# Overview

This book collects five related, but independent, sets of documentation:

# Entry-level 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 out-of-bounds 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 out-of-bounds 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.

## 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.

## 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:         }
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 2^31-1, 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:

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

2. 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.

3. 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 plain-text comments.

4. 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 single-state 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, and Apalache.

• 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 and Next == 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:

$apalache-mc 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 type Seq(Int), that is, it is a sequence of integers (sequences in TLA+ are indexed), and • INPUT_KEY has the type Int, 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 bit-width integers, we introduce another constant INT_WIDTH of type Int:  \* Bit-width 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: 4-bit, 8-bit, 16-bit, 32-bit, 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: $ apalache-mc 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 book-keeping for this variable.

Initialize variables. Having introduced the variables, we have to initialize them. That is, we want to specify lines 2-3 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 mountain-like 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 left-hand side of = is required to have the value specified in the right-hand 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 2-3)
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.

1

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:        }
17:    }


To this end, we redefine Next as follows:

\* Computation step (lines 5-16)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN          \* lines 6-14: 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 successor-state, 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 left-hand 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:

$apalache-mc 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 tool-agnostic 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: $ apalache-mc 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 non-empty. 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:

$apalache-mc 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+: AB~AA \/ BA /\ BA => B FALSEFALSETRUEFALSEFALSETRUE FALSETRUETRUETRUEFALSETRUE TRUEFALSEFALSETRUEFALSEFALSE TRUETRUEFALSETRUETRUETRUE Checking Postcondition. Let us check Postcondition on MC3_8.tla: $ apalache-mc 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 5-16)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN          \* lines 6-14
LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 9-10
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = mid -1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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 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 zero-based 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 9-10
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = mid -1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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 IF-THEN-ELSE 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:

$apalache-mc 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 --write-intermediate and --run-dir: $ apalache-mc check --inv=Postcondition --write-intermediate=1 --run-dir=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 bit-width).

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.

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:

$apalache-mc check --cinit=ConstInit --inv=Postcondition MC5_8.tla ... MC5_8.tla:39:22-39:29: 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 generators2. 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 with MAX_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. 2 If you know property-based testing, e.g., QuickCheck, Apalache generators are inspired by this idea. In contrast to property-based 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: $ apalache-mc 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:

$apalache-mc 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. 3 Instead of checking whether INPUT_SEQ is sorted in the post-condition, we could restrict the constant INPUT_SEQ to be sorted in every execution. That would effectively move this constraint into the pre-condition 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 6-14 /\ 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: $ apalache-mc 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: Fixed-width 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 6-14
/\ nSteps' = nSteps + 1
/\ LET mid == (low + high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 9-10
/\ low' = mid + 1
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = mid -1
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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 fixed-width 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 fixed-width integers:

\* Addition over fix-width integers.
\* 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 6-14
/\ nSteps' = nSteps + 1
/\ LET mid == IAdd(low, high) \div 2 IN
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 9-10
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = IAdd(mid, - 1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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:

$apalache-mc check --cinit=ConstInit --inv=PostconditionSorted MC7_8.tla ... State 2: state invariant 0 violated. ... Total time: 2.786 sec  $ apalache-mc check --cinit=ConstInit --inv=Progress MC7_8.tla
...
State 1: action invariant 0 violated.
...
Total time: 2.935 sec

$apalache-mc 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 out-of-domain 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 IF-THEN-ELSE ~isTerminated => ((low <= high) => (mid + 1) \in DOMAIN INPUT_SEQ)  Apalache finds a violation of this invariant in a few seconds: $ apalache-mc 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 ==
...


Now we check the four invariants: InBounds, PostconditionSorted, Termination, and Progress.

$apalache-mc check --cinit=ConstInit --inv=InBounds MC9_8.tla ... The outcome is: NoError ... Total time: 76.352 sec  $ apalache-mc check --cinit=ConstInit --inv=Progress MC9_8.tla
...
The outcome is: NoError
...
Total time: 63.578 sec

$apalache-mc check --cinit=ConstInit --inv=Termination MC9_8.tla ... The outcome is: NoError ... Total time: 72.682 sec  $ apalache-mc 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 5-16)
Next ==
IF ~isTerminated
THEN IF low <= high
THEN          \* lines 6-14
/\ nSteps' = nSteps + 1
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 9-10
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = IAdd(mid, - 1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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 6-14
/\ nSteps' = nSteps + 1
LET midVal == INPUT_SEQ[mid + 1] IN
\//\ midVal < INPUT_KEY \* lines 9-10
/\ UNCHANGED <<high, returnValue, isTerminated>>
\//\ midVal > INPUT_KEY \* lines 11-12
/\ high' = IAdd(mid, - 1)
/\ UNCHANGED <<low, returnValue, isTerminated>>
\//\ midVal = INPUT_KEY \* lines 13-14
/\ 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 5-16)
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 non-trivial 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 Test-driven development. It took us 2-3 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 non-determinism, 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

In this tutorial, we introduce the Snowcat ❄ 🐱 type checker We give concrete steps on running the type checker and annotating a specification with types.

