser:
 Any ASCII character except double quotes, control characters or backslash
\
 A backslash followed by another backslash, a single or double quote,
or one of the letters
f
,n
,r
ort
.
Examples. The following strings are examples of syntactically correct annotations:
@tailrec
@type("(Int, Int) => Int")
@require(Init)
@type: (Int, Int) => Int ;
@random(true)
@deprecated("Use operator Foo instead")
@range(0, 100)
The above examples are just syntactically correct. Their meaning, if there is any, is defined by the tool that is reading these annotations. Note that the example 3 is not following the syntax of Java annotations. We have introduced this format for oneargument annotations, especially, for type annotations. Its purpose is to reduce the visual clutter in annotations that accept a string as their only argument.
Currently, annotations are written in comments that precede a definition (see the explanation below). String arguments can span over multiple lines. For instance, the following examples demonstrate valid annotations inside TLA+ comments:
(*
@type: Int
=> Int
;
*)
\* @type: Int
\* => Int
\* ;
\* @hal_msg("Sorry,
\* I
\* CAN
\* do that,
\* Dave")
3. An annotated specification
The following specification shows how to write annotations, so they can be correctly parsed by the SANY parser and Apalache. Note the location of comments in front of: local operators, LETdefinitions, and recursive operators. Although these locations may seem to be suboptimal, this is how the SANY parser locates comments that precede declarations.
 MODULE Annotations 
EXTENDS Integers
CONSTANT
\* @type: Int;
N
VARIABLE
\* the singleargument annotation
\* @type: Set(Int);
set
\* @pure
\* using the Java annotations, a bit verbose:
\* @type(" Int => Int ")
Inc(n) == n + 1
\* @type: Int => Int;
LOCAL LocalInc(x) == x + 1
A(n) ==
LET \* @pure
\* @type: Int => Int;
Dec(x) == x + 1
IN
Dec(n)
RECURSIVE Fact(_)
\* @tailrec
\* @type: Int => Int;
Fact(n) ==
IF n <= 1 THEN 1 ELSE n * Fact(n  1)
\* @tailrec
\* @type: Int > Int;
FactFun[n \in Int] ==
IF n <= 1 THEN 1 ELSE n * FactFun[n  1]
===============================================================================
4. Implementation
The implementation of the annotation parser can be found in the class
at.forsyte.apalache.io.annotations.AnnotationParser
of the module
tlaio
, see AnnotationParser.
5. Discussion
Most likely, this topic does not deserve much discussion, as we are using the pretty standard syntax of Java annotations. So we are following the principle of the least surprise.
We also support the concise syntax for the annotations that accept a string as
a simple argument. For these annotations, we had to add the end marker ';'.
This is done because the SANY parser is pruning the linefeed character \n
,
so it would be otherwise impossible to find the end of an annotation.
ADR005: JSON Serialization Format
author  revision 

Jure Kukovec  1.1 
This ADR documents our decision on serializing the Apalache internal representation (IR) as JSON. The purpose of introducing such a serialization is to expose the internal representation in a standardized format, which can be used for persistent storage, or for analysis by thirdparty tools in the future.
1. Serializable classes
The following classes are serializable:

TLA+ expressions (see TlaEx) and subclasses thereof:
 Named expressions
NameEx
 Literal values
ValEx
for the following literals: Integers
TlaInt
 Strings
TlaStr
 Booleans
TlaBool
 Decimals
TlaDecimal
 Integers
 Operator expressions
OperEx
 LETIN expressions
LetInEx
 Named expressions

TLA+ declarations (see TlaDecl) and subclasses thereof:
 Variable declarations
TlaVarDecl
 Constant declarations
TlaConstDecl
 Operator declarations
TlaOperDecl
 Assumption declarations
TlaAssumeDecl
 Theorem declarations
TlaTheoremDecl
 Variable declarations

TLA+ modules, see TlaModule
2. Disambiguation field
Every serialization will contain a disambiguation field, named kind
. This field holds the name of the class being serialized. For example, the serialization of a NameEx
will have the shape
{
"kind": "NameEx"
...
}
3. Serializing tagged entities
Serializations of entities annotated with a TypeTag
will have an additional field named type
, containing the type of the expression (see ADR002, ADR004 for a description of our type system and the syntax for typesasstringannotations respectively), if the tag is Typed
, or Untyped
otherwise. For example, the integer literal 1
is represented by a ValEx
, which has type Int
and is serialized as follows:
{
"kind": "ValEx",
"type": "Int"
...
}
in the typed encoding, or
{
"kind": "ValEx",
"type": "Untyped"
...
}
in the untyped encoding.
4. Source information
Entities in the internal representation are usually annotated with source information, of the form {filename}:{startLine}:{startColumn}{endLine}:{endColumn}
, relating them to a file range in the provided specification (from which they may have been transformed as part of preprocessing).
JSON encodings may, but are not required to, contain a source
providing this information, of the following shape:
{
"source": {
"filename" : <FILENAME>,
"from" : {
"line" : <STARTLINE>,
"column" : <STARTCOLUMN>
},
"to" : {
"line" : <ENDLINE>,
"column" : <ENDCOLUMN>
}
}
}
or
{
"source": "UNKNOWN"
}
if no source information is available (e.g. for expressions generated purely by Apalache).
Serializations generated by Apalache are always guaranteed to contain a source
field entry.
Example:
{
"kind" : "NameEx",
"type" : "Int",
"name" : "myName",
"source": {
"filename" : "MyModule.tla",
"from" : {
"line" : 3,
"column" : 5
},
"to" : {
"line" : 3,
"column" : 10
}
}
}
5. Root wrapper
JSON serializations of one or more TlaModule
objects are wrapped in a root object with two required fields:
version
, the value of which is a string representation of the current JSON encoding version, shaped{major}.{minor}
, andmodules
, the value of which is an array containing the JSON encodings of zero or moreTlaModule
objects
It may optionally contain a field "name" : "ApalacheIR"
.
This document defines JSON Version 1.0. If and when a different JSON version is defined, this document will be updated accordingly.
Apalache may refuse to import, or trigger warnings for, JSON objects with obsolete versions of the encoding in the future.
Example:
{
"name": "ApalacheIR",
"version": "1.0",
"modules" = [
{
"kind": "TlaModule",
"name": "MyModule"
...
},
...]
}
6. General serialization rules
The goal of the serialization is for the JSON structure to mimic the internal representation as closely as possible, for ease of deserialization.
Concretely, whenever a class declares a field fld: T
, its serialization also contains a field named fld
, containing the serialization of the field value.
For example, if TlaConstDecl
declares a name: String
field, its JSON serialization will have a name
field as well, containing the name string.
If a class field has the type Traversable[T]
, for some T
, the corresponding JSON entry is a list containing serializations of the individual arguments. For example, OperEx
is variadic and declares args: TlaEx*
, so its serialization has an args
field containing a (possibly empty) list.
As a running example, take the expression 1 + 1
, represented with the correct type annotations as
OperEx(
oper = TlaArithOper.plus,
args = Seq(
ValEx( TlaInt( 1 ) )( typeTag = Typed( IntT1() ) ),
ValEx( TlaInt( 1 ) )( typeTag = Typed( IntT1() ) )
)
)( typeTag = Typed( IntT1() ) )
Since both subexpressions, the literals 1
, are identical, their serializations are equal as well:
{
"kind": "ValEx",
"type": "Int",
"value": {
"kind": "TlaInt",
"value": 1
}
}
Observe that we choose to serialize TlaValue
as a JSON object, which is more verbose, but trivial to deserialize. It has the following shape
{
"kind": <KIND> ,
"value": <VALUE>
}
The value
field depends on the kind of TlaValue
:
 For
TlaStr
: a JSON string  For
TlaBool
: a JSON Boolean  For
TlaInt(bigIntValue)
: If
bigIntValue.isValidInt
: a JSON number  Otherwise:
{ "bigInt": bigIntValue.toString() }
 If
 For
TlaDecimal(decValue)
: a JSON stringdecValue.toString
The reason for the nonuniform treatment of integers is that Apalache encodes its TLA+ integers as BigInt
, which means that it permits values for which .isValidInt
does not hold.
While it might seem more natural to encode the entire TlaValue
as a JSON primitive, without the added object layer we would have a much tougher time deserializing.
We would need a) a sensible encoding of BigInt
values, which are not valid integers, and b) a way to distinguish both variants of BigInt
, as well as decimals, when deserializing (since JSON is not typed).
We could encode all values as strings, but they would be similarly indistinguishable when deserializing. Importantly, the type
field of the ValEx
expression is not guaranteed to contain a hint, as it could be Untyped
Take jsonOf1
to be the serialization of ValEx( TlaInt( 1 ) )( typeTag = Typed( IntT1() ) )
shown above. The serialization of 1 + 1
is then equal to
{
"kind": "OperEx",
"type": "Int",
"oper": "PLUS",
"args": [jsonOf1, jsonOf1]
}
In general, for any given oper: TlaOper
of OperEx
, the value of the oper
field in the serialization equals oper.name
.
7. Implementation
The implementation of the serialization can be found in the class
at.forsyte.apalache.io.json.TlaToJson
of the module tlaimport
, see TlaToJson.
RFC006: Unit testing and propertybased testing of TLA+ specifications
authors  revision 

Igor Konnov, Vitor Enes, Shon Feder, Andrey Kuprianov, ...  2 
Abstract. This document discusses a framework for testing TLA+ specifications. Our first goal is to give the writers of TLA+ specifications an interactive approach to quickly test their specifications in the design phase, similar to unittesting in programming languages. Our second goal is to give the readers of TLA+ specifications a clear framework for dissecting TLA+ specifications, in order to understand them in smaller pieces. These ideas have not been implemented yet. We believe that the testing framework will enable the users of Apalache and TLC to write and read TLA+ specifications in a much more efficient manner than they do it today.
1. Long rationale
TLA+ is a specification language that was designed to be executable inside a human brain. Moreover, it was intended to run in the brains that underwent a specific software upgrade, called mathematical training. Many years have passed since then. We now have automatic tools that can run TLA+ in a computer (to some extent). Even more, these tools can prove or disprove certain properties of TLA+ specs.
Nowadays, we have two tools that aid us in writing a TLA+ spec: our brain and a model checker. Both these tools have the same problem. They are slow. Software engineers are facing a similar problem when they are trying to test their system against different inputs. Interestingly, software engineers have found a way around this problem. They first test the individual parts of the system and then they test the system as a whole. The former is done with unit tests, whereas the latter is done with integration tests. (Software engineers probably borrowed this approach from industrial engineers.) Unit tests are used almost interactively, to debug a small part of the system, while integration tests are run in a continuous integration environment, which is not interactive at all.
Actually, our brains also have a builtin ability of abstracting away from one part of a problem while thinking about the other part. That is why some of us can still win against automatic tools. Model checkers do not have this builtin ability. So it looks like when we are using TLC or Apalache, we are doing integration testing all the time. Unfortunately, when we are checking a specification as a whole, we rarely get a quick response, except for very small specs. This is hardly surprising, as we are interested in specifying complex systems, not the trivial ones.
Surprisingly, when we are writing large TLA+ specs, our interaction with the model checker looks more like an interaction with a Mainframe computer from the early days of computing than a modern interactive development cycle. We feed the model checker our specification and wait for hours in the hope that it gives us a useful response. If it does not, we have to make the specification parameters small enough for the model checker to do anything useful. If our parameters are already ridiculously small, we have to throw more computing power at the problem and wait for days. In contrast, verification tools for programs are meant to be much more interactive, e.g., see Dafny and Ivy.
Why cannot we do something like Unit testing in Apalache? We believe that we actually can do that. We can probably do it even better by implementing Propertybased testing, that is, test parts of our specifications against a large set of inputs instead of testing it against a few carefully crafted inputs.
2. A motivating example
Let's consider a relatively simple distributed algorithm as an example. The repository of TLA+ examples contains the wellknown leader election algorithm called LCR (specified in TLA+ by Stephan Merz). The algorithm is over 40 years old, but it is tricky enough to be still interesting. To understand the algorithm, check Distributed Algorithms by Nancy Lynch.
As the description suggests, when we fix N
to 6
and Id
to
<<27, 4, 42, 15, 63, 9>>
, TLC checks that the spec satisfies the invariant
Correctness
in just 11 seconds, after having explored 40K states.
Of course, had we wanted to check the property for all possible combinations
of six unique identifiers in the range of 1..6
, we would had to run TLC
6! = 720
times, which would take over 2 hours.
In Apalache, we can setup a TLA+ module instance, to check all instances of the algorithm that have from 2 to 6 processes:
 MODULE ChangRobertsTyped_Test 
(*
* A test setup for ChangRobertsTyped.
*)
EXTENDS Integers, Apalache
\* a copy of constants from ChangRobertsTyped
CONSTANTS
\* @type: Int;
N,
\* @type: Int > Int;
Id
\* a copy of state variables from ChangRobertsTyped
VARIABLES
\* @type: Int > Set(Int);
msgs,
\* @type: Int > Str;
pc,
\* @type: Int > Bool;
initiator,
\* @type: Int > Str;
state
INSTANCE ChangRobertsTyped
\* We bound N in the test
MAX_N == 6
\* we override Node, as N is not known in advance
OVERRIDE_Node == { i \in 1..MAX_N: i <= N }
\* initialize constants
ConstInit ==
/\ N \in 2..MAX_N
/\ Id \in [ 1..MAX_N > Int ]
\* The below constraints are copied from ASSUME.
\* They are not enforced automatically, see issue #69.
Assumptions ==
/\ Node = DOMAIN Id
/\ \A n \in Node: Id[n] >= 0
/\ \A m,n \in Node : m # n => Id[m] # Id[n] \* IDs are unique
InitAndAssumptions ==
Init /\ Assumptions
By running Apalache as follows, we can check Correctness
for all
configurations of 2 to 6 processes and all combinations of Id
:
apalache check cinit=ConstInit \
init=InitAndAssumptions inv=Correctness ChangRobertsTyped_Test.tla
Actually, we do not restrict Id
to be a function from 1..N
to 1..N
, but
rather allow Id
to be a function from 1..N
to Int
. So Apalache should
be able to check an infinite number of configurations!
Unfortunately, Apalache starts to dramatically slow down after having explored 6 steps of the algorithm. Indeed, it does symbolic execution for a nondeterministic algorithm and infinitely many inputs. We could try to improve the SMT encoding, but that would only win us several steps more. A more realistic approach would be to find an inductive invariant and let Apalache check it.
It looks like we are trapped: Either we have to invest some time in verification, or we can check the algorithm for a few data points. In case of LCR, the choice of process identifiers is important, so it is not clear at all, whether a few data points are giving us a good confidence.
This situation can be frustrating, especially when you are designing a large protocol. For instance, both Apalache and TLC can run for hours on Raft without finishing. We should be able to quickly debug our specs like software engineers do!
3. An approach to writing tests
What we describe below has not been implemented yet. Apalache has all the necessary ingredients for implementing this approach. We are asking for your input to find an ergonomic approach to testing TLA+ specifications. Many of the following ideas apply to TLC as well. We are gradually introducing Apalachespecific features.
A complete specification can be found in ChangRobertsTyped_Test.tla.
Our idea is to quickly check operators in isolation, without analyzing the whole specification and without analyzing temporal behavior of the specification. There are three principally different kinds of operators in TLA+:

Stateless operators that take input parameters and return the result. These operators are similar to functions in functional languages.

Action operators that act on a specification state. These operators are similar to procedures in imperative languages.

Temporal operators that act on executions, which are called behaviors in TLA+. These operators are somewhat similar to regular expressions, but they are more powerful, as they reason about infinite executions.
3.1. Testing stateless operators
Consider the following auxiliary operator in the specification:
succ(n) == IF n=N THEN 1 ELSE n+1 \* successor along the ring
While this operator is defined in the specification, it is clear that it is
well isolated from the rest of the specification: We only have to know the
value of the constant N
and the value of the operator parameter n
.
\* Note that succ(n) is not referring to state variables,
\* so we can test it in isolation.
\*
\* @require(ConstInit)
\* @testStateless
Test_succ ==
\* This is like a propertybased test.
\* Note that it is exhaustive (for the range of N).
\A n \in Node:
succ(n) \in Node
This test is very simple. It requires succ(n)
to be in the set Node
, for
all values n \in Node
. The body of the operator Test_succ
is pure TLA+.
We annotate the operator with @testStateless
, to indicate that it should
be checked in a stateless context.
We should be able to run this test via:
apalache test ChangRobertsTyped_Test.tla Test_succ
We pass the test name Test_succ
, as we expect the test
command to run all
tests by default, if no test name is specified. Also, we have to initialize the
constants with ConstInit
, which we specify with the annotation
@require(ConstInit)
.
3.2. Testing actions
Testing stateless operators is nice. However, TLA+ is built around the concept
of a state machine. Hence, we believe that most of the testing activity will be
centered around TLA+ actions. For instance, the LCR specification has two
actions: n0
and n1
. Let's have a look at n0
:
n0(self) == /\ pc[self] = "n0"
/\ IF initiator[self]
THEN /\ msgs' = [msgs EXCEPT ![succ(self)] = @ \union {Id[self]}]
ELSE /\ TRUE
/\ msgs' = msgs
/\ pc' = [pc EXCEPT ![self] = "n1"]
/\ UNCHANGED << initiator, state >>
Assume we like to test it without looking at the rest of the system, namely,
the predicates Init
and n1
. First of all, we have to describe the states
that could be passed to the action n0
. In this section, we will just use
TypeOK (see Section 5 for a more finegrained control over the
inputs):
TypeOK ==
/\ pc \in [Node > {"n0", "n1", "n2", "Done"}]
/\ msgs \in [Node > SUBSET {Id[n] : n \in Node}]
/\ initiator \in [Node > BOOLEAN]
/\ state \in [Node > {"cand", "lost", "won"}]
Further, we specify what kind of outcome we expect:
\* Assertion that we expect to hold true after firing Action_n0.
Assert_n0 ==
\E n, m \in Node:
msgs'[n] = msgs[n] \union {m}
(Do you think this condition actually holds true after firing n0
?)
Finally, we have to specify, how to run the action n0
. In fact, if you look
at Next
, this requires us to write a bit of code, instead of just calling
n0
:
\* Execute the action under test.
\* Note that we decouple Assert_n0 from TestAction_n0.
\* The reason is that we always assume that TestAction_n0 always holds,
\* whereas we may want to see Assert_n0 violated.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @ensure(Assert_n0)
\* @testAction
TestAction_n0 ==
\E self \in Node:
n0(self)
The operator TestAction_n0
carries several annotations:
 The annotation
@require(TypeOK)
tells the framework thatTypeOK
should act as an initialization predicate for testingTestAction_n0
.  The annotation
@testAction
indicates thatTestAction_n0
should be tested as an action that is an operator over unprimed and primed variable.  The annotation
@ensure(Assert_n0)
tells the framework thatAssert_n0
should hold afterTestAction_n0
has been fired.
We should be able to run this test via:
apalache test ChangRobertsTyped_Test.tla TestAction_n0
Importantly, we decompose the test in three parts:
 preparing the states by evaluating predicates
ConstInit
andTypeOK
(similar toInit
),  executing the action by evaluating the action predicate
TestAction_n0
(like a single instance ofNext
),  testing the next states against the previous states by evaluating
the predicate
Assert_n0
(like an action invariant).
3.3. Testing executions
Engineers often like to test a particular set of executions to support their intuition, or to communicate an example to their peers. Sometimes, it is useful to isolate a set of executions to make continuous integration break, until the protocol is fixed. Needless to say, TLA+ tools have no support for this standard technique, though they have all capabilities to produce such tests.
Similar to testing an action in isolation, we propose an interface for testing a restricted set of executions as follows:
\* Execute a sequence of 5 actions, similar to TestAction_n0.
\* We test a final state with Assert_n0.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @ensure(Assert_noWinner)
\* @testExecution(5)
TestExec_n0_n1 ==
\* in this test, we only execute actions by processes 1 and 2
\E self \in { 1, 2 }:
n0(self) \/ n1(self)
In this case, we are using a different assertion in the @ensure
annotation:
\* We expect no winner in the final state.
\* Note that Assert_noWinner is a predicate over a trace of states.
\*
\* @typeAlias: state = { msgs: Int > Set(Int), pc: Int > Str,
\* initiator: Int > Bool, state: Int > Str };
\* @type: Seq($state) => Bool;
Assert_noWinner(trace) ==
LET last == trace[Len(trace)] IN
\A n \in Node:
last.state[n] /= "won"
Similar to TestAction_n0
, the test TestExec_n0_n1
initialized the state
with the predicate Prepare_n0
. In contrast to TestAction_n0
, the
test TestExec_n0_n1
does two other steps differently:

Instead of firing just one action, it fires up to 5 actions in a sequence (the order and action are chosen nondeterministically).

Instead of testing a pair of states, the predicate
Assert_noWinner
tests the whole trace. In our example, we check the final state of the trace. In general, we could test every single state of the trace.
We should be able to run this test via:
apalache test ChangRobertsTyped_Test.tla TestExec_n0_n1
If the test is violated, a counterexample should be produced in the file
counterexample_TestExec_n0_n1.tla
.
3.4. Test executions with temporal properties
We see this feature to have the least priority, as you can do a lot by writing trace invariants. Actually, you can check bounded lassos as trace invariants. So for bounded model checking, you can always write a trace invariant instead of a temporal formula.
When we wrote the test TestExec_n0_n1
, we did not think about the
intermediate states of an execution. This test was a functional test: It is
matching the output against the input. When reasoning about state machines,
we often like to restrict the executions and check the properties of those
executions.
Fortunately, we have all necessary ingredients in TLA+ to do
exactly this. Test TestExec_correctness_under_liveness
.
\* @type: Seq($state) => Bool;
Assert_noWinner(trace) ==
LET last == trace[Len(trace)] IN
\A n \in Node:
last.state[n] /= "won"
\* Execute a sequence of 5 actions, while using temporal properties.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @require(Liveness)
Predicates Correctness
and Liveness
are defined in the spec as follows:
(***************************************************************************)
(* Safety property: when node n wins the election, it is the initiator *)
(* with the smallest ID, and all other nodes know that they lost. *)
(***************************************************************************)
Correctness ==
\A n \in Node : state[n] = "won" =>
/\ initiator[n]
/\ \A m \in Node \ {n} :
/\ state[m] = "lost"
/\ initiator[m] => Id[m] > Id[n]
Liveness == (\E n \in Node : state[n] = "cand") => <>(\E n \in Node : state[n] = "won")
Since Correctness
is a state predicate, we wrap it with a temporal operator
to check it against all states of an execution:
\* @ensure(GlobalCorrectness)
\* @testExecution(5)
3.5. Discussion
As you can see, we clearly decompose a test in three parts:
 Preparing the states (like a small version of
Init
),  Executing the action (like a small version of
Next
),  Testing the next states against the previous states (like an action invariant). Alternatively, you can write an assertion over a trace.
In the rest of this section, we comment on the alternative approaches.
3.5.1. But I can do all of that in TLA+
True. TLA+ is an extremely expressive language.
Let's go back to the test TestAction_n0
that was explained in Section
3.2:
\* Execute the action under test.
\* Note that we decouple Assert_n0 from TestAction_n0.
\* The reason is that we always assume that TestAction_n0 always holds,
\* whereas we may want to see Assert_n0 violated.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @ensure(Assert_n0)
\* @testAction
TestAction_n0 ==
\E self \in Node:
n0(self)
Can we rewrite this test in pure TLA+? Yes, but it is an errorprone approach. Let's do it stepbystep.
First of all, there is no simple way to initialize constants in TLA+, as we did
with ConstInit
(this is an Apalachespecific feature). Of course, one can
restrict constants with ASSUME(...)
. However, assumptions about constants
are global, so we cannot easily isolate constant initialization in one test.
The canonical way of initializing constants is to define them in a TLC
configuration file. If we forget about all these idiosyncrasies of TLC, we
could just use implication (=>
), as we normally do in logic. So our test
TestAction_n0_TLA
in pure TLA+ would look like follows:
TestAction_n0_TLA ==
ConstInit => (* ... *)
Second, we want to restrict the states with TypeOK
. That should be easy:
TestAction_n0_TLA ==
ConstInit =>
TypeOK (* ... *)
Third, we want to execute the action n0
, as we did in TestAction_n0
.
The intuitive way is to write it like follows:
TestAction_n0_TLA ==
ConstInit =>
/\ TypeOK
/\ \E self \in Node:
n0(self)
(* ... *)
Although the above code looks reasonable, we cheated. It combines two steps in
one: It initializes states with TypeOK
and it simultaneously executes the
action n0
. If we tried that in TLC (forgetting about ConstInit
), that would
not work. Though there is nothing wrong about this constraint from the
perspective of logic, it just restricts the unprimed variables and primed
variables. There is probably a way to split this code in two steps by applying
the operator \cdot
, which is implemented neither in TLC, nor in Apalache:
TestAction_n0_TLA ==
ConstInit =>
TypeOK
\cdot
(
\E self \in Node:
n0(self)
(* ... *)
)
In these circumstances, a more reasonable way would be to introduce a new file
like MCTestAction_n0.tla
and clearly specify TypeOK
as the initial
predicate and the action as the next predicate. But we do not want
stateoftheart dictate us our behavior.
Finally, we have to place the assertion Assert_n0
. Let's try it this way:
TestAction_n0_TLA ==
ConstInit =>
TypeOK
\cdot
(
/\ \E self \in Node:
n0(self)
/\ Assert_n0
)
Unfortunately, this is not the right solution. Instead of executing n0
and checking that the result satisfies Assert_n0
, we have restricted
the next states to always satisfy Assert_n0
!
Again, we would like to write something like the implication Action => Assertion
, but we are not allowed do that with the model checkers for TLA+.
We can use the operator Assert
that is supported by TLC:
TestAction_n0_TLA ==
ConstInit =>
TypeOK
\cdot
(
/\ \E self \in Node:
n0(self)
/\ Assert(Assert_n0, "assertion violation")
)
This time it should conceptually work. Once n0
has been executed, TLC could
start evaluating Assert(...)
and find a violation of Assert_n0
. There is
another problem. The operator Assert
is a purely imperative operator, which
relies on the order in which the formula is evaluated. Hence, Apalache does not
support this operator and, most likely, it never will. The imperative semantics
of the operator Assert
is simply incompatible with logical constraints.
Period.
Phew. It was not easy to write TestAction_n0_TLA
. In principle, we could
fix this pattern and extract the test in a dedicated file MC.tla
to run
it with TLC or Apalache.
Let's compare it with TestAction_n0
. Which one would you choose?
\* Execute the action under test.
\* Note that we decouple Assert_n0 from TestAction_n0.
\* The reason is that we always assume that TestAction_n0 always holds,
\* whereas we may want to see Assert_n0 violated.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @ensure(Assert_n0)
\* @testAction
TestAction_n0 ==
\E self \in Node:
n0(self)
Another problem of TestAction_n0_TLA
is that it has a very brittle structure.
What happens if one writes ~ConstInit \/ TypeOK ...
instead of ConstInit => TypeOK ...
? In our experience, when one sees a logical formula, they expect
that an equivalent logical formula should be also allowed.
In the defense of TLA+, the issues that we have seen above are not the issues
of TLA+ as a language, but these are the problems of the TLA+ tooling. There
is a very simple and aesthetically pleasing way of writing TestAction_n0
in
the logic of TLA+:
TestAction_n0_pure_TLA ==
(ConstInit /\ TypeOK) =>
(\E self \in Node: n0(self)) => Assert_n0
The operator TestAction_n0_pure_TLA
could be probably reasoned about in TLA+
Proof System. From the automation perspective, it would require a
completely automatic constraintbased solver for TLA+, which we do not have.
In practice, this would mean either rewriting TLC and Apalache from scratch, or
hacking them to enforce the right semantics of the above formula.
3.5.2. Why annotations instead of special operators
The annotations @require
and @ensure
are not our invention. You can find
them in Designbycontract languages. In particular, they are used as pre
and postconditions in code verification tools, e.g., JML, Dafny, QUIC
testing with Ivy.
You could ask a reasonable question: Why cannot we introduce operators such
as Require
and Ensure
instead of writing annotations? For instance,
we could rewrite TestAction_n0
as follows:
TestAction_n0_no_annotations ==
/\ Require(ConstInit)
/\ Require(TypeOK)
/\ \E self \in Node:
n0(self)
/\ Ensure(Assert_n0)
The above test looks selfcontained, no annotations. Moreover, we have probably
given more power to the users: They could pass expressions to Require
and
Ensure
, or they could combine Require
and Ensure
in other ways and do
something great... Well, we have actually introduced more problems to the users
than solutions. Since logical formulas can be composed in a lot of ways, we
could start writing interesting things:
Can_I_do_that ==
/\ ~Require(ConstInit)
/\ Require(TypeOK) => Ensure(ConstInit)
/\ \E self \in Node:
n0(self) /\ Require(self \in { 1, 2 })
/\ Ensure(Assert_n0) \/ Ensure(Assert_noWinner)
It is not clear to us how the test Can_I_do_that
should be understood.
But what is written is kind of legal, so it should work, right?
The annotations gives us a clear structure instead of obfuscating the requirements in logical formulas.
For the moment, we are using Apalache annotations in code comments. However, TLA+ could be extended with ensure/require one day, if they prove to be useful.
4. Using tests for producing quick examples
It is often nice to see examples of test inputs that pass the test. Apalache has all the ingredients to do that that. We should be able to run a command like that:
apalache example ChangRobertsTyped_Test.tla TestAction_n0
The above call would produce example_TestAction_n0.tla
, a TLA+ description of
two states that satisfy the test. This is similar to counterexample.tla
,
which is produced when an error is found.
In a similar way we should be able to produce an example of an execution:
apalache example ChangRobertsTyped_Test.tla TestExec_n0_n1
5. Bounding the inputs
The following ideas clearly stem from Propertybased testing, e.g., we use generators similar to Scalacheck. In contrast to propertybased testing, we want to run the test not only on some random inputs, but to run it exhaustively on all inputs within a predefined bounded scope.
5.1. Using Apalache generators
Let's go back to the example in Section 3.2.
In TestAction_n0
we used TypeOK
to describe the states that can be used as
the input to the test. While this conceptually works, it often happens that
TypeOK
describes a large set of states. Sometimes, this set is even infinite,
e.g., when TypeOK
refers to the infinite set of sequences Seq(S)
.
In Apalache, we can use the operator Gen
that produces bounded data structures,
similar to Propertybased testing. Here is how we could describe the set
of input states, by bounding the size of the data structures:
\* Preparing the inputs for the second test. Note that this is a step of its own.
\* This is similar to an initialization predicate.
Prepare_n0 ==
\* the following constraint should be added automatically in the future
/\ Assumptions
\* let the solver pick some data structures within the bounds
\* up to 15 messages
/\ msgs = Gen(3 * MAX_N)
/\ pc = Gen(MAX_N)
/\ initiator = Gen(MAX_N)
/\ state = Gen(MAX_N)
\* restrict the contents with TypeOK,
\* so we don't generate useless data structures
/\ TypeOK
In Prepare_n0
, we let the solver to produce bounded data structures with
Gen
, by providing bounds on the size of every set, function, sequence, etc.
Since we don't want to have completely arbitrary values for the data
structures, we further restrict them with TypeOK
, which we conveniently have
in the specification.
The more scoped version of TestAction_n0
looks like following:
\* Another version of the test where we further restrict the inputs.
\*
\* @require(ConstInit)
\* @require(Prepare_n0)
\* @ensure(Assert_n0)
\* @testAction
TestAction2_n0 ==
\E self \in Node:
n0(self)
5.2. Using TLC Random
Leslie Lamport has recently introduced a solution that allows one to run TLC
in the spirit of Propertybased testing. This is done by initializing
states with the operators that are defined in the module Randomization
. For
details, see Leslie's paper on Inductive invariants with TLC.
6. Test options
To integrate unit tests in the standard TLA+ development cycle, the tools
should remember how every individual test was run. To avoid additional
scripting on top of the commandline interface, we can simply pass the tool
options with the annotation @testOption
. The following example demonstrates
how it could be done:
TestExec_correctness_under_liveness ==
\E self \in Node:
n0(self) \/ n1(self)
GlobalCorrectness == []Correctness
\* A copy of TestExec_n0_n1 that passes additional flags to the model checker.
\*
\* @require(ConstInit)
\* @require(TypeOK)
\* @ensure(Assert_noWinner)
\* @testExecution(5)
\* @testOption("tool", "apalache")
The test options in the above example have the following meaning:

The annotation
testOption("tool", "apalache")
runs the test only if it is executed in Apalache. For example, if we run this test in TLC, it should be ignored. 
The annotation
testOption("search.smt.timeout", 10)
sets the toolspecific optionsearch.smt.timeout
to 10, meaning that the SMT solver should time out if it cannot solve a problem in 10 seconds. 
The annotation
testOption("checker.algo", "offline")
sets the toolspecific optionchecker.algo
tooffline
, meaning that the model checker should use the offline solver instead of the incremental one. 
The annotation
testOption("checker.nworkers", 2)
sets the toolspecific optionchecker.nworkers
to2
, meaning that the model checker should use two cores.
By having all test options specified directly in tests, we reach two goals:
 We let the users to save their experimental setup, to enable reproducibility of the experiments and later redesign of specifications.
 We let the engineers integrate TLA+ tests in continuous integration, to make sure that updates in a specification do not break the tests. This would allow us to integrate TLA+ model checkers in a CI/CD loop, e.g., at GitHub.
7. What do you think?
Let us know:

email: igor at informal.systems.
ADR007: Apalache Package Structure Guidelines
author  revision 

Jure Kukovec  1 
This ADR documents the design policies guiding the package and dependency structure of Apalache. When introducing new classes, use the guidelines defined below to determine which package to place them in.
1. Level Structure
We define Apalache architecture in terms of enumerated levels (L0, L1, L2, etc.). Each level may only hold dependencies belonging to a lower level.
Levels are split into two strata:
 Interface core: Classes within the interface core relate to general TLA+ concepts, and are intended to be usable by 3rd party developers.
 Apalalache core: Classes in the Apalache core relate to specific Apalache or modelchecking functionality
Staking the "core" strata together gives us the complete stack implementing Apalache.
2. Interface core
Together, L0L6 make up the interface core. Notably, these levels are Apalacheagnostic. The concepts of these levels are packaged as follows:
tlair
: L0L3tlatypes
: L4tlaaux
: L5L6
L0: TLA+ IR
In L0, we have implementations of the following concepts:
 The internal representation of TLA+, with the exception of:
 Apalachespecific operators (
ApalacheOper
)
 Apalachespecific operators (
 Any utility required to define the IR, such as:
UID
sTypeTag
s & TT1
L1: Auxiliary structures
In L1, we have implementations of the following concepts:
 Metainformation about the IR, such as:
 Annotations
 Source information
 Generic utilities, such as:
 Unique name generators
L2: IR IO
In L2, we have implementations of the following concepts:
 String printers for TLAIR
 Reading from or writing to files (
.tla
,.json
, TLCformats)
L3: Basic IR manipulation
In L3, we have implementations of the following concepts:
 TLA+ transformations that:
 do not introduce operators excluded from L0
 have typecorrectness asserted by manual inspection
Any implementation of a L3 transformation must be explicitly annotated and special care should be made during PR review to ascertain typecorrectness. Whenever possible, unit tests should test for type correctness.
L4: Type analysis
In L4, we have implementations of the following concepts:
 Any calculus in a TLA+ type system, including:
 Unification
 Sub/supertype relations
 Any typerelated static analysis, such as:
 Type checking
 Type inference
L5: Selftyping Builder:
In L5, we have implementations of the following concepts:
 The TLAIR builder
L6: Typeguaranteed IR manipulation
In L6, we have implementations of the following concepts:
 TLA+ transformations that:
 do not introduce operators excluded from L0
 do not manually introduce IR constructors and typetags, but use the selftyping builder instead
3. Apalache core
L7+ make up the Apalache core. The concepts of these levels are packaged as follows:
apabase
: L7L9apapass
: L10apatool
: L11+
If individual passes in L10 are deemed complex enough, they may be placed in their own package (e.g. apabmc
)
L7: Extensions of TLA+
In L7, we have implementations of the following concepts:
 Any Apalachespecific operators, introduced into the IR (e.g.
ApalacheOper
)  Extensions of any printer/reader/writer classes with support for the above specific operators
L8: Apalachespecific transformations
In L8, we have implementations of the following concepts:
 TLA+ transformations that introduce operators from L7
Like L3 transformations, if the builder is not used, the transformation should be explicitly marked and inspected for typecorrectness.
L9: Glue
In L9, we have implementations of the following concepts:
 Classes and traits used to define Apalache workflow or auxiliary functions, but not core functionality, such as:
 Exception handling
 Logging
 Classes and traits related to 3rd party technologies, such as:
 SMT
 TLC config
L10: Pass implementations
In L10, we have implementations of the following concepts:
 Apalache passes and related infrastructure
L11: Module implementations
In L11, we have implementations of the following concepts:
 Apalache modules
L12: Tool wrapper:
In L12, we have implementations of the following concepts:
 Apalache CMD interface
L13: GUI
In L13, we have implementations of the following concepts:
 Graphical interface, if/when one exists
4. Aliases, factories and exceptions
Any global alias (e.g type TlaExTransformation = TlaEx => TlaEx
) or factory belongs to the lowest possible level required to define it. For example, type uidToExMap = Map[UID, TlaEx]
belongs to L0, since UID
and TlaEx
are both L0 concepts, so it should be defined in an L0 alias package, even if it is only being used in a package of a higher level. Local aliases may be used wherever convenient.
Any exception belongs to the lowest possible level, at which it can be thrown. For example, AssignmentException
belongs to L10, as it is thrown in the TransitionPass
.
5. Visualization
A visualization of the architecture and the dependencies can be found below. Black arrows denote dependencies between components within a package, while red arrows denote dependencies on packages. Dotted arrows denote conditional dependencies, subject to the concrete implementation of components that have not yet been implemented (marked TBD). Classes within the level boxes are examples, but are not exhaustive.
ADR008: Apalache Exceptions
author  revision 

Jure Kukovec  1 
This ADR documents the various exception thrown in Apalache, and the circumstances that trigger them.
1. User input exceptions
Exceptions in this family are caused by incorrect input from the user. All of these exceptions should exit cleanly and should NOT report a stack trace. We should be able to statically enforce that none of these exceptions can be left unhandled.
1.1. Parserlevel
Since we depend on Sany for parsing, Apalache rejects any syntax which Sany cannot parse. If Sany produces an exception Apalache catches it and rethrows a class extending
ParserException
The exception should report the following details:
 Location of at least one parser issue
1.2. Apalachespecific input exceptions
This category of exceptions deals with problems triggered by incorrect or incomplete information regarding Apalache inputs. Examples include:
 Malformed config files
 Incorrect or missing
OVERRIDE_
 Incorrect or missing
UNROLL_
 Problems with
init/next/cinit/...
Exceptions thrown in response to these issues should extend
ApalacheInputException
The exceptions should report the following details:
 If an input is missing: the name of the expected input
 If an input is incorrect: the way in which it is incorrect
1.3. Typerelated exceptions
This category of exceptions deals with problems arising from the typesystem used by Apalache. Examples include:
 Missing or incorrect type annotations
 Incompatibility of argument types and operator types
Exceptions thrown in response to these issues should extend
TypeingException
The exceptions should report the following details:
 If an annotation is missing: the declaration with the missing annotation and its location
 If the types of the arguments at a builtin operator application site are incompatible with the operator type: Both the computed and expected types, and the location of the application site
 If the types of the arguments at a userdefined operator application site are incompatible with the operator annotation: Both the computed and expected types, the operator declaration, and the location of the application site
1.4. Staticanalysis exceptions
This category of exceptions deals with problems arising from the various static analysis passes performed by Apalache. Examples include:
 Missing or incorrect variable assignments
 Other analyses we might run in the future
Exceptions thrown in response to these issues should extend
StaticAnalysisException
The exceptions should report the following details:
 The location in the specification where the analysis failed and the expected result
1.5. Unsupported language exceptions
This category of exceptions deals with user input, which falls outside of the TLA+ fragment supported by Apalache. Examples include:
 Unbounded quantification
 (Unbounded)
CHOOSE
SelectSeq
 Fragments of community modules
Exceptions thrown in response to these issues should extend
UnsupportedFeatureException
The exceptions should report the following details:
 The location of the unsupported expression(s)
2. Tool failures
These exceptions are caused by bugs in Apalache. They are fatal and should throw with a stack trace.
2.1. Assumption violations
Whenever possible, it's recommended to test against the assumptions of a given pass/transformation. If the assumptions are violated, an
AssumptionViolationException
should be thrown. It should report the following details:
 The assumption being violated
 The pass/class/method in which the assumption is made
2.2. Pass/Transformationspecific exceptions
Depending on the pass/transformation, specialized exceptions may be thrown, to indicate some problem in either the pipeline, malformed input, missing or incomplete metadata or any other issue that cannot be circumvented. The exceptions should include a reasonable (concise) explanation and, whenever possible, source information for relevant expressions.
3. Exception explanations
On their own, exceptions should include concise messages with all the relevant information components, outlined above. In addition to that, we should implement an advanced variant of ExceptionAdapter
, called ExceptionExplainer
, that is enabled by default, but can be quieted if Apalache is invoked with the flag quietexceptions
.
The purpose of this class is to offer users a comprehensive explanation of the exceptions defined in Section 1. Whenever an exception is thrown, ExceptionExplainer
should offer:
 Inlined TLA+ code, in place of source location references
 Examples of similar malformed inputs, if relevant
 Suggestions on how to fix the exception
 A link to the manual, explaining the cause of the exception
ADR009: Apalache Outputs
authors  last revised 