## Setup

We assume that you have Apalache installed. If not, check the manual page on Apalache installation. The minimal required version is 0.15.0.

## Running example: Two-phase commit

As a running example, we are using the well-understood specification of Two-phase commit by Leslie Lamport (written by Stephan Merz). We recommend to reproduce the steps in this tutorial. So, go ahead and download two specification files: TwoPhase.tla and TCommit.tla.

## Step 1: Running Snowcat

Before we start writing any type annotations, let's run the type checker and see, whether it complains:

$apalache-mc typecheck TwoPhase.tla  The tool output is a bit verbose. Below, you can see the important lines of the output: ... PASS #1: TypeCheckerSnowcat > Running Snowcat .::. Typing input error: Expected a type annotation for VARIABLE tmPrepared ...  ## Step 2: Annotating tmPrepared In Step 1, Snowcat complained about the name tmPrepared. The reason for that is very simple: tmPrepared is declared as a variable, but Snowcat does not find a type annotation. From the comment next to the declaration of tmPrepared, we see that tmPrepared is supposed to be a subset of RM, which in turn is a set of resource managers. We have plenty of choices here of what a resource manager could be. Let's keep it simple and say that a resource manager is simply a name. Hence, we say that RM and tmPrepared are sets of strings. Let's add type annotations: CONSTANT \* @type: Set(Str); RM \* The set of resource managers  VARIABLES (* ... *) \* @type: Set(Str); tmPrepared, \* The set of RMs from which the TM has received$"Prepared"$\* messages.  Note that we had to put the annotation between the keyword CONSTANT and the identifier RM, and between VARIABLES and tmPrepared. We used the one-line TLA+ comment: \* @type: ...;. Alternatively, we could use the multi-line comment: (* @type: Set(Str); *). Importantly, the type annotation should end with a semi-colon: ;. Warning. If you want to write a type annotation on multiple lines, write it in a multi-line comment (* ... *) instead of starting multiple lines with a single-line comment \* .... See issue 718. ## Step 3: Running Snowcat again Having introduced the type annotations for RM and tmPrepared, let's run the type checker again: $ apalache-mc typecheck TwoPhase.tla


Snowcat does not complain about tmPrepared anymore. Now we get another message:

> Running Snowcat .::.
Typing input error: Expected a type annotation for VARIABLE tmState


## Step 4: Annotating tmState

Similar to Step 2, we are missing a type annotation. This time the type checker complains about the variable tmState:

VARIABLES
tmState,       \* The state of the transaction manager.


We can get a hint about the type of tmState from the type invariant TPTypeOK, where we see that tmState is just a string.

VARIABLES
(* ... *)
\* @type: Str;
tmState,       \* The state of the transaction manager.


## Step 5: Getting one more type error by Snowcat

Guess what? Run the type checker again:

$apalache-mc typecheck TwoPhase.tla  Snowcat does not complain about tmState anymore. But we are not done yet: > Running Snowcat .::. Typing input error: Expected a type annotation for VARIABLE rmState  ## Step 6: Annotating rmState This time we need a type annotation for the variable rmState. By inspecting TPTypeOK, we see rmState should be a function that, given a resource manager, produces one of the following strings: "working", "prepared", "committed", "aborted". So we need the function type: Str -> Str. Add the following type annotation: VARIABLES \* @type: Str -> Str; rmState, \*$rmState[rm]$is the state of resource manager RM.  ## Step 7: Running Snowcat to see another error Run the type checker again: $ apalache typecheck TwoPhase.tla


You must have guessed that the type checker complains about the variable msgs. Indeed, it just needs annotations for all CONSTANTS and VARIABLES:

> Running Snowcat .::.
Typing input error: Expected a type annotation for VARIABLE msgs


## Step 8: Annotating msgs

In the previous steps, it was quite easy to annotate variables. We would just look at how the variable is used, or read the comments, and add a type annotation. Figuring out the type of msgs is a bit harder.