Jure Kukovec, Shon Feder  20211214 
This ADR documents the various files produced by Apalache, and where they get written to.
1. Categories of outputs
Files produced by Apalache belong to one of the following categories:
 Counterexamples
 Log files
 Intermediate state outputs
 Run analysis files
Counterexamples (if there are any) and basic logs should always be produced, but the remaining outputs are considered optional.
Each optional category is associated with a flag: writeintermediate
for intermediate state outputs and an individual flag for each kind of analysis. At the time of writing, the only analysis is governed by profiling
, for profiling results.
All such optional flags should default to false
.
2. Output directory and run directories
Apalache should define an outdir
parameter, which defines the location of all outputs produced by Apalache. If unspecified, this value should default to the working directory, during each run, but it should be possible to designate a fixed location, e.g. <HOME>/apalacheout/
.
Each run looks for a subdirectory inside of the outdir
with the same name as
the principle file provided as input (or, for commands that do not read input
from a file, named after the executed subcommand). This subdirectory is called
the specification's (resp. command's) namespace within the outdir
. All
outputs originating from that input file (resp. command) will be written to this
namespace.
Each run produces a subdirectory in its namespace, with the following name:
<DATE>T<TIME>_<UNIQUEID>
based on the ISO 8601 standard.
Here, <DATE>
is the date in YYYYMMDD
format, <TIME>
is the local time in HHMMSS
format.
Example file structure for a run executed on a file test.tla
:
_apalacheout/
└── test.tla
├── 20211105T225455_810261790529975561
Custom run directories
The rundir
flag can be used to specify an output directory into which
outputs are written directly. When the rundir
flag is specified, all
content included in the run directory specified above will also be written
into the directories specified by this argument.
3. Structure of a run directory
Each run directory outlined in the previous section, should contain the following:
 A file
run.txt
, containing the command issued for this run, with all implicit parameters filled in, so it can be replicated exactly  0 or more counterexample files
 a prefilled bug report file
BugReport.md
, if the tool exited with aFailureMessage
 if
writeintermediate
is set, a subdirectoryintermediate
, containing outputs associated with each of the passes in Apalache  an Apalache log file
detailed.log
.  an SMT log file
log0.smt
 Files associated with enabled analyses, e.g.
profilerules.txt
4. Global Configuration File
Apalache should define a global configuration file apalache.cfg
, e.g. in the <HOME>/.tlaplus
directory, in which users can define the default values of all parameters, including all flags listed in section 1, as well as outdir
. The format of the configuration file is an implementation detail and will not be specified here.
Apalache should also look for a local configuration file .apalache.cfg
, within
current working directory or its parents. If it finds such file, any configured
parameters therein will override the parameters from the global config file.
If a parameter is specified in the configuration file, it replaces the default value, but specifying a parameter manually always overrides config defaults. In other words, parameter values are determined in the following way, by order of priority:
 If
<flag>=<value>
is given, use<value>
, otherwise  if a local
.apalache.cfg
file is found (or is specified with theconfigfile
argument) containing<flag>: <value>
, then use<value>
, otherwise  if the global
apalache.cfg
specifies<flag>: <value>
use<value>
, otherwise  Use the defaults specified in the
ApalacheConfig
class.
RFC010: Implementation of Transition Exploration Server
Table of Contents
Problem
Users of Apalache have voiced a need for the following kinds of behavior:
 incremental results (that can be iterated through to exhaustively enumerate all counterexamples)
 interactive selection of checking strategies
 interactive selection of parameter values
The discussion around these needs is summarized and linked in #79 .
The proximal use cases motivating this feature are discovered in the needs of or collaborators working on implementing model based testing (MBT) via the Modelator project. @konnov has given the following articulation of the general concern:
For MBT we need some way to exhaustively enumerate all counterexamples according to some strategy. There could be different strategies that vary in terms of implementation complexity or the number of produced counterexamples
The upshot: we can provide value by adding a utility that will allow users to interactively and incrementally explore the transition systems defined by a given TLA+ spec.
Proposal
Overview
In the current architecture, there is a single mode of operation in which
 the user invokes Apalache with an initial configuration, including an input specification,
 the specification and configurations are parsed and preprocessed,
 and then the model checker proper drives the TransitionExecutor to effect symbolic executions verifying the specified properties for the specified model.
This RFC proposes the addition of a symbolic transition exploration server.
The server will allow a client to interact with the various steps of the
verification process. The client is thus empowered to call upon the parser and
preprocessors at will, and to drive the TransitionExecutor
interactively, via
a simplified API.
The specific functionality that should be available for interaction is enumerated in the Requirements.
As per previous discussions, interactivity will be supported by running a daemon (or "service") that serves incoming requests. Clients will interact via a simple, well supported protocol, that provides an online RPC interface.
As a followup, we can create our own frontend clients to interact with this server. In the near term, we envision a CLI, a web frontend, and editor integrations. Many aspects of such clients should be trivial, once we can use the client code generated by the gRPC library . See the gRPC Example for details.
Requirements
The following requirements have been gathered through conversation and discussion on our GitHub issues:
 TRANSEX.1::QCHECK.1  : enable checking specs without repeated JVM startup costs 730#issue855835332
 TRANSEX.1::EXPLORE.1  : enable exploring model checking results for a spec without repeated preprocessing costs 730#issue855835332
 TRANSEX.1::LOAD.1  : enable loading and unloading specs 730#issuecomment818201654
 TRANSEX.1::EXTENSIBLE.1  : The transition explorer should be extensible in the following ways:
 TRANSEX.1::EXTENSIBLE.1::CLOUD.1  : extensible for cloudbased usage
 TRANSEX.1::EXTENSIBLE.1::CLI.1  : extensible for interactive terminal usage 
 TRANSEX.1::SBMC.1  : expose symbolic model checking 730#issue855835332
 TRANSEX.1::SBMC.1::ADVANCE.1  : can incrementally advance symbolic states
 TRANSEX.1::SBMC.1::ROLLBACK.1  : can incrementally rollback symbolic states
 TRANSEX.1::SBMC.1::TRANSITIONS.1  : can send data on available transitions
 TRANSEX.1::SBMC.1::SELECT.1  : can execute a specific transition given a selected action
 TRANSEX.1::SBMC.1::COUNTER.1  : supports enumerating counterexamples 79#issue534407916
 TRANSEX.1::SBMC.1::PARAMS.1 
: supports enumerating parameter values (CONSTANTS
) that lead to a
counterexample
79#issuecomment576449107
Architecture
The interactive mode will take advantage of the TransitionExecutor
's
abstraction for writing different model checking strategies, to give the user an
abstracted, interactive interface for dynamically specifying checking
strategies.
I propose the following highlevel architecture:
 Use an RPC protocol to allow the client and server mutually transparent interaction. (This allows us to abstract away the communication protocol and only consider the functional API in what follows.)
 Introduce a new module,
ServerModule
, into theapatool
package. This module will abstract over the relevant BMC passes which lead up to, and provide input for, theTransitionExplorer
, described below.  Introduce a new module,
TransitionExplorer
that enables the interactive exploration of the transition system.  Internally, the
TransitionExplorer
will make use of theTransitionExecutor
and relevant aspects of theSeqModelChecker
(or slightly altered versions of its methods).
NOTE: The highlevel sketch above assumes the new code organization proposed in ADR 7.
API
The following is a rough sketch of the proposed API for the transition explorer. It aims to present a highly abstracted interface, but in terms of existing data structures. Naturally, refinements and alterations are to be expected during implementation.
We refer to symbolic states as Frames
, which are understood as a set of
constraints, and we put this terminology in the API in order to help users
understand when they should be thinking in terms of constraint sets as opposed
to concrete states. Concrete states can be obtained by the functions suffixed
with Example
.
In essence, this proposed API is only a thin wrapper around the
TransitionExecutor
class.
During previous iterations of the proposed API we discussed exposing a
higherlevel API, targeted at meeting the requirements more directly. However,
discussion revealed that the expensive computational costs of SAT solving in
most cases made it infeasible to meet the requirements in this way. Instead, we
must expose most of the underlying logic of the TransitionExecutor
, and task
the users with building their own exploration strategies with these primitives.
It is likely that we will be able to provide some higherlevel functionality to users by way of wrapper libraries we implement on top of the proposed API, but that work should be left to a subsequent iteration.
NOTE: This interface is intended as an abstract API, to communicate the mappings from request to reply. See the gRPC Example for a sketch of what the actual implementation may look like.
/** A State is a map from variable names to values, and represents a concrete state. */
type State = Map[String, TlaEx]
/** An abstract representation of a transition.
*
* These correspond to the numbered transitions that the `TransitionExectur` uses
* to advance frames. But we likely want to present some more illuminating metadata
* associated with the transitions to the users. E.g., ability to view a
* representation as a `TlaEx` or see an associated operator name (if any)? */
type Transition
/** An execution is an alternating sequence of states and transitions,
* terminating in a state with no transition.
*
* It is a highlevel representation of the `EncodedExecution` maintained by the
* transition executor).
*
* E.g., an execution from `s1` to `sn` with transitions `ti` for each `i` in `1..n1`:
*
* List((s1, Some(t1)), (s2, Some(t2)), ..., (sn, None))
*/
type Execution = List[(State, Option[Transition])]
trait UninitializedModel = {
val spec: TlaModule
val transitions: List(Transition)
}
trait InitializedModel extends UninitializedModel {
val constInitPrimed: Map[String, TlaEx]
}
/** An abstract representation of the `CheckerInput`
*
* A `Model` includes all static data representing a model.
*
* An `UninitializedModel` is missing the information that would be needed in
* order to actually explore its symbolic states (such as the initial values of
* its constants).
*
* An `InitializedModel` has all data needed to explore its symbolic states. */
type Model = Either[UninitializedModel, InitializedModel]
/** The type of errors that can result from failures when loading a spec. */
type LoadErr
/** The type of errors that can result from failures to assert a constraint. */
type AssertErr
/** The type of errors that can result from checking satisfiability of a frame. */
type SatError
/** Maintains session and connection information */
type Connection
trait TransEx {
/** Used internally */
private def connection(): Connection
/** Reset the state of the explorer
*
* Returns the explorer to the same state as when the currently loaded model
* was freshly loaded. Used to restart exploration from a clean slate.
*
* [TRANSEX.1::LOAD.1]
*/
def reset(): Unit
/** Load a model for exploration
*
* If a model is already loaded, it will be replaced and the state of the exploration
* [[reset]].
*
* [TRANSEX.1::QCHECK.1]
* [TRANSEX.1::LOAD.1]
* [TRANSEX.1::SBMC.1::TRANSITIONS.1]
*
* @param spec the root TLA+ module
* @param aux auxiliary modules that may be required by the root module
* @return `Left(LoadErr)` if parsing or loading the model from `spec` goes
* wrong, or `Right(UninitializedModel)` if the model is loaded successfully.
*/
def loadModel(spec: String, aux: List[String] = List()): Either[LoadErr, UninitializedModel]
/** Initialize the constants of a model
*
* This will always also reset an exploration back to its initial frame. */
def initializeConstants(constants: Map[String, TlaEx]): Either[AssertErr, InitializedModel]
/** Prepare the loaded modle with the given `transition` */
def prepareTransition(transition: Transition): Either[AssertErr, Unit]
/** The transitions that have been prepared. */
def preparedTransitions(): Set[Transition]
/** Apply a (previously prepared) transition to the current frame.
*
* Without any arguments, a previously prepared transition is selected
* nondeterministically.
*
* When given a `transition`, apply it if it is already prepared, or prepare
* it and then apply it, if not.
*
* Interfaces to `assumeTransition` and `pickTransition`, followed by
* `nextState`. */
def applyTransition(): Either[AssertErr, Unit]
def applyTransition(transition: Transition): Either[AssertErr, Unit]
/** Assert a constraint into the current frame
*
* Interface to `assertState` */
def assert(assertion: TlaEx): Either[AssertErr, Unit]
/** The example of an execution from the an initial state up to the current symbolic state
*
* Additional executions can be onbtained by asserting constraints that alter the
* search space. */
def execution: Either[SatErr, Execution]
/**
* Check, whether the current context of the symbolic execution is satisfiable.
*
* @return Right(true), if the context is satisfiable;
* Right(false), if the context is unsatisfiable;
* Left(SatErr) inidicating if the solver timed out or reported *unknown*.
*/
def sat(): Either[SatErr, Boolean]
/** Terminate the session. */
def terminate(): Unit
}
object TransEx {
/** Create a transition explorer
*
* Establishes the connection and a running session
*
* The channel is managed by the gRPC framework. */
def apply(channel: Channel): TransEx = ...
}
Constructing the IR
In order to form assertions (represented in the spec as values of TlaEx
),
users will need a way of constructing the Apalache IR. Similarly, they'll need a
way of deconstructing and inspecting the IR in order to extract useful content
from the transitions.
To meet this need, the gRPC libraries we generate for client code will also include ASTs in the client language along with generated serialization and deserialization of JSON represents into and out of the AST. This will enable users to construct and inspect expressions as needed.
Protocol
We have briefly discussed the following options:
 Custom protocol on top of HTTP
 JSONrpc
 gRPC
I propose use of gRPC for the following reasons:
 It will automate most of the IO and protocol plumbing we'd otherwise have to do ourselves.
 It is battle tested by industry
 It is already used in Rust projects within Informal Systems. This should make it easier to integrate into modelator.
 The Scala library appears to be well documented and actively maintained.
 Official support is provided in many popular languages, and we can expect wellmaintained library support in most languages.
 The gRPC libraries include both the RPC protocol and plumbing for the transport layer, and these are decomposable, in case we end up wanting to use different transport (i.e., sockets) or a different protocol for some purpose down the line.
For a discussion of some comparison between JSONrpc and gRPC, see
 https://www.mertech.com/blog/knowyourapiprotocols
 https://stackoverflow.com/questions/58767467/whatthedifferencebetweenjsonrpcwithhttp2vsgrpc
I have asked internally, and engineers on both tendermintrs
and hermes
have
vouched for the ease of use and reliability of gRPC.
Using gRPC can help satisfy [TRANSEX.1::EXTENSIBLE.1] in the following ways:
 [TRANSEX.1::EXTENSIBLE.1::CLOUD.1] should be satisfied out of the box, since HTTP is the default transport for gRPC.
 [TRANSEX.1::EXTENSIBLE.1::CLI.1] can be satisfied by implementing a CLI client that we can launch via an Apalache subcommand.
gRPC Example
Here is as simple example of what it would actually look like to configure gRPC for the server (adapted from ScalaPB grpc):
We define our messages in a proto
file:
syntax = "proto3";
package com.transex.protos;
service TransExServer {
rpc LoadModel (LoadRequest) returns (LoadReply) {}
}
message LoadRequest {
required string spec = 1;
optional repeated string aux = 2;
}
message LoadReply {
enum Result {
OK = 0;
PARSE_ERROR = 1;
TYPE_ERROR = 2;
// etc...
};
required Result result;
required Model model;
}
The generated Scala will look roughly as follows:
object TransExGrpc {
// Abstract class for server
trait TransEx extends AbstractService {
def serviceCompanion = TransEx
def loadModel(request: LoadRequest): Future[LoadReply]
}
// Abstract class for block client
trait TransExBlockingClient {
def serviceCompanion = TransEx
def loadModel(request: LoadRequest): LoadReply
}
// Abstract classes for asynch client + various other boilerplate
}
A sketch of implementing a server using the gRPC interface:
class TransExImpl extends TransExGrpc.Greeter {
override def loadModel(req: LoadRequest) = {
val rootModule = req.model
val auxModules = req.aux
// run incomming specs through parsing passes
val model : UninitializedModel = Helper.loadModel(rootModule, auxModules)
val reply = LoadReply(result = Ok, model = model)
Future.successful(reply)
}
}
A sketch of using the client (e.g., to implement our CLI client):
// Create a channel
val channel = ManagedChannelBuilder.forAddress(host, port).usePlaintext(true).build
// Make a blocking call
val request = LoadRequest(spec = <loaded module as string>)
val blockingStub = TransExGrpc.blockingStub(channel)
val reply: LoadReply = blockingStub.loadModel(request)
reply.result match {
ParseError => // ...
Ok =>
val model: UninitializedModel = reply.model
// ...
}
NOTE: To ensure that we are able to maintain a stable API, we should version the API from the start.
ADR011: alternative SMT encoding using arrays
author  revision 

Rodrigo Otoni  1.8 
This ADR describes an alternative encoding of the KerA+ fragment of TLA+ into SMT. Compound data structures, e.g. sets, are currently encoded using the core theory of SMT, with the goal being to encode them using arrays with extensionality instead. The hypothesis is that this will lead to increased solver performance and more compact SMT instances. We target the Z3 solver and will use the SMTLIB Standard (Version 2.6) in conjunction with Z3specific operators, e.g. constant arrays.
For an overview of the current encoding check the TLA+ Model Checking Made Symbolic paper, presented at OOPSLA'19. In the remainder of the document the use of the new encoding and the treatment of different TLA+ operators are described. For further details on the new encoding check the Symbolic Model Checking for TLA+ Made Faster paper, presented at TACAS'23.
1. CLI encoding option
The encoding using arrays is to be an alternative, not a replacement, to the already existing encoding.
Given this, a new option is to be added to the check
command of the CLI. The default encoding will be
the existing one. The option description is shown below. The envvar SMT_ENCODING
can also be used to
set the encoding, see the model checking parameters for details. In addition to the arrays
encoding,
which uses SMT arrays to encode TLA+ sets and functions, we also have the funArrays
encoding, which
restricts itself to encoding only TLA+ functions as SMT arrays.
smtencoding : the SMT encoding: oopsla19, arrays (experimental), funArrays (experimental), default: oopsla19 (overrides envvar SMT_ENCODING)
Code changes
The following changes will be made to implement the new CLI option:
 Add new string variable to class
CheckCmd
to enable the new option.  Add new
smtEncoding
field toSolverConfig
.  Add new class
SymbStateRewriterImplWithArrays
, which extends classSymbStateRewriterImpl
.  Use the new option to set the
SolverConfig
encoding field and select between differentSymbStateRewriter
implementations in classesBoundedCheckerPassImpl
andSymbStateRewriterAuto
.  The infrastructure changes made for the
funArrays
encoding mirror the ones made for thearrays
one. See PR 2027 for details.
2. Testing the new encoding
The new encoding should provide the same results as the existing one, the available test suite
will thus be used to test the new encoding. To achieve this, the unit tests needs to be made parametric
w.r.t. the SolverConfig
encoding field and the implementations of SymbStateRewriter
, and the
integration tests need to be tagged to run with the new encoding.
Code changes
The following changes will be made to implement the tests for the new encoding:
 Refactor the classes in
tlabmcmt/src/test
to enable unit testing with different configurations ofSolverConfig
and implementations ofSymbStateRewriter
.  Add unit tests for the new encoding, which should be similar to existing tests, but use a
different solver configuration and
SymbStateRewriterImplWithArrays
instead ofSymbStateRewriterImpl
.  Add integration tests for the new encoding by tagging existing tests with
arrayencoding
, which will be run by the CI with envvarSMT_ENCODING
set toarrays
.
3. Encoding sets
Sets are currently encoded in an indirect way. Consider a sort some_sort
and distinct elements elem1
,
elem2
, and elem3
of type someSort
, as defined below.
(declaresort some_sort 0)
(declareconst elem1 some_sort)
(declareconst elem2 some_sort)
(declareconst elem3 some_sort)
(assert (distinct elem1 elem2 elem3))
A set set1
containing elem1
, elem2
, and elem3
is currently represented by a constant of type
set_of_some_Sort
and three membership predicates, as shown below.
(declaresort set_of_some_Sort 0)
(declareconst set1 set_of_some_Sort)
(declareconst elem1_in_set1 Bool)
(declareconst elem2_in_set1 Bool)
(declareconst elem3_in_set1 Bool)
(assert elem1_in_set1)
(assert elem3_in_set1)
(assert elem2_in_set1)
The new encoding has each set encoded directly as an array whose domain and range equal the set's sort
and the Boolean sort, respectively. SMT arrays can be thought of as a functions, as this is exactly how
they are represented internally in Z3. Set membership of an element elem
is thus attained by simply
setting the array at index elem
to true
.
One important point in the new encoding is the handling of set declarations, since declaring an empty set requires the setting of all array indexes to false. This can be easily achieved for finite sets by explicitly setting each index, but falls outside the quantifierfree fragment of firstorder logic in the case of infinite sets, e.g. the set of integers. To handle declarations of infinite sets we rely on Z3's constant arrays, which map all indexes to a fixed value. Below is an example using the new encoding.
(declareconst set2_0 (Array some_sort Bool))
(declareconst set2_1 (Array some_sort Bool))
(declareconst set2_2 (Array some_sort Bool))
(declareconst set2_3 (Array some_sort Bool))
(assert (= set2_0 ((as const (Array some_sort Bool)) false)))
(assert (= set2_1 (store set2_0 elem1 true)))
(assert (= set2_2 (store set2_1 elem2 true)))
(assert (= set2_3 (store set2_2 elem3 true)))
The store
operator handles array updates and receives the array to be updated, the index, and the new
value, returning the updated array. For array access, the select
operator can be used, which receives
an array and an index and returns the value at the given index, as shown below.
(assert (= (select set2_2 elem1) true)) ; SAT
(assert (= (select set2_2 elem2) true)) ; SAT
(assert (= (select set2_2 elem3) true)) ; UNSAT
(assert (= (select set2_3 elem1) true)) ; SAT
(assert (= (select set2_3 elem2) true)) ; SAT
(assert (= (select set2_3 elem3) true)) ; SAT
For consistency, the new encoding uses constant arrays to declare both finite and infinite arrays.
Code changes
The following changes will be made to implement the new encoding of sets:
 Add alternative rewriting rules for sets when appropriate, by extending the existing rules.
 All alternative rules will be suffixed with
WithArrays
.  The new rules will not rely on
LazyEquality
and will aim to use SMT equality directly.  Only the generation of SMT constraints will be modified by the new rules, the other Arena elements will remain unchanged.
 All alternative rules will be suffixed with
 In class
SymbStateRewriterImplWithArrays
, add the new rules toruleLookupTable
by overriding the entries to their older versions.  Add four new Apalache IR operators in
ApalacheOper
,Builder
,ConstSimplifierForSmt
, andPreproSolverContext
, to represent the array operations. The
selectInSet
IR operator represents the SMTselect
.  The
storeInSet
IR operator represents the SMTstore
.  The
unchangedSet
IR operator represents an equality between the current and new SSA array representations. This is required to constraint the array representation as it evolves. It is important to note that this operator assumes that all arrays are initially empty, so an element not explicitly added is assumed to not be in the array. To check absence of an element,selectInSet
should be used with negation.  The
smtMap
IR operator represents the use of SMT map.
 The
 In class
Z3SolverContext
, add/change appropriate methods to handle SMT constraints over arrays. The main changes will de done in
declareCell
and the newmkSelect
,mkStore
, andmkUnchangedSet
methods, as these methods are directly responsible for creating the SMT constraints representing sets and set membership.  With the new IR operators, the "inrelation" concept, which underpins
declareInPredIfNeeded
andgetInPred
, will not be applied to the new encoding. Cases for the new IR operators will be added totoExpr
, which will default toTlaSetOper.in
andTlaSetOper.notin
for the existing encoding.  The
smtMap
IR operator will be used to encode the TLA+ set filter operation. It constructs a temporary array that contains the evaluation of the filter's predicate for each set element and uses SMT map to compute the intersection of the set being filtered and the set represented by the temporary array constructed.  Cases for
FinSetT
andPowSetT
will be added togetOrMkCellSort
, as these types are no longer represented by uninterpreted constants. cellCache
will be changed to contain a list of cells, in order to handle the effects ofpush
andpop
in the SSA assignment of sets. The following examples illustrates this need.(assert (= set_0 ((as const (Array Int Bool)) false))) (assert (= set_1 (store set_0 5 true))) (push) (assert (= set_2 (store set_1 6 true))) (push) (assert (= set_3 (store set_2 7 true))) (assert (= (select set_3 7) true)) (pop 2) (assert (= (select set_1 7) false)) ; Without the list we would query set_3 here
 The main changes will de done in
4. Encoding functions and sequences
Functions are currently encoded as sets of pairs, with each pair representing a mapping present in
the function. The first element of a pair is a tuple containing some function arguments and the second
element is the return value given by such arguments. The handling of functions is thus given by
operations over sets and tuples. Sequences of type T
are currently encoded as tuples of form
⟨start,end,fun⟩
, where start
and end
are integers and fun
is a function from integers to T
.
The new encoding of functions will thus encompass sequences, as their tuple representations is
intended to be kept.
The new encoding will, like the current one, also map tuples of arguments to return values, but
will do so natively instead of simply relying on sets. A function will be represented by two SMT
arrays. The first array will store the domain of the function and will be encoded as a standard
TLA+ set. The second array will store the mappings, having sort <S1,...,Sn>
as its domain, with
Si
being the sort of argument i
, and the sort of the function's codomain as its range.
The sorts of the array domain and range can be infinite, but the domain of the function itself,
and by implication the number of mappings tuples, will always be finite.
To encode the TLA+ function finSucc = [x \in {1,2,3} > x + 1 ]
, which computes the successors
of integers from 1
to 3
, we first have to declare its domain, as shown below; tuples are
represented here as per the OOPSLA'19 encoding.
(declaresort Tuple_Int 0) ; Sort of <Int>
(declareconst tuple_with_1 Tuple_Int) ; <1>
(declareconst tuple_with_2 Tuple_Int) ; <2>
(declareconst tuple_with_3 Tuple_Int) ; <3>
(declareconst finSucc_domain_0 (Array Tuple_Int Bool))
(declareconst finSucc_domain_1 (Array Tuple_Int Bool))
(declareconst finSucc_domain_2 (Array Tuple_Int Bool))
(declareconst finSucc_domain_3 (Array Tuple_Int Bool))
(assert (= finSucc_domain_0 ((as const (Array Tuple_Int Bool)) false))) ; {}
(assert (= finSucc_domain_1 (store finSucc_domain_0 tuple_with_1 true))) ; {<1>}
(assert (= finSucc_domain_2 (store finSucc_domain_1 tuple_with_2 true))) ; {<1>,<2>}
(assert (= finSucc_domain_3 (store finSucc_domain_2 tuple_with_3 true))) ; {<1>,<2>,<3>}
The array storing the function's domain is used to guard the definition of the array storing the
function's mappings, since mappings should only be present for values in the domain. The array
storing the mappings of finSucc
is shown below.
(declareconst finSucc_0 (Array Tuple_Int Int))
(declareconst finSucc_1 (Array Tuple_Int Int))
(declareconst finSucc_2 (Array Tuple_Int Int))
(declareconst finSucc_3 (Array Tuple_Int Int))
(assert (ite (select finSucc_domain_3 tuple_with_1)
(= finSucc_1 (store finSucc_0 tuple_with_1 2))
(= finSucc_1 finSucc_0)))
(assert (ite (select finSucc_domain_3 tuple_with_2)
(= finSucc_2 (store finSucc_1 tuple_with_2 3))
(= finSucc_2 finSucc_1)))
(assert (ite (select finSucc_domain_3 tuple_with_3)
(= finSucc_3 (store finSucc_2 tuple_with_3 4))
(= finSucc_3 finSucc_2)))
Note that, unlike with the new encoding for sets, we do not use constant arrays. The reason is that the function's domain cannot be altered, so the array has to constrain only the values in said domain. Function application can be done by simply accessing the array at the index of the passed arguments. A function application with arguments outside the function's domain leads to an unspecified result in TLA+, which is perfectly captured by unconstrained entries in the SMT array. Below are some examples of function application.
(assert (= (select finSucc_3 tuple_with_1) 2)) ; SAT
(assert (= (select finSucc_3 tuple_with_2) 3)) ; SAT
(assert (= (select finSucc_3 tuple_with_3) 4)) ; SAT
(declareconst tuple_with_4 Tuple_Int) ; <4>
(assert (= (select finSucc_3 tuple_with_4) 16)) ; SAT
Although a function's domain cannot be altered, its image can be changed via the TLA+ function update operator. The update will be encoded as a guarded array update, as illustrated below; attempting to update an entry outside the function's domain will lead to no change happening.
(declareconst finSucc_4 (Array Tuple_Int Int))
(declareconst finSucc_5 (Array Tuple_Int Int))
(assert (ite (select finSucc_domain_3 tuple_with_1) ; [finSucc EXCEPT ![1] = 9]
(= finSucc_4 (store finSucc_3 tuple_with_1 9))
(= finSucc_4 finSucc_3)))
(assert (ite (select finSucc_domain_3 tuple_with_4) ; [finSucc EXCEPT ![4] = 25]
(= finSucc_5 (store finSucc_4 tuple_with_4 25))
(= finSucc_5 finSucc_4)))
(assert (= (select finSucc_5 tuple_with_1) 2)) ; UNSAT
(assert (= (select finSucc_5 tuple_with_1) 9)) ; SAT
(assert (= (select finSucc_5 tuple_with_4) 16)) ; SAT
In contrast to the current encoding, which produces a number of constraints that is linear in the size of the set approximating the function when encoding both function application and update, the new encoding will produce a single constraint for each operation. This will potentially lead to a significant increase in solving performance.
Code changes
The following changes will be made to implement the new encoding of functions:
 Add alternative rewriting rules for functions when appropriate, by extending the existing rules. The
same caveats stated for the rewriting rules for sets will apply here.
 The sets of pairs used in the current encoding are the basis for the counterexample generation in
SymbStateDecoder
. In order to continue having counterexamples, these sets will keep being produced, but will not be present in the SMT constraints. They will be carried only as metadata in theArena
.
 The sets of pairs used in the current encoding are the basis for the counterexample generation in
 Update class
SymbStateRewriterImplWithArrays
with the rules for functions.  Update the
storeInSet
IR operator to also store function updates. It will have the value resulting from the update as an optional argument. Since functions will be encoded as SMT arrays, the
selectInSet
,storeInSet
, andunchangedSet
IR operators will be used when handling them. A future refactoring may rename these operators.
 Since functions will be encoded as SMT arrays, the
 Update class
Z3SolverContext
to handle the new SMT constraints over arrays. A case for
FunT
will be added togetOrMkCellSort
.  In
declareCell
, functions will be declared as arrays, but will be left unconstrained.  The
mkStore
method will be updated to also handle functions. It will have an additional optional argument containing the value to be stored in the range of the array. The new argument's default value istrue
, for the handling of sets.  The
mkNestedSelect
method is added to support set membership in function sets, i.e.,f \in [S > T]
. The nesting has firstfunAppRes = f[s \in S]
, followed byfunAppRes \in T
.
 A case for
5. Encoding the remaining TLA+ features
The use of SMT arrays will be restricted to TLA+ sets and functions for the moment. The encoding of additional features using SMT arrays, or potentially ADTs, will be left for the future.
ADR012: Adopt an ADR Template
authors  last revised 

Shon Feder  20211205 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of our development of Apalache facing the need to communicate and record our significant decisions we decided for adopting an ADR template adapted from the "Alexandrian Form" to achieve concise and consistent records of our architectural decisions accepting the regimentation and loss of unexpected possibility that comes with adopting a template.
Context
The development of Apalache is picking up momentum. We have more contributors joining us immanently, and hope to welcome and support external OSS contributors soon. As the number of contributors grows, so does the importance of establishing supports to encourage communication between individuals and accross time.
Maintaining records of architectural decisions (aka "ADR"s) is advised by informal.systems company policy, but the details of how such records should be written, kept, or used, have not been settled. I hypothesize that we have much to gain by experimenting with a consistent, well reasoned format for our ADRs. I think it will help us be mindful of their purpose, make them more useful as diagnostic and prognostic tools, and help reduce the amount of time needed for drafting and approval.
Options
While considering approaches to ADRs, I consulted the following resources, and many of the children links to found therein, :
 https://adr.github.io/
 https://github.com/joelparkerhenderson/architecturedecisionrecord
 https://en.wikipedia.org/wiki/Architectural_decision
I was surprised by the amount of literature surrounding this topic, and wanted to select something that would help focus and clarify our ADRs, while avoiding any undue burden that might come from associated management or development practices.
Each approach to ADRs can inspire a family of templates. I found most of them to be too involved or intimidating, and I opted for the most light weight approach I could find, while making some changes to clarify the language and content to support our context and existing styles of communication.
Solution
I propose adopting this simple articulation of ADRs and their purpose as our guide:
An architecture decision record (ADR) is a document that captures an important architecture decision made along with its context and consequences.
(see https://github.com/joelparkerhenderson/architecturedecisionrecord#whatisanarchitecturedecisionrecord)
Following the Teamwork advice offered in that same document, I propose adopting an ADR template that puts all emphasis on the key purposes of the communication, leaving it up to each author to fill in the template with as much or as little detail as they think necessary to support the particular decision in question.
To this end, I propose this template, which is adapted from the Alexandrian pattern. This template is itself adapted from the socalled "Alexandrian form". Martin Fowler has a succinct summary of its qualities in its native context of "design patterns".
Consequences
ADR013: Configuration Management Component
authors  last revised 

Shon Feder  20220815 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of using Apalache from other programs, different environments, and
in different planned modes (see, e.g., #703),
facing the need to supply different configurations for different use cases
we decided for adopting the PureConfig library and for introducing a small component to
integrate PureConfig
with our CLI parsing
in order to achieve maintainable, reasonable, and extensible management of configurations
accepting the additional external dependency and development costs.
Context
As our application grows more flexible, gains more adoption and usage in programmatic pipelines, and strives to provide more functionality, we inevitably need to make it more configurable.
Recent additions that have extended configurability include:
 #1081, introducing
the
rundir
flag.  #1036, introducing
the
writeintermediate
, andprofiling
,outdir
configuration parameters, exposed both via CLI and configuration file.  #1054, introducing
the
smtencoding
flag.
The ongoing work for the server mode is expected to require introducing several more configurable paramters.
As discussed in #1069 and #1929 we have at least 5 different sources from which we need to load configuration parameters, and the loading must cascade, with the first listed source taking priority:
 CLI arguments OR data supplied by RPC
 environment variables
 A local configuration files (perhaps with the location overridden by a CLI flag)
 A global configuration
 Predetermined defaults
We are currently managing this configuration in an ad hoc way, with a bespoke configuration loading system, and various ad hoc methods for effecting overrides.
Options
The problem can be decomposed into three parts:
 Reading parameters from CLI and environment variables (currently done through our CLI library).
 Reading parameters from configuration files (currently done in an ad hoc way)
 Cascade loading these paramters in the correct order, to end up with the correct intended configuration.
To address (2) and (3), we should use an existing configuration management library, since this will save us development time, and allow us to take advantage of other developer's careful engineering around this problem, freeing us to focus on our core problem domain.
There are some configuration libraries that aim to provide an integrated solutions to all three problems, but I have dismissed them for reasons described below.
Comparison of configuration management libraries
I considered four actively maintained libraries focused on application configuration. This section reports my findings.
Activity
Library  Contributors  Last Release  GitHub Stars  Build Status 

config  89  20201022  5.5k  passing 
profig  5  20210114  23  failing 
metaconfig  23  20210531  29  passing 
PureConfig  58  20211121  1.2k  passing 
Features
Library  Formats  File Merging  Envvar Fallback  CLI Arg Merging  Language  Typing  Documentation 

config  java properties, JSON, HOCON  yes  yes  manual  Java  dynamic  excellent 
profig  java properties, JSON, YAML, HOCON, XML  yes  yes  automatic  Scala  dynamic  decent 
metaconfig  JSON, HOCON  ?  ?  automatic  Scala  static  poor 
PureConfig  java properties, JSON, HOCON  yes  semi  automatic  Scala  static  excellent 
Additional notes
 [conifg][]
 Integrates with Guice
 Lots of support due to Java usage
 profig
 only apparent advantage over config is automatic CLI parsing, but that also requires swapping out our CLI library.
 metaconfig
 PureConfig
 Type safe wrapper around config, so should inherit all features of that basis (including Guice integration)
 Will automatically merge configs based on a priority list of files.
 Support optional configuration fallback
 Supports writing out configs (can be used in bug reports or populating default config to help guide users)
Evaluation
I discount profig because it has nothing significant to recommend it over config.
metaconfig is attractive due its support for type safe configuration, generation of markdown documentation, but the poor documentation and relatively small user base counts against it. Those other factors are not sufficiently attractive to outweigh the risks.
The choice between config and PureConfig is easy: PureConfig includes everything provided by config, but exposes a types safe, Scalanative API. Moreover, it's got a substantial userbase and excellent documentation.
Solution
We will adopt PureConfig as our configuration management library. It will enable us to cascade load configuration files from many exernal sources (including a json blob passed in via CLI inputs) and provide typesafe access to the configured values.
We will continue to rely on clist
for CLI parsing for the time being, which
takes care of loading environment variable settings and CLI arguments with our
desired overriding precedence. This will require we add a thin abstraction that
will ensure the CLI arguments end up overriding the configured values. This
abstraction will replace the more ad hoc process we are currently employing to
this end.
Here's a short example of how basic usage should look (approximately), allowing
us to replace dozens of lines of code in the OutputManager
implementing our
current adhoc configuration parsing:
import pureconfig._
import pureconfig.generic.auto._
// Setting a defaul value
case class Port(number: Int = 8080) extends AnyVal
sealed trait SmtEncoding
case class Arrays extends SmtEncoding
case class OOPSLA19 extends SmtEncoding
case class ApalacheConfig(
runDir: Option[Path] = None,
serverPort: Port = Port(),
writeIntermediate: Boolean = false,
profiling: Boolean = false,
outDir: Path = Path("."),
smtEncding: SmtEncoding = OOPSLA19,
)
case classs ExampleUseOfConfigs() = {
val cli = CliParseResults()
val localConfig = ConfigSource.file(Path.cwd.resolve(".aplache.config"))
val globalConfig = ConfigSource.file(ApalacheHome.resolve("apalache.config"))
val loadedConfig: ConfigReader.Result[ApalacheConfig] = globalConfig
.withFallback(localConfig)
.load[ApalacheConfig]
// Finally, override with CLI arguments
// Unfortunatley, I've not found a robust way to automate this yet
val config = loadedConfig.copy(
runDir = cli.runDir.getOrElse(loadedConfig.runDir),
serverPort = cli.runDir.getOrElse(loadedConfig.serverPort),
// etc..
)
}
This ApalacheConfig
class can then be passed around to all parts of the
program that need to read such configurations.
Consequences
After utilizing the approach proposed here for nearly a year, we were able to introduce several additional configurations easily, and we found the local configuration files useful for tweaking program behavior. We subsequently decided to further extend the configuration system by integrating the CLI within the configuration system and use it as the basis for statically representing all program options. See ADR 022.
ADR014: Precise type inference for records and variants
authors  last revised 

Igor Konnov  20211212 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
1. Summary
This ADR extends ADR002 on types and type annotations.
Virtually every user of Snowcat has faced the issue of record type checking being imprecise. Some people call it "unsound", though soundness depends on the type system. This is due to our decision to support the variant pattern that can be found in untyped TLA+ specifications. In this ADR, we are proposing a plan of action for introducing precise type inference for records and variants (discriminated unions) in the type checker. This would deliver the most asked feature. On the downside, we would have to:

Increase the complexity of the type checker.

Slightly update the rewriting rules in the model checker.

Require the users to modify their specs to use the variant operators.
As much as possible, we have tried to make the type annotations nonintrusive and compatible with TLC. After precisely specifying the requirements for the variant type, we have found that it would be impossible to do sound type checking without introducing additional operators.
We believe that the benefits outweigh the downsides in the long run. Moreover, it will improve user experience, as this is the most requested feature.
2. Context
As discussed in ADR002, the type checker is not checking the record types precisely. Consider the following operator:
Foo ==
LET S == {[ type > "A", a > 1], [ type > "B", b > 2 ]} IN
\E m \in S:
m.a > m.b
The type checker assigns the type Set([type: Str, a: Int, b: Int])
to S
. As
a result, one can write the expression m.a > m.b
, which does not make a lot
of sense. This may lead to unexpected results in a large specification. In the
above example, the model checker would just produce some values for m.a
or
m.b
, which will probably result in a spurious counterexample.
Further, multiple related issues and potential solutions were underlined in #401 and #789.
There are two main patterns of record use in TLA+:

Plain records. A record with a fixed number of fields is passed around.