Let's look at the definitions of Messages and TPTypeOK:

Message ==
...
[type : {"Prepared"}, rm : RM]  \union  [type : {"Commit", "Abort"}]

TPTypeOK ==
...
/\ msgs \subseteq Message


Now you should be able to see that msgs is a set that may contain three kinds of records:

1. The record [type |-> "Commit"],
2. The record [type |-> "Abort"],
3. A record [type |-> "Prepared", rm |-> r], for some r \in RM.

This looks like an issue for the type checker, as we always require the elements of a set to have the same type.

Actually, the type checker allows records to be generalized to a type that contains additional fields. In the above definition of Messages, the set of records [type: {"Prepared"}, rm: RM] has the type Set([type: Str, rm: Str]). (Note that the record has the field called "type", which has nothing to do with our types.) Likewise, the set of records [type: {"Commit", "Abort"}] has the type Set([type: Str]). Both of these types can be unified to the common type:

Set([type: Str, rm: Str])


The above type is actually what we need for the variable msgs. Let's annotate the variable with this type:

VARIABLES
(* ... *)
\* @type: Set([type: Str, rm: Str]);
msgs


## Step 9: Running Snowcat and seeing no errors

Let's see whether Snowcat is happy about our types now:

$apalache-mc typecheck TwoPhase.tla  The type checker is happy. It has computed the types of all expressions:  > Running Snowcat .::. > Your types are great! > All expressions are typed  # Discussion To see the complete code, check TwoPhase.tla. Note that we have not touched the file TCommit.tla at all! The type checker has figured out all the types in it by itself. We have added five type annotations for 248 lines of code. Not bad. It was quite easy to figure out the types of constants and variables in our example. As a rule, you always have to annotate constants and variables with types. Hence, we did not have to run the type checker five times to see the error messages. 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: \* @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. ## 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 modules EXTENDS 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: $ apalache-mc 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:

$apalache-mc typecheck BakeryTyped.tla ... [BakeryWoTlaps.tla:66:17-66:20]: 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: $ apalache-mc 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:

$apalache-mc apalache-mc 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: $ apalache-mc apalache-mc 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:

$apalache-mc apalache-mc 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: $ apalache-mc apalache-mc 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 define-block. 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 define-block 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.