Variants. Records of various shapes are collected in a single set and passed around. The precise record shape is controlled with a special field (discriminator), which is usually called
type
in TLA+ specs.
2.1. Untyped plain records
When it comes to records, it is clear that users expect the type checker to complain about missing record fields. Indeed, it is very easy to introduce a spurious record field by mistyping the field name. It happened to all of us.
Interestingly, plain records are used less often than variants. Perhaps, if records are required, the specification is quite complex already, so it would also need variants.
Occurences in tlaplusexamples:
LamportMutex is an interesting borderline case, in which the spec uses the variant pattern, but it could be typed with a single record type. Here are the interesting pieces in this spec:
ReqMessage(c) == [type > "req", clock > c]
AckMessage == [type > "ack", clock > 0]
RelMessage == [type > "rel", clock > 0]
Message == {AckMessage, RelMessage} \union {ReqMessage(c) : c \in Clock}
...
Broadcast(s, m) ==
[r \in Proc > IF s=r THEN network[s][r] ELSE Append(network[s][r], m)]
...
Request(p) ==
...
/\ network' = [network EXCEPT ![p] = Broadcast(p, ReqMessage(clock[p]))]
...
Although, every message record is accompanied with the field type
, all
records have the same shape, namely, they have two fields: A string field
type
and an integer field clock
.
2.2. Untyped variants
Most of the benchmarks stem from the Paxos specification. They all follow the
same pattern. Messages are represented with records of various shapes. Every
record carries the field type
that characterizes the record shape. For instance,
here is how records are used in Paxos:
Message == [type : {"1a"}, bal : Ballot]
\cup [type : {"1b"}, acc : Acceptor, bal : Ballot,
mbal : Ballot \cup {1}, mval : Value \cup {None}]
\cup [type : {"2a"}, bal : Ballot, val : Value]
\cup [type : {"2b"}, acc : Acceptor, bal : Ballot, val : Value]
...
Send(m) == msgs' = msgs \cup {m}
...
Phase1b(a) == /\ \E m \in msgs :
/\ m.type = "1a"
/\ m.bal > maxBal[a]
...
/\ Send([type > "1b", acc > a, bal > m.bal,
mbal > maxVBal[a], mval > maxVal[a]])
...
Occurences in tlaplusexamples:
More advanced code patterns over variants can be found in Raft:
Receive(m) ==
LET i == m.mdest
j == m.msource
IN \* Any RPC with a newer term causes the recipient to advance
\* its term first. Responses with stale terms are ignored.
\/ UpdateTerm(i, j, m)
\/ /\ m.mtype = RequestVoteRequest
/\ HandleRequestVoteRequest(i, j, m)
\/ /\ m.mtype = RequestVoteResponse
/\ \/ DropStaleResponse(i, j, m)
\/ HandleRequestVoteResponse(i, j, m)
\/ /\ m.mtype = AppendEntriesRequest
/\ HandleAppendEntriesRequest(i, j, m)
\/ /\ m.mtype = AppendEntriesResponse
/\ \/ DropStaleResponse(i, j, m)
\/ HandleAppendEntriesResponse(i, j, m)
...
Next ==
...
\/ \E m \in DOMAIN messages : Receive(m)
\/ \E m \in DOMAIN messages : DuplicateMessage(m)
\/ \E m \in DOMAIN messages : DropMessage(m)
...
3. Other issues
As can be seen from the Paxos example, we should take care of sets of records that are used as sets of variants:
Message == [type : {"1a"}, bal : Ballot]
\cup [type : {"1b"}, acc : Acceptor, bal : Ballot,
mbal : Ballot \cup {1}, mval : Value \cup {None}]
\cup [type : {"2a"}, bal : Ballot, val : Value]
\cup [type : {"2b"}, acc : Acceptor, bal : Ballot, val : Value]
4. Options
There are several solutions to the issue of precise record typing.
4.1. Support only plain records
In this case, we would only allow to mix records that have exactly the same shape. As a result, when we use variants, we would have to add extra fields to all of them. This solution is not very different from the current implementation of record type checking, though it would allow us to quickly detect spelling errors.
Blocker. This does not look like a real solution, as it would immediately render the existing examples invalid. Moreover, they would be no obvious way to repair these examples.
4.2. Support plain records and variants, but no row typing
In this scenario, the type checker would issue an error, if a record expression accesses a field that is outside of its type:
FieldAccess ==
LET m == [ a > 2, b > "B" ] IN
/\ m.a > 1 \* type OK
/\ m.b = "B" \* type OK
/\ m.c = { 1, 2 } \* should flag a type error
Moreover, the following example would require a type annotation:
RowAccess(m) ==
m.a > 0 \* should flag a type error
In the above example, the type checker would not be able to infer the type of
m
and would require an explicit type annotation:
\* @type: [ a: Int, b: Str ];
RowAccessAnnotated(m) ==
m.a > 0 \* should not flag a type error
Blocker. It is not clear to me, what type we would assign to the record
access operator. Although the type of m
in FieldAccess
is obvious to a
human reader, as it simply requires us to do the toptothebottom type
propagation, Snowcat constructs a set of equality constraints that are solved
by unification. This approach requires that we capture the record access
operator .
as an equality over type variables and types.
4.3. Support plain records and variants, including row typing
This solution is inspired by the approach outlined by Leijen05, but is much more limited. The following features discussed in Leijen05 are neither needed needed nor supported:
 extension for records
 extension for variants
 record restriction
 scoped labels
Our use is limited to a few special cases that give support for subtyping and incremental inference of anonymous record types, in which we cannot know the full set of fields up front.
5. Solution
In the following, we present row types as type terms. We discuss the userfacing syntax of the type system later in the text.
5.1. Plain records
By using Row types, we should be able to infer a polymorphic record type for m
in the unannotated
RowAccess
operator:
Rec(RowCons("a", Int, z))
In this example, RowCons("a", Int, z)
indicates a row indicating that the type of the record enclosing it
has the field a
of type Int
. On top of that, this row
extends a parametric type z
, which either contains a nonempty sequence of
rows, or is an empty sequence, that is, RowNil
. Importantly, RowCons
is
wrapped with the term Rec
, so no additional fields can be added to the type.
The example FieldAccess
contains a record constructor
[ a > 2, b > "B" ]
. We can write a general type inference rule for it:
e_1: t_1, ..., e_n: t_n
 [rec]
[ f_1 > e_1, ..., f_n > e_n]:
Rec(
RowCons(f_1, t_1,
RowCons(f_2, t_2,
...
RowCons(f_n, t_n, RowNil)
...)
)
)
In FieldAccess
, we use row types to construct a series of type equations
(over free type variables k
, ..., q
):
// from LET m == [ a > 2, b > "B" ] IN
m_type = Rec("a", Int, RowCons("b", Str, RowNil))
// from m.a > 1
m_type = Rec(RowCons("a", k, l))
k = Int
// from m.b = "B"
m_type = Rec(RowCons("b", m, n))
m = Str
// from m.c = { 1, 2 }
m_type = Rec(RowCons("c", p, q))
p = Set(Int)
To solve the above equations, one has to apply unification rules. Precise
unification rules for rows are given in the paper by Leijen05. Importantly,
their unification rules allow RowCons(f1, t1, RowCons(f2, t2, r3))
to be
unified with RowCons(f2, t2, RowCons(f1, t1, r3))
. Hence, fields may bubble
up to the head, and it should be possible to isolate a single field and assign
the rest to a type variable. By partially solving the above equation, we would
arrive at contradiction. This supports our intuition that the operator
FieldAccess
is illtyped.
Hence, we formulate the type inference rule for record access in our type system as follows:
r: Rec(RowCons("f", t_1, t_2))
 [rec_acc]
r.f: t_1
Note that the above rule can be rewritten into a series of equalities over types variables and type terms, which is how this would be implemented in the type checker.
If we only had to deal with records, that would be a complete solution. Unfortunately, variants introduce additional complexity.
5.2. Variants
Example 5.2.1. Now we have to figure out how to deal with TLA+ expressions like:
{ [ tag > "1a", bal > 3 ], [ tag > "2a", bal > 4, val > 0 ] }
Obviously, we cannot fit both of the records into a single plain record type,
provided that we want to precisely track the fields that are present in a
record. So the type checker should report a type error, if we only implement
type inference for the case explained in Section 5.1. To support this important
pattern, we introduce variants. They are similar to unions in
TypeScript.
In contrast to TypeScript, we fix one field to designate the record type in a
variant. Also, we are using the word "variant", to avoid any confusion
with the TLA+ operators UNION
and \union
.
We reserve the field name tag
for the record discriminator.
Variant constructor. We introduce a special TLA+ operator Variant
that extracts the tag from a record and wraps the record into a variant.
We need this operator to distinguish between plain records and records that
belong to a variant. The operator Variant
is defined in TLA+ as follows:
Variant(r) ==
\* fallback untyped implementation
r
This operator does not change its argument, but it provides the type checker with a hint that the record should be treated as a member of a variant.
Consider the record constructor of n+1
fields, one of them being the field
"tag"
:
[ tag > "<TAG>", f_1 > e_1, ..., f_n > e_n ]
Here, "<TAG>"
stands for a string literal such as "1a"
or "2a"
. The
general rule for Variant([ tag > "<TAG>", f_1 > e_1, ..., f_n > e_n ])
looks as follows:
e_1: t_1, ..., e_n: t_n
z is a fresh type variable
 [variant]
Variant([ tag > "<TAG>", f_1 > e_1, ..., f_n > e_n ]):
Variant(
RowCons("<TAG>",
Rec(
RowCons("tag", Str,
RowCons(f_1, t_1,
...
RowNil)...)
),
z
)
)
According to the rule [variant]
, the operator Variant
wraps a record
constructor that contains a string literal for the field tag
. The variant
contains the record that was passed in the constructor, whereas the other
alternatives of the variant are captured with a fresh type variable z
, which must
be a row.
Importantly, we use rows at two levels:

To construct a single record, whose shape is defined precisely.

To construct a variant, whose only record is known at the time, while the rest is captured with the type variable
z
.
Going back to Example 5.2.1, the set constructor would produce a set of equalities:
a = Set(b)
a = Set(d)
b = Variant(RowCons("1a",
Rec(RowCons("tag", Str, RowCons("bal", Int, RowNil))),
c))
d = Variant(RowCons("2a",
Rec(RowCons("tag", Str,
RowCons("bal", Int,
RowCons("val", Int, RowNil)))),
e))
By solving these equalities with unification, we will arrive at the following variant:
Set(Variant(RowCons(
"1a",
Rec(RowCons("tag", Str, RowCons("bal", Int, RowNil))),
RowCons(
"2a",
Rec(RowCons("tag", Str,
RowCons("bal", Int,
RowCons("val", Int, RowNil)))),
t
))))
Note that we still do not know the precise shape of the variant, as it
closes with the type variable t
. This is actually what we expect, as the set
may be combined with records of other shapes. Normally, the final shape of a
variant propagates via state variables of the TLA+ specification.
Type annotations for variants. Snowcat requires that all state variables
are annotated. What shall we write for variants? We introduce the common type
notation for variants that separates records with a pipe, that is, 
. For
instance, consider the following variable declaration in a TLA+ specification:
VARIABLES
\* @type: Set([ tag: "1a", bal: Int ]  [ tag: "2a", bal: Int, val: Int ]);
msgs
Note that even though the syntax of individual elements of a variant is very similar to that of a record, there is small difference: The tag field is not declared as a string type, but carries the values of the tag itself.
As variants can grow large very quick, it is more convenient to introduce them via a type alias. For instance:
VARIABLES
(*
@typeAlias: MESSAGE =
[ tag: "1a", bal: Int ]
 [ tag: "2a", bal: Int, val: Int ]);
@type: Set(MESSAGE);
*)
msgs
Filter a set of variants. As we have seen, the following pattern is quite common in TLA+ specifications, e.g., it is met in Paxos:
\E m \in msgs:
/\ m.type = "1a"
...
LET Q1b == { m \in msgs : m.type = "1b" /\ ... }
We introduce the operator FilterByTag
that is a typesafe version of this
pattern:
FilterByTag(Set, tag) == { e \in Set: e.tag = tag }
We introduce a special type inference rule for FilterByTag
:
set: Set(Variant(RowCons("<TAG>", r, z)))
 [variant_filter]
FilterByTag(set, "<TAG>"): Set(r)
Importantly, FilterByTag
returns a set of records that carry the tag <TAG>
,
so we can access record fields of every individual record in the set.
Match by tag. In rare cases, we do not have a set to filter. For instance, we could have a sequence of log messages:
VARIABLE
\* @type: Seq([ tag: "EventA", val: Int ]  [ tag: "EventB", src: Str ]);
log
In this case, we would not be able to easily use FilterByTag
. Of course, we
could wrap a variant into a singleton set and then apply FilterByTag
to it
and CHOOSE
on top of it. However, this looks clunky and does not guarantee
type safety. A much simpler solution is to introduce another special operator:
MatchTag(variant, tag, ThenOper(_), ElseOper(_)) ==
\* fallback untyped implementation
IF variant.tag = tag
THEN ThenOper(variant)
ELSE ElseOper(variant)
The idea of MatchTag
is that it passes the extracted record to ThenOper
,
when its tag value matches tag
; otherwise, it passes the reduced variant to
ElseOper
. This is precisely captured by the inference rule:
variant: Variant(RowCons("<TAG>", r, t))
ThenOper: r => z
ElseOper: Variant(t) => z
 [variant_match]
MatchTag(variant, "<TAG>", ThenOper, ElseOper):
(Variant(RowCons("<TAG>", r, t)),
Str,
r => z,
Variant(t) => z
) => z
The above rule looks menacing. Here is an example of matching a record
in the above example with the variable log
:
IsDefined(eventAOrB) ==
LET ElseB(onlyB) ==
MatchTag(onlyB, "EventB", LAMBDA b: b.src /= "", LAMBDA elseValue: FALSE)
IN
MatchTag(event, "EventA", LAMBDA a: a.val /= 1, ElseB)
The operator ElseB
looks redundant, as we know that onlyB
is a singleton
variant. To this end, we introduce the operator MatchOnly
:
MatchOnly(variant, ThenOper(_)) ==
\* fallback untyped implementation
ThenOper(variant)
For completeness, we give the inference rule for MatchOnly
:
variant: Variant(RowCons("<TAG>", r, RowNil))
ThenOper: r => z
 [variant_match_only]
MatchOnly(variant, ThenOper):
(Variant(RowCons("<TAG>", r, RowNil)),
r => z
) => z
It looks like the solution with MatchTag
and MatchOnly
introduce a lot of
boilerplate. However, this is probably the best solution that we can have,
unless we can extend the grammar of TLA+.
5.3. Changes in the model checker
Having precise types for variants, we have two options:

Keep the current encoding, that is, a variant is encoded as a superrecord that contains all possible fields of the member records. The type checker will guarantee that we do not access the fields of the superrecord that are not present in the actual record type.

Implement the suggestion by Shon Feder.
Although Option 2 looks nicer, we prefer keeping Option 1. The reason is that the current implementation introduces the minimal number of constraints by mashing all possible fields into a superrecord. The alternative solution (option 2) would introduce additional constraints, when the spec requires us to extract an element from a set.
Recall the example with a set of messages:
VARIABLES
(*
@typeAlias: MESSAGE =
[ tag: "1a", bal: Int ]
 [ tag: "2a", bal: Int, val: Int ]);
@type: Set(MESSAGE);
*)
msgs
Consider an existential quantifier over the variable msgs
:
Next ==
\E m \in msgs:
P
In the current encoding, m
is a superrecord that contains three fields:
tag
, bal
, and val
, even if some of these fields are not required by the
actual type of m
. In the alternative encoding, m
is a tuple tup
,
which equals to one of the following tuples, depending on the value of the
field m.tag
:
tup = IF m.tag = "1a"
THEN << { [ tag > "1a", bal > b ] }, {} >>
ELSE << {}, [ tag > "2a", bal > b, val > v ] }, {} >>
Since it is impossible to statically compute the actual type of m
, the
rewriting rules would have to replicate the structure of both elements of the
tuple tup
. This would lead to a blowup in the number of constraints.
5.4. Additional requirements
Given the decisions in Section 5.3, we additionally require that all records in
a variant type have compatible field types. In more detail, if a variant type
contains two record types [ tag: "A", value: a, ... ]
and [ tag: "B", value: b, ...]
, then the types a
and b
must be unifiable. In practice, this
often implies that a
and b
are simply the same type.
5.5. The Variants module
We introduce a new module that is called Variants.tla
. It contains the
operators Variant
, FilterByTag
, MatchTag
, and MatchOnly
. This module
will be distributed with Apalache. As is custom in the TLA+ community, the
users should be also able to copy Variants.tla
next to their specification.
6. Consequences
This will be a relatively big change in the types and the type checker. Additionally, it would render many existing specifications illtyped, as the proposed solution imposes a stricter typing discipline. On the positive, the proposed solution is backwardscompatible with TLC, as we are proposing the default untyped implementation for the operators.
ADR015: Informal Trace Format in JSON
authors  proposed by  last revised 

Igor Konnov  Vitor Enes, Andrey Kupriyanov  20230914 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
We propose a simple format for counterexamples (traces) in JSON. Although Apalache already supports serialization to JSON in ADR005, it is a general serialization format for all the constructs of TLA+ that are supported by Apalache. This makes tool integration harder. It also make it hard to communicate counterexamples to engineers who are not familiar with TLA+. This ADR015 contains a very simple format that does not require any knowledge of TLA+ and can be easily integrated into other tools.
Revisions
Rev. 20230914. Each integer value num
is now represented as { #bigint: "num" }
.
The use of JSON numbers is not allowed anymore. This simplifies custom parsers for ITF,
see Consequences.
Context
A TLA+ execution (called a behavior in TLA+) is a very powerful concept. It can represent virtually any execution of a state machine, including sequential programs, concurrent programs, and distributed systems. Counterexamples that are produced by TLC and Apalache are just executions of a TLA+ state machine. These counterexamples have two shapes:

A finite execution, that is, a sequence of states.

A lasso execution, that is, a finite sequence of states (prefix) followed by an infinitely repeated sequence of states (loop). Any infinite execution of a finitestate system can be represented by a lasso. (In general, executions of infinitestate systems cannot be represented by lassos.)
Although the concept of an execution in TLA+ is quite simple, it builds upon the vocabulary of TLA+. Moreover, TLA+ counterexamples are using the expression language of TLA+.
To illustrate the problem, consider a very simple TLA+ specification of the MissionariesAndCannibals puzzle (specified by Leslie Lamport). We use a typed version of this specification, see MissionariesAndCannibalsTyped. Consider the following instance of the specification:
 MODULE MC_MissionariesAndCannibalsTyped 
Missionaries == { "m1_OF_PERSON", "m2_OF_PERSON" }
Cannibals == { "c1_OF_PERSON", "c2_OF_PERSON" }
VARIABLES
\* @type: Str;
bank_of_boat,
\* @type: Str > Set(PERSON);
who_is_on_bank
INSTANCE MissionariesAndCannibalsTyped
NoSolution ==
who_is_on_bank["E"] /= {}
=============================================================================
By checking the invariant NoSolution
, we obtain the following counterexample
in TLA+:
 MODULE counterexample 
EXTENDS MC_MissionariesAndCannibalsTyped
(* Constant initialization state *)
ConstInit == TRUE
(* Initial state *)
State0 ==
bank_of_boat = "E"
/\ who_is_on_bank
= "E"
:> { "c1_OF_PERSON", "c2_OF_PERSON", "m1_OF_PERSON", "m2_OF_PERSON" }
@@ "W" :> {}
(* Transition 0 to State1 *)
State1 ==
bank_of_boat = "W"
/\ who_is_on_bank
= "E" :> { "c1_OF_PERSON", "m1_OF_PERSON" }
@@ "W" :> { "c2_OF_PERSON", "m2_OF_PERSON" }
(* Transition 0 to State2 *)
State2 ==
bank_of_boat = "E"
/\ who_is_on_bank
= "E" :> { "c1_OF_PERSON", "m1_OF_PERSON", "m2_OF_PERSON" }
@@ "W" :> {"c2_OF_PERSON"}
(* Transition 0 to State3 *)
State3 ==
bank_of_boat = "W"
/\ who_is_on_bank
= "E" :> {"c1_OF_PERSON"}
@@ "W" :> { "c2_OF_PERSON", "m1_OF_PERSON", "m2_OF_PERSON" }
(* Transition 0 to State4 *)
State4 ==
bank_of_boat = "E"
/\ who_is_on_bank
= "E" :> { "c1_OF_PERSON", "c2_OF_PERSON" }
@@ "W" :> { "m1_OF_PERSON", "m2_OF_PERSON" }
(* Transition 0 to State5 *)
State5 ==
bank_of_boat = "W"
/\ who_is_on_bank
= "E" :> {}
@@ "W"
:> { "c1_OF_PERSON", "c2_OF_PERSON", "m1_OF_PERSON", "m2_OF_PERSON" }
(* The following formula holds true in the last state and violates the invariant *)
InvariantViolation == who_is_on_bank["E"] = {}
================================================================================
(* Created by Apalache on Wed Dec 22 09:18:50 CET 2021 *)
(* https://github.com/informalsystems/apalache *)
The above counterexample looks very simple and natural, if the reader knows TLA+. In our experience, these examples look alien to engineers, who are not familiar with TLA+. It is unfortunate, since the counterexamples have a very simple shape:

They are simply sequences of states.

Every state is a mapping from variable names to expressions that do not refer to other variables.

The expressions are using a very small subset of TLA operators:

Integer and string literals.

Set constructor, sequence/tuple constructor, record constructor.

TLC operators over functions:
:>
and@@
.
In hindsight, the above expressions are not very far from the JSON format. As many engineers know JSON, it seems natural to write these counterexamples in JSON.
Options

Use the TLA+ format:

Pros:
 easy to understand, if you know TLA+.
 it looks amazing in PDF.

Cons:
 quite hard to understand, if you don't know TLA+.
 quite hard to parse automatically.


Use the JSON serialization as in ADR005:

Pros:
 easy to parse automatically.

Cons:
 almost impossible to read.
 too detailed and too verbose.
 requires the knowledge of Apalache IR and of TLA+.


Use the Informal Trace Format, which is proposed in this ADR:

Pros:
 almost no introduction is required to read the traces.
 relatively compact.
 easy to parse automatically.
 uses the idioms that are understood by the engineers.
 not bound to TLA+.

Cons:
 consistency of the format is in conflict with the ease of writing.

Solution
In this ADR, we propose a very simple format that represents executions of
state machines that follows the concepts of TLA+ and yet avoids complexity of
TLA+. It is so simple that we call it "Informal Trace Format" (ITF).
(Obviously, it is formal enough to be machinereadable). By convention, the
files in this format should end with the extension .itf.json
.
The ITF Format
Trace object. A trace in ITF is a JSON object:
{
"#meta": <optional object>,
"params": <optional array of strings>,
"vars": <array of strings>,
"states": <array of states>,
"loop": <optional int>
}
The field #meta
is an arbitrary JSON object, whose purpose is to provide
the reader with additional comments about the trace. For example, it may look
like:
"#meta": {
"description": "Generated by Apalache",
"source": "MissionariesAndCannibalsTyped.tla"
}
The optional field params
is an array of names that must be set in the
initial state (if there are any parameters). The parameters play the same role
as CONSTANTS
in TLA+. For example, the field may look like:
"params": [ "Missionaries", "Cannibals" ]
The field vars
is an array of names that must be set in every state.
For example, the field may look like:
"vars": [ "bank_of_boat", "who_is_on_bank" ]
The field states
is an array of state objects (see below). For example,
the field may look like:
"states": [ <state0>, <state1>, <state2> ]
The optional field loop
specifies the index of the state (in the array of
states) that starts the loop. The loop ends in the last state. For example,
the field may look like:
"loop": 1
State object. A state is a JSON object:
{
"#meta": <optional object>,
"<var1>": <expr>,
...
"<varN>": <expr>
}
As in the trace object, the field #meta
may be an arbitrary object.
Different tools may use this object to write their metadata into this object.
The names <var1>, ..., <varN>
are the names of the variables that are
specified in the field vars
. Each state must define a value for every specified variable. The syntax of <expr>
is specified below.
Expressions. As usual, expressions are inductively defined. An expression
<expr>
is one of the following:

A JSON Boolean: either
false
ortrue
. 
A JSON string literal, e.g.,
"hello"
. TLA+ strings are written as strings in this format. 
A big integer of the following form:
{ "#bigint": "[][09]+" }
. We are using this format, as many JSON parsers impose limits on integer values, see RFC7159. Big and small integers must be written in this format. 
A list of the form
[ <expr>, ..., <expr> ]
. A list is just a JSON array. TLA+ sequences are written as lists in this format. 
A record of the form
{ "field1": <expr>, ..., "fieldN": <expr> }
. A record is just a JSON object. Field names should not start with#
and hence should not pose any collision with other constructs. TLA+ records are written as records in this format. 
A tuple of the form
{ "#tup": [ <expr>, ..., <expr> ] }
. There is no strict rule about when to use sequences or tuples. Apalache differentiates between tuples and sequences, and it may produce both forms of expressions. 
A set of the form
{ "#set": [ <expr>, ..., <expr> ] }
. A set is different from a list in that it does not assume any ordering of its elements. However, it is only a syntax form in our format. Apalache distinguishes between sets and lists and thus it will output sets in the set form. Other tools may interpret sets as lists. 
A map of the form
{ "#map": [ [ <expr>, <expr> ], ..., [ <expr>, <expr> ] ] }
. That is, a map holds a JSON array of twoelement arrays. Each twoelement arrayp
is interpreted as follows:p[0]
is the map key andp[1]
is the map value. Importantly, a key may be an arbitrary expression. It does not have to be a string or an integer. TLA+ functions are written as maps in this format. 
An expression that cannot be serialized:
{ "#unserializable": "<string representation>" }
. For instance, the set of all integers is represented with{ "#unserializable": "Int" }
. This should be a very rare expression, which should not occur in normal traces. Usually, it indicates some form of an error.
ITF as input to Apalache
To be able to read ITF traces, Apalache demands the following:
 The
#meta
field must be present  The
#meta
field must contain a fieldvarTypes
, which is an object, the keys of which are the variables declared invars
, and the values are string representations of their types (as defined in ADR002).
For example:
"#meta": {
"varTypes": {
"bank_of_boat": "Str",
"who_is_on_bank": "Str > Set(PERSON)"
}
}
Example
The counterexample to NoSolution
may be written in the ITF format as follows:
{
"#meta": {
"source": "MC_MissionariesAndCannibalsTyped.tla",
"varTypes": {
"bank_of_boat": "Str",
"who_is_on_bank": "Str > Set(PERSON)"
}
},
"vars": [ "bank_of_boat", "who_is_on_bank" ],
"states": [
{
"#meta": { "index": 0 },
"bank_of_boat": "E",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ "c1_OF_PERSON", "c2_OF_PERSON",
"m1_OF_PERSON", "m2_OF_PERSON" ] } ],
[ "W", { "#set": [] } ]
]
}
},
{
"#meta": { "index": 1 },
"bank_of_boat": "W",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ "c1_OF_PERSON", "m1_OF_PERSON" ] } ],
[ "W", { "#set": [ "c2_OF_PERSON", "m2_OF_PERSON" ] } ]
]
}
},
{
"#meta": { "index": 2 },
"bank_of_boat": "E",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ "c1_OF_PERSON",
"m1_OF_PERSON", "m2_OF_PERSON" ] } ],
[ "W", { "#set": [ "c2_OF_PERSON" ] } ]
]
}
},
{
"#meta": { "index": 3 },
"bank_of_boat": "W",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ "c1_OF_PERSON" ] } ],
[ "W", { "#set": [ "c2_OF_PERSON", "m1_OF_PERSON", "m2_OF_PERSON" ] } ]
]
}
},
{
"#meta": { "index": 4 },
"bank_of_boat": "E",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ "c1_OF_PERSON", "c2_OF_PERSON" ] } ],
[ "W", { "#set": [ "m1_OF_PERSON", "m2_OF_PERSON" ] } ]
]
}
},
{
"#meta": { "index": 5 },
"bank_of_boat": "W",
"who_is_on_bank": {
"#map": [
[ "E", { "#set": [ ] } ],
[ "W", { "#set": [ "c1_OF_PERSON", "c2_OF_PERSON",
"m1_OF_PERSON", "m2_OF_PERSON" ] } ]
]
}
}
]
}
Compare the above trace format with the TLA+ counterexample. The TLA+ example looks more compact. The ITF example is heavier on the brackets and braces, but it is also designed with machinereadability and tool automation in mind, whereas TLA+ counterexamples are not. However, the example in the ITF format is also selfexplanatory and does not require any understanding of TLA+.
Note that we did not output the operator InvariantViolation
of the TLA+
example. This operator is simply not a part of the trace. It could be added in
the #meta
object by Apalache.
Discussion
Shon Feder @shonfeder flagged important concerns about irregularity of the proposed format in the PR comments. In a regular approach we would treat all expressions uniformly. For example:
// proposed form:
"hello"
// regular form:
{ "#type": "string", "#value": "hello" }
// proposed form:
{ "#set": [ 1, 2, 3] }
// regular form:
{
"#type": "set",
"#value": [
{ "#type": "int", "#value": "1" },
{ "#type": "int", "#value": "2" },
{ "#type": "int", "#value": "3" }
]
}
The more regular approach is less concise. In the future, we might want to add a flag that lets the user choose between the regular output and the output proposed in this ADR, which is more ad hoc.
Another suggestion is to use JSON schema. For the moment, it seems to be a heavyweight solution with no obvious value. However, we should keep it in mind and use schemas, when the need arises.
Consequences
We have found that the ITF format is easy to produce and relatively easy to parse.
Ambiguity in the representation of integers. As it was brought up in the initial
discussions, the choice between representing integers
as JSON numbers, e.g., 123
and objects, e.g., { #bigint: "123" }
, makes it harder to
write a parser of custom ITF traces. Hence, we have decided to keep only the object format,
as the more general of the two representations.
ADR16: ReTLA  Relational TLA
author  revision 

Jure Kukovec  1 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
We propose introducing support for a severely restricted fragment of TLA+, named Relational TLA (reTLA for short), which covers uninterpreted firstorder logic. The simplicity of this fragment should allow Apalache to use a more straightforward encoding, both in SMT, as well as potentially in languages suited for alternative backend solvers.
Running Apalache with this encoding would skip the modelchecking pass and instead produce a standalone file containing all of the generated constraints, which could be consumed by other tools.
Context
Apalache currently supports almost the full suite of TLA+ operators. Consequently, the standard encoding of TLA+ into SMT is very general, and a lot of effort is spent on encoding data structures, such as sets or records, and their evolution across the states. Additionally, we need to track arenas; bookkeeping auxiliary constructions, which are a byproduct of our encoding approach, not TLA+ logic itself.
However, it is often the case that, with significant effort on the part of the specification author, expressions can be rewritten in a way that avoids using the more complex structures of TLA+. For example, consider the following snippet of a messagepassing system:
CONSTANT Values
VARIABLE messages
SendT1(v) == messages \union { [ type > "t1", x > v ] }
ReadT1 ==
\E msg \in messages:
/\ msg.type = "t1"
/\ F(msg.x) \* some action
Next ==
\/ \E v \in Values: SendT1(v)
\/ ReadT1
The central object is the set messages
, which is modified at each step, and contains records. This makes it one of the more expensive expressions, in terms of the underlying SMT encodings in Apalache. However, as far as the use of messages
goes, there is a major insight to be had: it is not necessary to model messages
explicitly for ReadT1
, we only need to specify the property that certain messages of type "t1" with given payloads have been sent. Naturally, modeling messages
explicitly is sufficient for this purpose, but if one wanted to avoid the use of sets and records, one could write the following specification instead:
CONSTANT Values
VARIABLE T1messages
SendT1(v) == [T1messages EXCEPT ![v] = TRUE]
ReadT1 ==
\E v \in Values:
/\ T1messages[v]
/\ F(v) \* some action
Next ==
\/ \E v \in Values: SendT1(v)
\/ ReadT1
By encoding, for example, set membership checks as predicate evaluations, one can write some specifications in a fragment of TLA+ that avoids all complex data structures, setsofsets, records, sequences, and so on, and replaces them with predicates (functions). Rewriting specifications in this way is nontrivial, and shouldn't be expected of engineers, however, should a specification author undertake such a transformation, we should be able to provide some payoff. If we only limit ourselves to specifications in this restricted fragment (defined explicitly below), the current SMT encoding is needlessly complex. We can implement a specialized encoding, which does not use arena logic of any kind, but is much more direct and even lends itself well to multiple kinds of solvers (e.g. IVy or VMT, in addition to standard SMT).
Options

Reuse most of the existing implementation and encoding, with a modified language watchdog, then output an SMT file from the context, instead of solving.

Pros:
 Little work

Cons:
 Locked to the SMT format
 Unnecessary additional SMT constraints produced


Write custom rewriting rules that generate constraints symbolically

Pros:
 Fewer constraints
 Higher level of abstraction
 Can support multiple output formats

Cons:
 More work

Solution
We propose option (2), and give the following categorization of the reTLA fragment:
 Boolean, integer and uninterpreted literals (including strings)
 Restricted sets:
Int
,Nat
orBOOLEAN
, orCONSTANT
declared and has a typeSet(T)
, for some uninterpreted typeT
, or
 Boolean operators (
/\, \/, =>, <=>, ~
)  Quantified expressions (
\E x \in S: P, \A x \in S: P
), on the condition thatP
is in reTLA andS
is a restricted set.  Functions:
 Definitions (
[x1 \in S1, ..., xn \in Sn > e]
), on the condition that:e
is in reTLA and has anInt
,Bool
or uninterpreted type All
Si
are restricted sets.
 Updates (
[f EXCEPT ![x] = y]
), ify
is in reTLA  Applications (
f[x]
)
 Definitions (
 (In)equality and assignments:
a = b
anda /= b
if botha
andb
are in reTLAx' = v
ifx
is aVARIABLE
andv
is in reTLA
 Control flow:
IF p THEN a ELSE b
ifp,a,b
are all in reTLA
In potential future versions we are likely to also support:
 Standard integer operators (
+, , u, *, %, <, >, <=, >=
)  ranges
a..b
, where botha
andb
are in reTLA.  Tuples
Consequences
Reserved for the future.
PDR017: Checking temporal properties
authors  last revised 

Igor Konnov, Philip Offtermatt  20220401 
This is a preliminary design document. It will be refined and it will mature into an ADR later.
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
The "TLA" in TLA+ stands for Temporal Logic of Actions, whereas the plus sign (+) stands for the rich syntax of this logic. So far, we have been focusing on the plus of TLA+ in Apalache. Indeed, the repository of TLA+ Examples contains a few occurrences of temporal properties.
In this PDR, we lay out a plan for implementing support for model checking of temporal properties in Apalache.
Context
Apalache supports checking of several kinds of invariants: state, action, and trace invariants. Some of the TLA+ users do not want to be limited by invariants, but want to write temporal properties, which let them express safety and liveness more naturally. A detailed description of temporal properties can be found in Specifying Systems (Sections 16.2.34). In short, temporal formulas are Boolean combinations of the following kinds of subformulas:

State predicates:
 Boolean formulas that do not contain primes.

Action predicates:

Stutter
A
:[A]_e
, for an action formulaA
and an expressione
(usually a tuple of variables). This predicate is equivalent toA \/ e' = e
. 
Nostutter
A
:<A>_e
, for an action formulaA
and an expressione
, which is equivalent toA /\ e' /= e
. 
ENABLED A
, for an action formulaA
, is true in a states
if there is a statet
such thats
is transformed tot
via actionA
.


Temporal formulas:

Eventually:
<>F
, for a temporal formulaF
. 
Always:
[]F
, for a temporal formulaF
. 
Weak fairness:
WF_e(A)
, for an expressione
and a formulaF
, which is equivalent to[]<>~(ENABLED <A>_e) \/ []<><A>_e
. 
Strong fairness:
SF_e(A)
, for an expressione
and a formulaF
, which is equivalent to<>[]~(ENABLED <A>_e) \/ []<><A>_e
. 
Leadsto:
F ~> G
, which is equivalent to[](F => <>G)
.

Technically, TLA+ also contains four other operators that are related to
temporal properties: A \cdot B
, \EE x: F
, \AA x: F
, and F +> G
. These
very advanced operators appear so rarely that we ignore them in this work.
Most likely, For their semantics, check Specifying Systems (Section
16.2.34).
Design space
SAT encodings of bounded model checking for LTL are a textbook material. A
linear encoding for LTL is presented in the paper Biere et al. 2006. It is
explained in Handbook of Satisfiability (Chapter 14). Hence, we do not have
to do much research around this. However, we have to adapt the standard
techniques to the domain of TLA+. For instance, we have to understand how to
deal with ENABLED
, WF
, and SF
, which deviate from the standard setting of
model checking.
Viktor Sergeev wrote the first prototype for liveness checking at University of Lorraine in 2019. Since his implementation was tightly integrated with the exploration algorithm, which was refactored several times, this implementation has not been integrated in the main branch. We have learned from this prototype and discuss our options under this angle.
There are two principally different approaches to the implementation of temporal model checking:

Tight integration with Transition Executor.
In this approach, we would extend the transition executor to incrementally check LTL properties via the encoding by Biere et al. 2006. This approach would let us implement various optimizations. However, it would be harder to maintain and adapt, as we have seen from the first prototype.

Specification preprocessing.
In this approach, given a specification
S
and a temporal propertyP
, we would produce another specificationS_P
and an invariantI_P
that has the following property: the temporal propertyP
holds on specificationS
if and only if the invariantI_P
holds on specificationS_P
. In this approach, we encode the constraints by Biere et al. 2006 directly in TLA+. The potential downside of this approach is that it may be less efficient than the tight integration with the transition executor.
We choose the second approach via specification preprocessing. This will simplify maintenance of the codebase. Additionally, the users will be able to see the result of preprocessing and optimize it, if needed. When the new implementation is well understood, we can revisit it and consider Option 1, once ADR010 is implemented.
Work plan
The work plan is tracked in the issue on temporal properties.
We propose to split this work into two big subtasks:

Task 1. Temporal operators: Support for
<>P
,[]P
,<A>_e
, and[A]_e
via preprocessing. 
Task 2. Fairness: Support for
ENABLED A
,WF_e(A)
, andSF_e(A)
via preprocessing.
The task on Temporal operators is wellunderstood and poses no technical
risk. By having solved Task 1, we can give users a relatively complete toolset
for safety and liveness checking. Indeed, even fairness properties can be
expressed via <>
and []
.
To support temporal reasoning as it was designed by Leslie Lamport, we have to solve Task 2. Most likely, we will have to introduce additional assumptions about specifications to solve it.
1. Temporal operators
This task boils down to the implementation of the encoding explained in Biere et al. 2006.
In model checking of temporal properties, special attention is paid to lasso
executions. A lasso execution is a sequence of states s[0], s[1], ..., s[l], ..., s[k]
that has the following properties:
 the initial state
s[0]
satisfiesInit
,  every pair of states
s[i]
ands[i+1]
satisfiesNext
, fori \in 0..k1
, and  the loop closes at index
l
, that is,s[l] = s[k]
.
The lasso executions play an important role in model checking, due to the lasso property of finitestate systems:
Whenever a finitestate transition system `M` violates a temporal property
`F`, this system has a lasso execution that violates `F`.
You can find a proof in the book on Model Checking. As a result, we can focus on finding a lasso as a counterexample to the temporal property. Importantly, this property holds only for finitestate systems. For example, if all variable domains are finite (and bounded), then the specification is finitestate. However, if a specification contains integer variables, it may produce infinitely many states. That is, an infinitestate system may still contain lassos as counterexamples but it does not have to, which makes this technique incomplete. An extension to infinitestate systems was studied by Padon et al. 2021. This is beyond the scope of this task.
There are multiple ways to encode the constraints by Biere et al. 2006. The different ways are demonstrated on the EWD998 spec, which specifies a protocol for termination detection, using token passing in a ring.
Trace Invariants
The lasso finding problem can be encoded as a trace invariants. See e.g. the EWD998 protocol with trace invariants. Roughly, a loop is encoded by demanding there exists a loop index at which point the state is identical to the state at the end of the execution.
Implementation details:
 Instead of quantifying over indices, one could use an additional Boolean variable starting out FALSE that nondeterministically guesses when the execution enters the loop and is set to TRUE at that point. Experiments suggest this negatively impacts performance, but it can help understand counterexamples, since the loop is immediately visible in the states.
Advantages:
 The predicate in the spec is very close to the semantic meaning of the temporal operators, e.g.
[] x >= 2
becomes\A step \in DOMAIN hist: hist[step].x >= 2
 Only very few new variables are added (none, but depending on implementation choices maybe one/two).
Disadvantages:

Trace invariants require Apalache to pack the sequence of states. This sometimes produces unnecessary constraints.

When a trace invariant is violated, the intermediate definitions in this invariant are not printed in the counterexample. This will make printing of the counterexamples to liveness harder, e.g. see an example
Encoding with auxiliary variables
The loop finding problem can alternatively be approached
by adding extra variables: One variable InLoop
which
determines whether the execution is currently on the loop,
and for each variable foo
of the original spec an extra variable loop_foo
,
which, once InLoop
is true, stores the state of foo
at the start of the loop.
Then, the loop has been completed if vars = loop_vars
.
Apart from the variables for finding the loop, this approach also needs extra variables for determining the satisfaction of the temporal property to be checked. There again exist multiple ways of concretely implementing this:
Encoding with Buchi automata
One can extend the spec with a Buchi automaton which is updated in each step. The Buchi automaton encodes the negation of the temporal property, thus if the automaton would accept, the property does not hold. By checking whether an accepting state of the automaton is seen on the loop, it can be determined whether the automaton accepts for a looping execution. The encoding is described in Biere et al. 2002 See e.g. the EWD998 protocol with a Buchi automaton.
Implementation details:
 An implementation of this encoding would need an implementation of an algorithm for the conversion from LTL to Buchi automata. This could be an existing tool, e.g. Spot or our own implementation.
Advantages:
 Buchi automata for very simple properties can be simple to understand
 Underlying automata could be visualized
 Only needs few extra variables  the state of the Buchi automaton can easily be encoded as a single integer
Disadvantages:
 Can be slow: Buchi automata generally exhibit either nondeterminism or can get very large
 Hard to understand: Engineers and even experts have a hard time intuitively understanding Buchi automata for mildly complicated properties
Tableau encoding
One can instead extend the spec with auxiliary Boolean variables roughly corresponding to all nodes in the syntax tree who have temporal operators beneath them. The value of each variable in each step corresponds to whether the formula corresponding to that node in the syntax tree is satisfied from that point forward. The encoding is described in Section 3.2 of Biere et al. 2006 See e.g. the EWD998 protocol encoded with a tableau.
Implementation details:
 Naming the auxiliary variables is very important, since they are supposed to represent the values of complex formulas (ideally would simple have that formula as a name, but this is not syntactically possible for most formulas), and there can be many of them.
Advantages:
 Very clear counterexamples: In each step, it is clearly visible which subformulas are or are not satisfied.
 Relatively intuitive specs: The updates to the auxiliary variables correlate with the intuitive meaning of their subformulas rather directly in most cases
Disadvantages:
 Many variables are added: The number of variables is linear in the number of operators in the formula
 Specifications get long: The encoding is much more verbose than that for Buchi automata
Decision  which encoding should be used?
We chose to implement the tableau encoding, since it produces the clearest counterexamples. Buchi automata are hard to understand. For trace invariants, the lack of quality in counterexamples makes it very hard to debug and understand invariant violations.
2. Fairness
WF_e(A)
and SF_e(A)
use ENABLED(A)
as part of their definitions. Hence,
ENABLED(A)
is of ultimate importance for handling WF
and SF
. However, we
do not know how to efficiently translate ENABLED(A)
into SMT. A
straightforward approach requires to check that for all combinations of state
variables A
does not hold.
This work requires further research, which we will do in parallel with the first part of work. To be detailed later...
Consequences
ADR018: Inlining in Apalache
author  revision  last revised 

Jure Kukovec  1  20220421 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
This ADR defines the various kinds of inlining considered in Apalache and discusses the pros and cons of their implementations. Since we have recently reworked the inliner in #1569, we saw it fit to document exactly how inlining is supposed to work and we have chosen the transformations performed in the inlining pass.
Context
TLA+ allows the user to define their own operators (e.g. A
), in addition to the standard ones built into the language itself (e.g. \union
).
This can be done either globally, where the module directly contains a definition, or locally via LETIN, where a local operator is defined within the body of another operator. For example:
GlobalA(p, q) ==
LET LocalB(r) == r * r
IN LocalB(p + q)
defines a global operator GlobalA
, within which there is a locally defined LocalB
.
Suppose we are given an invariant GlobalA(1,2) = 9
. How do we evaluate whether or not this invariant holds? To do that, we need to evaluate
LET LocalB(r) == r * r IN LocalB(p + q)
, and to do that, we need to evaluate LocalB(p + q)
.
However, we cannot evaluate LocalB(p + q)
in a vacuum, because p
and q
are not values we can reason about, but instead formal parameters.
What we need to do, is determine the value of "LocalB(p + q)
, if p = 1
and q = 2
". In other words, we need to apply the substitution {p > 1, q >2}
to LocalB(p + q)
, which gives us LocalB(1 + 2)
.
Repeating this process, we apply the substitution {r > 1 + 2}
to
r * r
, the body of LocalB
, to obtain the following equivalence:
GlobalA(1,2) = 9 <=> (1 + 2) * (1 + 2) = 9
The process of applying these substitutions as syntactic transformations is called inlining.
More precisely, suppose we are given a nonrecursive operator A
with the following definition:
A(p1,...,pn) == body
The term "inlining" (of A
) typically refers to the process of replacing instances of operator application A(e1,...,en)
with body[e1/p1,...,en/pn]
, i.e. the expression obtained by replacing each instance of pi
with ei
within body
.
We elect to use the term in a broader sense of "replacing an operator with its definition", and define two "flavors" of inlining:
 Standard inlining: the instantiation described above
 Nonnullary inlining: the instantiation described above, except the inlining skips nullary LETdefined operators
 Passbyname inlining: replacing an operator name
A
used as an argument to a higherorder operator with a local LETdefinition:LET A_LOCAL(p1,...,pn) == A(p1,...,pn) IN A_LOCAL
The reason for doing (1) is that, at some point, a rewriting rule would have to generate constraints from A(e1,...,en)
.
To do this, we couldn't just separately encode body
and e1,...,en
, because the richness of the data structures allowed in TLA+ makes it difficult to combine independently generated constraints, in cases where the operator parameters are complex expressions (e.g. e1
is some record with varied nested constructs).
This is mostly due to the fact that there is no 1to1 correspondence between TLA+ objects and SMT datatypes, so encoding object equality is more complicated (which would be needed to express that ei
instantiates pi
).
Therefore we must, no later than at the point of the rewriting rule, know body[e1/p1,...,en/pn]
.
While inlining nonnullary operators is strictly necessary, inlining nullary operators is not, because nullary operators, by definition, do not have formal parameters.
Therefore, in a wellconstructed expression, all variables appearing in a nullary operator are scoped, i.e. they are either specificationlevel variables (defined as VARIABLE
), or bound in the context within which the operator is defined, if local. An example of the latter would be i
in
\E i \in S: LET i2 == i * i in i2 = 0
which is not bound in the nullary operator i2
, but it is defined in the scope of the \E
operator, under which i2
is defined. Therefore, any analysis of i2
will have i
in its scope.
The nonnullary variant of (1) is therefore strictly better for performance, because it allows for a sort of caching, which avoids repetition. Consider for example:
A1(p) ==
LET pCached == p
IN F(pCached,pCached)
A2(p) == F(p,p)
If we apply the substitution {p > e}
, for some complex value e
, to the bodies of both operators, the results are
LET pCached == e
IN F(pCached,pCached)
and
F(e,e)
In the first case, we can translate pCached
to a cell (Apalache's SMT representation of TLA+ values, see this
paper for details) and reuse the cell expression twice, whereas in the second, e
is rewritten twice, independently.
So in the case that we perform (1), we will always perform the nonnullary variant, because it is strictly more efficient in our cellarena framework fo rewriting rules.
The reason for doing (2) is more pragmatic; in order to rewrite expressions which feature any of the higherorder (HO) builtin operators, e.g. ApaFoldSet(A, v, S)
, we need to know, at the time of rewriting, how to evaluate an application of A
(e.g. A(partial, current)
for folding).
Performing (2) allows us to make the rewriting rule local, since the definition becomes available where the operator is used, and frees us from having to track scope in the rewriting rules.
Examples
Suppose we have an operator
A(p,q) == p + 2 * q
Then, the result of performing (1) for A(1, 2)
would be 1 + 2 * 2
.
The constant simplification could take the inlined expression and simplify it to 5
, whereas it could not do this across the application boundary of A(1,2)
.
The result of performing (2) for ApaFoldSet(A, 0, {1,2,3})
would be
ApaFoldSet(LET A_LOCAL(p,q) == p + 2 * q, 0, {1,2,3})
While this resulting expression isn't subject to any further simplification, notice that it does contain all the required information to fully translate to SMT, unlike ApaFoldSet(A, 0, {1,2,3})
, which requires external information about A
.
Options
Knowing that we must perform (1) at some point, what remains is to decide whether we perform inlining ondemand as part of rewriting, or whether to isolate it to an independent inliningpass (or as part of preprocessing), i.e. performing a syntactic transformation on the module, that replaces A(e1,...,en)
with body[e1/p1,...,en/pn]
, or merely generating rewriting rules that encode A(e1,...,en)
equivalently as body[e1/p1,...,en/pn]
, while preserving the specification syntax.
Additionally, if we do isolate inlining to a separate pass, we can choose whether or not to perform (2).

Perform no inlining in preprocessing and inline only as needed in the rewriting rules.
 Pros: Spec intermediate output remains small, since inlining increases the size of the specificaiton
 Cons:
 Fewer optimizations can be applied, as some are only applicable to the syntactic forms obtained after inlining (e.g.
ConstSimplifier
can simplifyIF TRUE THEN a ELSE b
, but notIF p THEN a ELSE b
)  Rewriting rules for different encodings have to deal with operators in their generality.
 Fewer optimizations can be applied, as some are only applicable to the syntactic forms obtained after inlining (e.g.

Independently perform only standard (nonnullary) inlining (1), but no passbyname inlining (2)
 Pros: Allows for additional optimizations after inlining (simplification, normalization, keramelization, etc.)
 Cons: Rewriting rules still need scope, to resolve higherorder operator arguments in certain builtin operators (e.g. folds)
 Recall that the nonnullary variant of (1) is strictly better than the simple one (while being trivial to implement), because nullary inlining is prone to repetition.

Independently perform nonnullary inlining and passbyname inlining
 Pros:
 Enables further optimizations (simplification, normalization, keramelization, etc.)
 Using nonnullary inlining has all of the benefits of standard inlining, while additionally being able to avoid repetition (e.g. not inlining
A
inA + A
)  Passbyname inlining allows us to keep rewriting rules local
 Cons: Implementation is more complex
 Pros:
Solution
We elect to implement option (3), as most of its downsides are developer burdens, not theoretical limitations, and its upsides (in particular the ones of nonnullary inlining) are noticeable performance benefits. Maintaining local rewriting rules is also a major technical simplification, which avoids potential bugs with improperly tracked scope.
Consequences
TBD
ADR019: Harmonize changelog management
authors  last revised 

Shon Feder  20220506 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of maintaining a changelog to communicate salient changes about
our software, facing the friction caused by frequent merge conflicts in our
UNRELEASED
changelog, we decided to find a lightweight, conflict free,
changelog process in order to achieve a smoother development process. We have
accepted the development cost and learning curve required by adopting a new
process.
Context
We maintain a changelog
to make it easier for users and contributors to see precisely what notable changes have been made between each release (or version) of the project.
Any changeset that introduces observable changes to the behavior of our software
should include additions to the changelog. Currently, additions are made by
updating the UNRELEASED.md
file. The changes recorded there are then added to
the CHANGES.md
file during our automated release process.
This process creates a race condition over UNRELEASED.md
between any
concurrent pull requests. This consequently results in developers constantly
having to resolve merge conflicts. This busywork slows down development and
adds no value.
Options
We now enumerate and consider the various options we've conceived for addressing this problem.
With the exception of option (0), all of the following options would resolve the problem of merge conflicts and could be integrated into our existing automated deployment pipeline. However each option has different costs w/r/t dependencies, processes, and learning curve.
All of the following options will likely require ongoing maintenance, even option (0): this is because we already have a set of scripts that manage our changelog entries in order to support our fully automated releases. So this is not a space where we are introducing automation for the first time, but is instead a situation where we are changing an existing automated system.
0. Don't change anything
We could do nothing. Developers would continue to resolve changelog conflicts manually.
Pros
 Requires no development time on a solution.
 Would require no additional maintenance.
Cons
 We'll continue having developer slowdown and busy work when merge conflicts arise.
1. Use one of the automated changelog generators
There is a superabundance of tools for automatically generating changelogs. These tools extract changelogs from various different sources such as commit messages, issues, or pull requests.
Pros
 Requires no new development
 Would eliminate the need to add changelog entries in a separate file
Cons
 Developers have to learn the annotation format required by the tool, and they have to remember to deploy it in their commits/PRs/issues.
 Infrastructure maintainers have to learn how to configure and maintain the tool.
 We pickup a dependency external to both the company and our project
 Changelogs are meant to communicate a very specific kind of information to a
specific audience. Git commit messages, pull requests, and issues are for
communicating quite different information to an entirely different audience.
Requiring that these communication channels be overloaded risks degrading the
quality of communications in all the conflated channels.
 However, it's possible we could work out some conventions to add empty commits or designate special PR labels to work around this interference?
2. Use Unclog
@thanethomson has written the unclog utility, which is used in both informalsystems/tendermintrs and informalsystems/ibcrs. The tool stores changelog entries in a directory structure, with each entry in its own file, ensuring merge conflicts can't arise.
Entries are added via the CLI. E.g.:
unclog add id somenewfeature \
issue 23 \
section breakingchanges \
message "Some *new* feature"
If the messages
argument is omitted, it will open your configured editor for
authoring the message.
The tool has subcommands to generate and update the CHANGES.md
, and it supports
a variety of options via its TOML configuration.
Pros
 Requires no new development
 It is developed internally to informal
Cons
 Adds cognitive and procedural overhead to adding changelog entries
 Would add a rust dependency to our devenv
 Requires external contributors to either install a CLI tool or figure out a relatively complex file and folder structure to add changelog entries
3. Use a custom git merge driver
If we wrote a custom git merge driver to work on the simple changelog format we could continue our current practice, and just fall back to the custom merge driver in case of merge conflicts.
Pros
 Allows us to keep our current process of simply updating a markdown file, with which, moreover, most devs are already familiar
Cons
 Could require external dependencies to be installed, depending on implementation.
 It's not possible to configure custom merge drivers for github, so we'd either need to develop a github action or bot to monitor for merge conflicts in the changelog and merge them automatically, or devs would be back to having to resolve merge conflicts locally.
4. Write some lightweight tooling
We could author some simple tooling that would capture the core behavior of
unclog
, using files in directories, but neither require external dependencies
nor knowledge of a new CLI.
Pros
 No new tooling to learn
 No external dependencies to integrate
Cons
 Would require some development time (I estimate approx. 1 day, see the design sketch below for details on the implementation)
 Would require contributors to understand the folder structure
Solution
We have opted for option 4. Despite the added cost of development, we think the simplicity of use and maintenance will offset the development cost.
Design
We will introduce a new directory tree to the project:
.unreleased/
├── breakingchanges/
├── bugfixes/
├── documentation/
└── features/
Each change will be added as a single markdown file in the appropriate directory.
Two sbt
targets will be added to facilitate our continuous deployment:
releaseNotes
, merges the unreleased notes into a single new file of the expected format in ourtarget
directory. This can be used for upload to github releases.changeLog
updatesCHANGES.md
by prepending the output ofreleaseNotes
. It then removes all notes from the.unreleased
directory so that the next iteration of development starts from a clean slate.
Consequences
TBD
ADR020: Introduce static membership in arenas
authors  last revised 

Igor Konnov  20220601 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
We discuss an extension of the model checker arenas. The main application of this extension is a more efficient implementation of powersets and functions sets. Potentially, this extension will let us optimize the number of SMT constraints and thus improve performance of the model checker in general.
1. Context
We give only a brief introduction to arenas. A detailed exposition can be found in KKT19.
1.1. Short introduction to arenas
The model checker heavily relies on the concept of arenas, which are a static overapproximation of the data structures produced by symbolic execution of a TLA+ specification. Here we give a very short recap. In a nutshell, all basic values of TLA+ (such as integers, strings, and Booleans) and data structures (sets, functions, records, tuples, and sequences) are translated into cells. Cells are SMT constants, which can be connected to other cells by edges. Currently, we have three kinds of edges:

has. A membership edge
(c_p, c_e)
represents that a parent cellc_p
potentially contains an elementc_e
(e.g., ifc_p
represents a set). These edges encode manytomany relations. 
dom. A domain edge
(c_f, c_d)
represents that a function cellc_f
has the domain represented with a cellc_d
. These edges encode manytoone relations.Likewise, a domain edge
(c_F, c_c)
represents that a function set cellc_F
has the domain represented with a cellc_c
. 
cdm. A codomain edge
(c_F, c_c)
represents that a function set cellc_F
has the codomain represented with a cellc_c
. These edges encode manytoone relations.For historic reasons, functions are also encoded with the edges called dom and cdm, though the cdm edge points to the function relation, not its codomain. We would prefer to call label the relation edge with rel, not cdm. As these edges are manytoone, we can map them from their kinds
kind > (c_p, c_e)
. This requires simple refactoring, so we are not going to discuss the dom and cdm any further.
There is a need for refactoring and extension of the hasedges. We summarize the issues with the current implementation of this kind of edges:

Originally, every edge
(c_S, c_e)
of the kind has was encoded as a Boolean constantin_${c_e.id}_${c_S.id}
in SMT. Hence, every time we introduce a copyc_T
of a setc_S
, we introduce a new edge(c_T, c_e)
in the arena, and thus we have to introduce another Boolean constantin_${c_e.id}_${c_T.id}
in SMT. Alternatively, we could use a single Boolean variable both for(c_S, c_e)
and(c_T, c_e)
. 
Later, when translating records and tuples, we stopped introducing Boolean constants in SMT for the hasedges. However, we do not track in the arena the fact that these edges are presented only in the arena, not in SMT. Hence, we have to be careful and avoid expressing membership in SMT when working with these edges.

As every hasedge directly refers to its parent in the edge name (that is,
in_${c_e.id}_${c_S.id}
), we cannot share edges when encodingSUBSET S
and[S > T]
. As a result, we have to introduce a massive number of Boolean constants and constraints, which are not necessary. 
We keep adding edges and SMT constants to the solver context, even when we know exactly that an element belongs to a set, e.g., as in
{ 1, 2, 3 }
.
1.2. Arena examples
To introduce the context in more detail, we give an example of how several TLA+ expressions are represented in arenas and SMT.
Consider the following expression:
{ a, b, c } \union { d, e }
Let's denote the arguments of the set union to be S
and T
. In the current arena representation, the rewriting
rule SetCtorRule
creates the following SMT constants (assuming that a, ..., e
were translated to arena cells):

Two cells
c_l
andc_r
to represent the setsS
andT
. These cells are backed with two SMT constants of an uninterpreted sort, which corresponds to the common type ofS
andT
. 
Five SMT constants of the Boolean sort that express set membership of
a, b, c
andd, e
inS
andT
, respectively. The setsS
andT
are backed with SMT constants of the sort ofS
andT
. 
One cell
c_u
to represent the setS \union T
. 
Five Boolean constants of the Boolean sort that express set membership of
a, b, c, d, e
inS \union T
.
It is obvious that 10 Boolean constants introduced for set membership are completely unnecessary, as we know for sure
that the respective elements belong to the three sets. Moreover, when constructing S \union T
, the rule SetCupRule
creates five SMT constraints:
;; a, b, c belong to the union, when they belong to S
(= in_a_u in_a_l)
(= in_b_u in_b_l)
(= in_c_u in_c_l)
;; d and e belong to the union, when they belong to T
(= in_d_u in_d_r)
(= in_e_u in_e_r)
2. Options

Keep things as they are.

Implement the extension of membership edges presented below.
3. Solution
3.1. Pointers to the elements
Instead of the current solution in the arenas, which maps a parent cell to a
list of element cells, we propose to map parent cells to membership pointers of
various kinds. To this end, we introduce an abstract edge (the Scala code can
be found in the package at.forsyte.apalache.tla.bmcmt.arena
):
https://github.com/informalsystems/apalache/blob/81e397fadc6f3ce346d8f8a709ebb3715ac57391/tlabmcmt/src/main/scala/at/forsyte/apalache/tla/bmcmt/arena/ElemPtr.scala#L8L24
Having an abstract edge, we introduce various case classes. The simplest case
is the FixedElemPtr
, which always evaluates to a fixed Boolean value:
https://github.com/informalsystems/apalache/blob/81e397fadc6f3ce346d8f8a709ebb3715ac57391/tlabmcmt/src/main/scala/at/forsyte/apalache/tla/bmcmt/arena/ElemPtr.scala#L26L39
Instances of FixedElemPtr
may be used in cases, when the membership is
statically known. For instance, set membership for the sets {1, 2, 3}
and
1..100
is static (always true
) and thus it does not require any additional
variables and constraints in SMT. The same applies to records and tuples.
The next case is element membership that is represented via a Boolean constant in SMT:
https://github.com/informalsystems/apalache/blob/81e397fadc6f3ce346d8f8a709ebb3715ac57391/tlabmcmt/src/main/scala/at/forsyte/apalache/tla/bmcmt/arena/ElemPtr.scala#L41L64
Instances of SmtConstElemPtr
may be used in cases, when set membership can be
encoded via a Boolean constant. Typically, this is needed when the membership
is either to be defined by the solver, or when this constant caches a complex
SMT constraint. For instance, it can be used by CherryPick
.
The most general case is represented via an SMT expression, which is encoded in TLA+ IR:
https://github.com/informalsystems/apalache/blob/81e397fadc6f3ce346d8f8a709ebb3715ac57391/tlabmcmt/src/main/scala/at/forsyte/apalache/tla/bmcmt/arena/ElemPtr.scala#L66L77
Instances of SmtExprElemPtr
may be used to encode set membership via SMT
expressions. For instance:

Evaluating an array expression, e.g., via
apalacheStoreInFun
. 
Combining several pointers. For instance, when computing
{ x \in S: P }
, we would combine set membership inS
and the value ofP
for everyx
.
3.2. Optimization 1: constant propagation via membership pointers
One immediate application of using the new representation is completely SMTfree construction of some of the TLA+ expressions.
Recall the example in Section 1.2. With the new representation, the set constructor would simply
create five instances of
FixedElemPtr
that carry the value true
, that is, the elements unconditionally belong to S
and T
. Further, the
rule SetCupRule
would simply copy the five pointers, without propagating anything to SMT.
As a result, we obtain constant propagation of set membership, while keeping the general spirit of the arenabased encoding.
3.3. Optimization 2: sharing membership in a powerset
Consider the TLA+ operator that constructs the powerset of S
, that is, the set
of all subsets of S
:
SUBSET S
Let c_S
be the cell that represents the set S
and c_1, ..., c_n
be the
cells that represent the potential elements of S
. Note that in general,
membership of all these cells may be statically unknown. For example, consider
the case when the set S
is constructed from the following TLA+ expression:
{
x \in 1..100:
\E y \in 1..10:
y * y = x
}
In the above example, computation of the predicate is delegated to the SMT solver.
The code in PowSetCtor
constructs 2^Cardinality(S)
sets that contain all subsets of S
. The tricky part here is
that some of the elements of S
may be outside of S
. To deal with that, PowSetCtor
constructs cells for each
potential combinations of c_1, ..., c_n
and adds membership tests for each of them. For instance, consider the
subset T
that is constructed by selecting the indices 1, 3, 5
of 1..n
. The constructor will introduce three
constraints:
(= in_c_1_T in_c_1_S)
(= in_c_3_T in_c_3_S)
(= in_c_5_T in_c_5_S)
Hence, the current encoding introduces 2^n
SMT constants for the subsets and
n * 2^(n  1)
membership constraints in SMT (thanks to Jure @Kukovec for
telling me the precise formula).
With the new representation, we would simply copy the respective membership
pointers of the set S
. This would require us to introduce zero constraints in
the SMT, though we would still introduce 2^n
SMT constants to represent the
subsets themselves. Note that we would still have to introduce n * 2^(n  1)
pointers in the arena. But this would be done during the process of rewriting.
3.4. Feature: computing the set of functions via pointer sharing
Sometimes, it happens that the model checker has to expand a set of functions
[S > T]
. Such a set contains T^S
functions. Since the model checker works with arenas, it can only construct an arena representation of [S > T]
. To this end, assume that the set S
is encoded via cells s_1, ..., s_m
,
and the set T
is encoded via cells t_1, ..., t_n
.
If we wanted to construct [S > T]
in the current encoding, we would have to
introduce a relation for each function in the set [S > T]
. That is, for
every sequence i[1], ..., i[n]
over 1..n
, it would construct the relation
R
:
{ <<s_1, t_i[1]>>, ..., <<s_m, t_i[m]>> }
Let's denote with p_j
the pair <<s_j, t_i[j]>>
for 1 <= j <= m
.
Moreover, we would add m
membership constraints (per function!) in SMT:
(= in_p_1_R (and in_s_1_S in_t_i[1]_T))
...
(= in_p_m_R (and in_s_m_S in_t_i[m]_T))
As a result, this encoding would introduce m * n^m
constants in SMT and the same number of membership constraints. For
instance, if we have m = 10
and n = 5
, then we would introduce 90 million constants and constraints!
Using the approach outlined in this ADR, we can simply combine membership pointers of S
and T
via SmtExprElemPtr
.
This would neither introduce SMT constants, nor SMT constraints. Of course, when this set is used in expressions
like \E x \in S: P
or \A x \in S: P
, the edges will propagate to SMT as constraints.
4. Consequences
RFC021: Prioritization of Work
authors  last revised 

Shon Feder, Gabriela Moreira  20220524 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of the distributed development of Apalache
facing repeated friction around unplanned work and lack of agreement on priorities
we decided to adopt an Action Priority Matrix
to achieve shared understanding and agreement on prioritization
accepting the additional overhead required by scoring and evaluating our work items.
Context
In recent months, we have repeatedly encountered conflicts over prioritization of our work. Different members within our team, or different stakeholders outside of the team, have voiced opposing views on what work items deserve focused and urgent attention. In reflecting on these occasions in our meetings, we have come to agree that these conflicts are due, at least in part, to lack of a clearly established framework for assessing, communicating, and recording the priority of work.
In particular, when plans change, or when urgent, unplanned work needs attention, we would like a lightweight framework for reaching consensus on the new priorities, and deriving a new ordering of the work to be done.
The decision to adopt an action priority matrix for prioritizing our work was reached in December of 2021, but we weren't clear at that time how to we would determine the scores for each work item. This ADR aims to outline and codify the approach we will use.
In this document we talk about "items of work" or "work items". These are just some division of work into a conceptually unified unit. In general, any work item can be further divided into smaller units of work. Work items are represented by "tickets" or "issues" that track the task, e.g., GitHub issues.
Options
We considered 3 different approaches to prioritization:
 Action Priority Matrix: We score each work item based on the expected effort it will require to complete and the anticipated impact of the results. We then place these tasks on a matrix and coordinate the scores to optimize for the highest value delivery with the least effort.
 Cost of Delay (CoD): Similar to the action priority matrix, this is based on assigning two scores to work items: value and urgency. These scores are then correlated to minimize the negatives financial impacts of delays. It basically works by asking "What is the (financial) impact of this not being completed today?"
 Voting: Finally, we discussed a more subjective strategy based on having people vote on tasks they think are more important and using that as a basis for prioritizing.
References
More information on various approaches to prioritization can be found in these sources we consulted:
 http://www.tarrani.net/linda/prioritizing.pdf
 https://www.mindtools.com/pages/article/newHTE_95.htm
 https://www.productplan.com/glossary/actionprioritymatrix/
 https://ciowiki.org/wiki/Action_Priority_Matrix_(APM)
 https://kanbanize.com/leanmanagement/valuewaste/costofdelay
Solution
Decision
We felt that the CoD approach was too dependent on the need to determine shortterm financial returns. This is hard to square with our role as an open source, R&D center serving Informal Systems and the aims of correct software development in the community at large.
We felt that the unstructured voting approach was too subjective. Moreover, while it would work to surface our individual preferences, it doesn't help establish a common ground for shared understanding about priorities.
We finally decided to adopt the Action Priority Matrix. We feel that the process is lightweight enough that we can implement it without slowing down our development but rigorous enough to root our shared understanding in the objective needs and constraints we face in our work.
Process
We follow an adapted version of the approach to APM described in the CIO Wiki.
Categories
APM leads us to divide work items into four categories, as follows:
Impact \ Effort  Low Effort  High Effort 

High Impact  "Quick Wins"  "Major Projects" 
Low Impact  "Fillins"  "Thankless tasks" 
The category within which a work item falls determines the general approach taken towards the work:
Category  Approach 

Quick wins  Do ASAP 
Major projects  Sustain long term focus 
Fillins  Do in spare time, but never taking time from above 
Thankless tasks  Avoid, unless there's literally nothing else to do 
Scoring
We determine which category a work item falls under based on the impact and effort scores assigned to the work. We therefore need a shared understanding of how to assess the impact and effort of a work item. We register only 3 levels on each axis to keep the cognitive load of reckoning scores minimal.
Scores are recorded by affixing labels to the github issue tracking the work item.
Effort
Effort is scored best on a rough estimate of the amount of focused time it would take to complete a work item:
Effort  Meaning  Label 

Easy  Can be completed within about 1 day  efforteasy 
Medium  Can be completed within 3 days  effortmedium 
Difficult  Will take 5 or more days  efforthard 
Ideally, as soon as we recognized that an actively planned ticket will take more than 5 days of focused work, we would factor it into smaller tickets, allowing us to avoid perpetually prolonged monster tickets. But some people may prefer to track "major projects" with a single issue rather than a milestone, and this doesn't pose a significant problem.
Impact
Impact is scored based on considering impact in the follow 4 domains:
 User / customer: The benefit of the results of a work item to our users and customers
 Mission: Furtherance of our organization's mission and objectives
 Invention: Advancement of state of the art in formal verification and specification
 Development: Improvements to our development capacities and bandwidth (which supports advancing the other three factors).
Here are the meaning of the impact levels, as related to each domains:
 High:
 User / customer: Unblocks critical work
 Mission: Advances critical organizational objectives
 Invention: Opens up novel verification and specifications abilities
 Development: Saves > 3 hours per week
 Medium:
 User / customer: Unblocks noncritical work
 Mission: Advances noncritical organizational objectives
 Invention: Makes incremental improvement to verification and specification abilities
 Development: Saves between 1 to 3 hours per week
 Low:
 User / customer: Improves functionality, but an easy workaround exists
 Mission: No significant advance of organizational objectives
 Invention: No significant improvement to verification and specification abilities
 Development: Saves < 1 hours per week
These are the labels to use on issues:
Impact  Label 

Low  impactlow 
Medium  impactmedium 
High  impacthigh 
Prioritization and evaluation
The initial priority of a requirement should be established at the time work is agreed upon, including when setting plans quarterly and when triaging unplanned work. All collaborators with a stake in the work are responsible for ensuring the work is scored in a way that they feel to be correct.
When unplanned work is introduced, the stakeholders involved should determine its score (using informal language, when needed), and this should be used to decided whether it is worth interrupting any ongoing or planned work.
When deciding which work item to take on next, we should favor work that is
nearest to scoring minimum effort and maximum impact: (efforteasy
,
impacthigh
). Ties should be resolved based on the worker's inclination or
discussions within other stakeholders.
The priority of work should be reevaluated as the situation changes. E.g.:
 When goals change, then impact should be reconsidered.
 If we discover work was incorrectly scoped, then the effort should be reevaluted.
We can generate views into the 4 quadrants via filters in our project board, or any other tooling or visualization we find suitable.
When reviewing work in progress or queued up next at our weekly meetings, we should always be sure that the highest priority work on the board is being addressed. We should also take this time to estimate priorities for anything new that came up on the week before that hasn't been prioritized already.
Consequences
TBD
ADR022: Unify Configuration Management and "Pass Options"
authors  last revised 

Shon Feder  20220815 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of extending Apalache's functionality and adding new features
facing the need to reason about our program configuration and make execution thread safe
we decided for unifying our CLI input with our configuration management system
to achieve a more modular architecture and greater static guarantees
accepting the development costs and possible increased cost of introducing new CLI inputs in the future.
Context
As work has proceeded on Shai, following the design laid out in RFC 010, it has been revealed that we can provide value to the MBT team by exposing the current command line interface via RPC calls (see #2013). This functionality will require a way of receiving program configuration input from a gRPC call and forwarding that configuration on to various pass executors.
Additionally, for a long while we have been aware of the limitations and brittleness of our current system for storing and communicating configured options throughout the paths of the program (see #1174) and we have recognized the value we might derive by unifying our CLI inputs with our configuration management system (see #1177).
Options
We considered trying to implement the needed RPCs without addressing #1174 and #1177, but two factors convinced us that these were legitimate prerequisites:
 If we try to expose the current CLI functionality via RPC calls without first introducing a unifying abstraction, any flag or option introduced on one side (e.g., for the CLI) will require duplicated work the other side (e.g., for the RPC). The high maintenance cost is liable to cause the two endpoints to drift.
 The current method of communicating configured values to the rest of the program is through an untyped, mutable, singleton map. This means the configurations are not threadsafe, since mutation of the option map in a concurrent RPC could change the configurations of another RPC call in process.
Thus, we resolved to proceed with unifying the CLI with configuration system described in ADR 013, and replacing the mutable, untyped option map with an immutable, statically typed data structure representing the possible configurable of our various routines.
Solution
Following ADR 013, we introduced support for a limited set of configurable
values that could be read from either a config file or the CLI, and recorded in
an instance of the small ApalacheConfig
class. However, the most CLI inputs
were fed directly into the options map, without interacting at all with the
configuration system.
To address factor (1) above, we have decided to make the communication of all
configurable inputs pass through an ApalacheConfig
. As a special case, the CLI
is reworked to produce an instance of ApalacheConfig
, which is then merged
with configurations from other sources, before being passed along to the various
process executors. Requiring that all program configuration be mapped through
the ApalacheConfig
will enable us to automatically derive configurations from
incoming RPC data, and since any relevant updates to the CLI inputs will have to
be reflected in changes to the ApalacheConfig
, we can be assured that the two
input methods will stay in sync.
To address (2), it would be enough replace the PassOptions
map with the
ApalacheConfig
, which could then be supplied directly to the process
executors. But two further considerations have lead us to adopt an additional
level of abstraction:
 In order to support merging of partial configurations (with fallback to
defaults), the values of the
ApalacheConfig
must all beOptional
. However, before we begin executing a process, we know what data the process will require. If we just pass theApalacheConfig
directly, each process would have to validate the presence of the needed data every time it wanted to access a value.  Most processes only require a subset of the settings represented in
ApalacheConfig
. If we pass the entire configuration to every process, we would have no way of reasoning about what configurations affect which process.
To address these considerations, we will introduce a family of case classes
representing the sets of options required for a process. By narrowing down the
interface of the options required for an executor, we can specify statically
which configurations it depends on. In the process of mapping from
ApalacheConfig
into the options classes, we can validate that all needed
values are present. As a result, by the time we begin executing the program
logic of a process, we'll have a static guarantee that all needed configurable
values are available.
The following diagram represents the data flow dependencies of the proposed configuration system:
Consequences
TBD
ADR023: Trace evaluation
authors  last revised 

Jure Kukovec  20220928 
Table of Contents
 Summary (Overview)
 Context (Problem)
 Options (Alternatives)
 Solution (Decision)
 Consequences (Retrospection)
Summary
In the context of improving usability
facing difficulties understanding counterexample traces
we decided to implement trace evaluation
to achieve a better user experience
accepting the development costs.
Context
As explained in #1845, one often runs into the problem of unreadable counterexamples;
for complex specifications, it is often the case that either the state contains many variables, or the invariant contains many conjunctive components (or both).
In that case, trying to determine exactly why e.g. A1 /\ ... /\ An
was violated basically boils down to manually evaluating each formula Ai
with whatever variables are present in the state.
This is laborious and errorprone, and should be automated.
Options
 Call REPL in each state:
 No convenient REPL integration at the moment
 No clear way of saving outputs
 Encode trace traversal as an Apalache problem and use the solver
 No additional rules or IO needed
Solution
We choose option (2).
Suppose we are given a trace s_0 > s_1 > ... > s_n
over variables x_1,...,x_k
as well as expressions E_1,...,E_m
, such that all free variables of E_1,...,E_m
are among x_1,...,x_k
. W.l.o.g. assume all constants are inlined.
The above defines a trace t_0 > t_1 > ... > t_n
over variables v_1,...,v_m
, such that
t_i.v_j = E_j[s_i.x_1/x_1,...,s_i.x_k/x_k]
for all i \in 0..n, j \in 1..m
. In other words, v_j
in state t_i
is the evaluation of the expression E_j
in state s_i
.
By using transition filtering instead of a generic Nextdecomposition, this can be encoded as a specification free of controlnondeterminism, instate computation, or invariants, and is thus incredibly efficient to represent in SMT.
Then, the solver will naturally return an ITF trace containing the evaluations of E_1,...,E_m
in each state s_0,...,s_n
(the values of v_1,...,v_m
).
Input
The invocation of the trace evaluation command should look like this:
$ apalachemc tracee trace=<trace> expressions=<exprs> <source>
where:
<trace>
is a trace produced by apalache, in either.tla
,.json
or.itf.json
formats<exprs>
is a commaseparated list of expression names, to be evaluated over the trace provided bytrace
Note that <source>
can just be the file used to produce the trace in the first place.
Output
The above command should produce an Apalache trace (in all available formats), with the following properties:
 The output trace length is equal to the input trace length
 If
expressions=E_1,...,E_m
is used, the variables of the output trace areE_1,...,E_m
.  For all
i,j
, the value ofE_i
in statej
of the output trace is equal to the evaluation ofE_i
, as defined in<source>
, using the values the variables of the input trace hold in statej
of the input trace.
Recall that the output trace will only display the expressions E_1,...E_m
as the output state variables. Should you wish to view the original trace variables, you need to add an expression, like one of the ones below for example:
E_single == x_1
E_state == [ x1 > x_1, ..., xk > x_k ]
E_vars == <<x_1, ..., x_k>>
Optionally, we could investigate one of the following two alternatives to the output format:
 The output trace variables are
x_1,...,x_k,E_1,...,E_m
instead, wherex_1,...,x_k
are the variables of the original trace. The value of each variable from the input trace has the same value in every state of the output trace, as it does in the corresponding state of the input trace. This is perhaps preferable to use with the ITF trace viewer.  The output contains both the input trace and the output trace (as it would have been produced in the original suggestion) in the same file, but separately.
Example
Assume we are given the source test.tla
Source
 MODULE test 
EXTENDS Integers
VARIABLE
\* @type: Int;
x
A == x * x
B == IF x < 3 THEN 0 ELSE 1
C == [y \in {1,2,4} > {y} ][x]
D == x % 2 = 0
Init == x = 1
Next == x' = x + 1
Inv == TRUE
=========================================================================
and trace testTrace.json
(length 5, x=1 > ... > x=5
).
Trace
{
"name": "ApalacheIR",
"version": "1.0",
"description": "https://apalache.informal.systems/docs/adr/005adrjson.html",
"modules": [
{
"kind": "TlaModule",
"name": "counterexample",
"declarations": [
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "ConstInit",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Untyped",
"kind": "ValEx",
"value": {
"kind": "TlaBool",
"value": true
}
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "State0",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "OperEx",
"oper": "AND",
"args": [
{
"type": "Bool",
"kind": "OperEx",
"oper": "EQ",
"args": [
{
"type": "Int",
"kind": "NameEx",
"name": "x"
},
{
"type": "Int",
"kind": "ValEx",
"value": {
"kind": "TlaInt",
"value": 1
}
}
]
}
]
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "State1",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "OperEx",
"oper": "AND",
"args": [
{
"type": "Bool",
"kind": "OperEx",
"oper": "EQ",
"args": [
{
"type": "Int",
"kind": "NameEx",
"name": "x"
},
{
"type": "Int",
"kind": "ValEx",
"value": {
"kind": "TlaInt",
"value": 2
}
}
]
}
]
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "State2",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "OperEx",
"oper": "AND",
"args": [
{
"type": "Bool",
"kind": "OperEx",
"oper": "EQ",
"args": [
{
"type": "Int",
"kind": "NameEx",
"name": "x"
},
{
"type": "Int",
"kind": "ValEx",
"value": {
"kind": "TlaInt",
"value": 3
}
}
]
}
]
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "State3",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "OperEx",
"oper": "AND",
"args": [
{
"type": "Bool",
"kind": "OperEx",
"oper": "EQ",
"args": [
{
"type": "Int",
"kind": "NameEx",
"name": "x"
},
{
"type": "Int",
"kind": "ValEx",
"value": {
"kind": "TlaInt",
"value": 4
}
}
]
}
]
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "State4",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "OperEx",
"oper": "AND",
"args": [
{
"type": "Bool",
"kind": "OperEx",
"oper": "EQ",
"args": [
{
"type": "Int",
"kind": "NameEx",
"name": "x"
},
{
"type": "Int",
"kind": "ValEx",
"value": {
"kind": "TlaInt",
"value": 5
}
}
]
}
]
}
},
{
"type": "() => Bool",
"kind": "TlaOperDecl",
"name": "InvariantViolation",
"formalParams": [
],
"isRecursive": false,
"body": {
"type": "Bool",
"kind": "ValEx",
"value": {
"kind": "TlaBool",
"value": true
}
}
}
]
}
]
}
After running tracee trace=testTrace.json expressions=A,B,C,D test.tla
, we should see
...
Constructing an example run I@16:06:59.450
Check the trace in: <PATH>/example0.tla, ... I@16:06:59.563
The outcome is: NoError I@16:06:59.571
Trace successfully generated.
where example0.tla
looks like
Result
 MODULE counterexample 
EXTENDS test
(* Constant initialization state *)
ConstInit == TRUE
(* Initial state *)
State0 == A = 1/\ B = 0/\ C = {1}/\ D = FALSE
(* Transition 0 to State1 *)
State1 == A = 4/\ B = 0/\ C = {2}/\ D = TRUE
(* Transition 1 to State2 *)
State2 == A = 9/\ B = 1/\ C = {}/\ D = FALSE
(* Transition 2 to State3 *)
State3 == A = 16/\ B = 1/\ C = {4}/\ D = TRUE
(* Transition 3 to State4 *)
State4 == A = 25/\ B = 1/\ C = {}/\ D = FALSE
(* The following formula holds true in the last state and violates the invariant *)
InvariantViolation == TRUE
================================================================================
(* Created by Apalache on Mon Oct 17 16:06:59 CEST 2022 *)
(* https://github.com/informalsystems/apalache *)
Consequences
TBD
ADR024: Arena computation isolation
authors  last revised 

Jure Kukovec  20230420 
Table of Contents