# 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 anti-pattern.
*)

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 non-negative 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${N}.tla for N=3,5,7,10. For instance, MC_FoldExcept3.tla looks as follows: --------------------- MODULE MC_FoldExcept3 --------------------------------- Proc == { "p1", "p2", "p3" } VARIABLES \* Process clocks \* \* @type: Str -> Int; clocks, \* Drift between pairs of clocks \* \* @type: <<Str, Str>> -> Int; drift INSTANCE FoldExcept ==============================================================================  We run Apalache for different instances of N: apalache-mc check --next=NextSlow --inv=DriftInv MC_FoldExcept${N}.tla
apalache-mc 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 higher-level: 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 array-like 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: 1. State-spaces and transition systems 2. What is a symbolic state? 3. What are symbolic traces? 4. How do I interpret Apalache counterexamples? A glossary of notation and definitions can be found below ## On state-spaces 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 markdown-syntax 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 initial-state predicate Init, a transition-predicate 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 specification-variables 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 well-typed, parseable specification), which evaluates to either TRUE or FALSE. We say $$P_{S_0}(s)$$ is the evaluation of this formula. By the subset-predicate 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 single-argument 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 subset-predicate 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 $$i\in \{0,\dots,k-1\}$$. 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: 1. $$R = \{0,1,\dots,6\}$$ (the set of remainders modulo 7), since those are the only values x', which is defined as a % 7 expression, can take. 2. for any $$k = 0,\dots,5$$, it is the case that $$r(k) = \{0,\dots,k\} \ne R$$, so the diameter is not in $$\{1,\dots,5\}$$ 3. 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 $$R \setminus I$$! ## Explicit-state model checking The idea behind explicit-state model checking is to simply perform the following algorithm (in pseudocode, $$\leftarrow$$ represents assignment): Compute $$S_0$$ and set $$Visited \leftarrow \emptyset, ToVisit \leftarrow S_0$$ 1. 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} 2. 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, k-1)) 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 first-order 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{k-1}) Next == /\ CondN(A1,...,Ak) /\ A1' \in G1(A1,...,Ak) /\ A2' \in G2(A1,...,Ak, A1') ... /\ Ak' \in Gk(A1,...,Ak, A1',...,A{k-1}')  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 syntax-imposed 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 state-space 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{k-1}>>  This specification will have $$C^k$$ distinct states, despite its rather simplistic behavior. ## Explicit-state bounded model checking Adapting the general explicit-state 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 $$Visited \leftarrow \emptyset, ToVisit \leftarrow \{ (s,0)\mid s \in S_0 \}$$ 1. While $$ToVisit \ne \emptyset$$, pick some $$(s,j) \in ToVisit$$: 1. If $$\neg I(s)$$ then terminate, since a witness is found. 2. 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 2. 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 explicit-state 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 $P_R^k(s) := \exists s_{k-1} \in S \ .\ P_R^{k-1}(s_{k-1}) \land s_{k-1} \to s$ Which can be expanded to $Pk(s) = \exists s_0,\dots,s_{k-1} \in S \ .\ P_{S_0}(s_0) \land s_0 \to s_1 \land s_1 \to s_2 \land \dots \land s_{k-1} \to s$ Then, the formula describing invariant violation in exactly $$k$$ steps, $$\exists s_k \in R(k) \setminus I$$, becomes $\exists s_0,\dots,s_k \in S \ .\ P_{S_0}(s_0) \land \neg I(s_k) \land \bigwedge_{i=0}^{k-1} s_i \to s_{i+1}$ 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 subset-predicate 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 sub-invariant 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 subset-predicate 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_{m-1}(s) \land p_m(s) \qquad& | \\ \neg p_1(s) \land \neg p_2(s) \land \dots \land \neg p_{m-1}(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 sub-invariants, 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 $$i = 0,\dots,k-1$$, $$Y_{i+1}$$ 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 $$s_i$$ (0-indexed). 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 $$T / R = \{ \lbrack t\rbrack_R \mid t \in T \} \subseteq 2^T$$. • Subset-predicate 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: 1. The traffic light can switch from red to green, 2. The traffic light can switch from green to red, or 3. 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: apalache-mc 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: apalache-mc 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/_apalache-out/TrafficLight.tla/2022-05-30T18-04-13_3349613574715319837/counterexample1.tla, /home/user/apalache/docs/src/tutorials/_apalache-out/TrafficLight.tla/2022-05-30T18-04-13_3349613574715319837/MC1.out, /home/user/apalache/docs/src/tutorials/_apalache-out/TrafficLight.tla/2022-05-30T18-04-13_3349613574715319837/counterexample1.json, /home/user/apalache/docs/src/tutorials/_apalache-out/TrafficLight.tla/2022-05-30T18-04-13_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/_apalache-out/TrafficLight.tla/2022-05-30T18-04-13_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: 1. The initial state formula appears twice, once as a comment and once in TLA+. 2. 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 human-readable 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 finite-state 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 unroll-variable 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 unroll-variable ♢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 nitty-gritty details and see the encoding in action, you can look at the intermediate TLA specifying the encoding. Run apalache-mc check --next=StutteringNext \ --write-intermediate=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. # Getting Started Apalache is a symbolic model checker for the specification language TLA+. As such, it is a recent alternative to the explicit-state 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 command-line and to help us discover bugs. ## Assumptions Apalache is working under the following assumptions: 1. 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 co-domains. 2. 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. 3. 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: 1. Prebuilt package: download a prebuilt package and run it in the JVM. 2. Docker: download and run a Docker image. 3. 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/apalache-mc, or, better, add the ./bin directory to your PATH and run apalache-mc. 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 command-line 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 <your-spec-directory>:/var/apalache ghcr.io/informalsystems/apalache <args>


The following docker parameters are used:

• --rm to remove the container on exit

• -v <your-spec-directory>:/var/apalache bind-mounts <your-spec-directory> 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 <your-spec-directory>. 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 <your-spec-directory>:/var/apalache:z

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

• <args> are the tool arguments as described in Running the Tool.

We provide a convenience wrapper for this docker command in script/run-docker.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/run-docker.sh <args>


To specify a different image, set APALACHE_TAG like so:

$APALACHE_TAG=foo$APALACHE_HOME/script/run-docker.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 unstable

$alias apalache='docker run --rm -v$(pwd):/var/apalache ghcr.io/informalsystems/apalache:unstable'


## Using the unstable version of Apalache

The development of Apalache proceeds at a high pace, and we introduce a substantial number of improvements in the unstable branch before the next stable release. Please refer to the change log and manual on the unstable branch for the description of the newest features. We recommend using the unstable version if you want to try all the exciting new features of Apalache. But be warned: It is called "unstable" for a reason. To use unstable, just type ghcr.io/informalsystems/apalache:unstable 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:unstable


Run it with the following command:

$docker image build -t apalache:0.7.0 .  # Building from source 1. Install git. 2. Install the Eclipse Temurin or Zulu builds of OpenJDK 17. 3. Install sbt. • On Debian Linux or Ubuntu, follow this guide • On Arch: sudo pacman -Syu sbt • On macOS / Homebrew: brew install sbt 4. Clone the git repository: git clone https://github.com/informalsystems/apalache.git. 5. Change into the project directory: cd apalache. 6. Install direnv and run direnv allow, or use another way to set up the shell environment. 7. To build and package Apalache for development purposes, run make. • To skip running the tests, you can run make package. 8. To build and package Apalache for releases and end-user distribution, run make dist. 9. The distribution package will be built to ./target/universal/apalache-<VERSION>, and you can move this wherever you'd like, and ensure that the <dist-package-location>/bin directory is added to your PATH. ## 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 your PATH, which will make apalche-mc available. • Install direnv and run direnv allow, which will put apalche-mc in your path when you are inside of the Apalache repo directory. • Run ./bin/apalache-mc directly. # Running the Tool Opt-in statistics programme: if you opt-in 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: $ apalache-mc --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 to parse 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 of parse 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 of typecheck 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 the check command.

• check performs all of the operations of typecheck 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 the --length parameter. It terminates successfully, if there are no invariant violations.

• test performs all of the operations of check in a mode that is designed to test a single action.

## 1. Model checker and simulator command-line parameters

### 1.1. Model checker command-line parameters

The model checker can be run as follows:

$apalache-mc check [--config=filename] [--init=Init] [--cinit=ConstInit] \ [--next=Next] [--inv=Inv] [--length=10] [--temporal=TemporalProp] [--algo=(incremental|offline)] \ [--discard-disabled] [--no-deadlock] \ [--tuning-options-file=filename] [--tuning-options=key1=val1:...:keyn=valn] \ [--smt-encoding=(oopsla19|arrays)] \ [--out-dir=./path/to/dir] \ [--write-intermediate=(true|false)] \ [--config-file=./path/to/file] \ [--profiling=false] \ <myspec>.tla  The arguments are as follows: • General parameters: • --config specifies the TLC configuration file • --init specifies the initialization predicate, Init by default • --next specifies the transition predicate, Next by default • --cinit specifies the constant initialization predicate, optional • --inv specifies the invariant to check, optional • --length specifies the maximal number of Next steps, 10 by default • --temporal specifies the temporal property to check, optional • Advanced parameters: • --algo lets you to choose the search algorithm: incremental is using the incremental SMT solver, offline is using the non-incremental (offline) SMT solver • --smt-encoding lets you choose how the SMT instances are encoded: oopsla19 (default) uses QF_UFNIA, and arrays (experimental) uses arrays with extensionality. This parameter can also be set via the SMT_ENCODING environment variable. See the alternative SMT encoding using arrays for details. • --discard-disabled does a pre-check 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 pre-check may be slower than checking the invariant. Default: true. • --max-error <n> instructs the search to stop after n errors, see Enumeration of counterexamples. Default: 1. • --view <name> sets the state view to <name>, see Enumeration of counterexamples. • --no-deadlock disables deadlock-checking, when --discard-disabled=false is on. When --discard-disabled=true, deadlocks are found in any case. • --tuning-options-file specifies a properties file that stores options for fine tuning • --tuning-options can pass and/or override these fine tuning options on the command line • --out-dir set location for outputting any generated logs or artifacts, ./_apalache-out by default • --write-intermediate if true, then additional output is generated. See Detailed output. false by default • --run-dir=DIRECTORY write all outputs directly into the specified DIRECTORY • --config-file 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 of profile-rules.txt used in profiling. The file is only created if profiling is set to True. Setting profiling to False is incompatible with the --smtprof flag. The default is False. Options passed with --tuning-options have priority over options passed with --tuning-options-file. 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: 1. Command-line value 2. Environment variable value 3. Configuration file value ### 1.2. Simulator command-line parameters The simulator can be run as follows: $ apalache-mc simulate
[all-checker-options] [--max-run=NUM] [--save-runs] <myspec>.tla


The arguments are as follows:

• Special parameters:

• --max-run=NUM: but produce up to NUM simulation runs (unless --max-error errors have been found), default: 100

• --save-runs: save an example trace for each simulated run, default: false

### 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" apalache-mc <args>


If you are running Apalache via docker directly (instead of using the script in $APALACHE_HOME/script/run-docker.sh), you'll also need to expose the environment variable to the docker container: $ JVM_ARGS="-Xmx1G" docker run -e JVM_ARGS --rm -v <your-spec-directory>:/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_{k-1}/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:

$apalache-mc check --init=Init --inv=IndInv --length=0 <myspec>.tla  • IndNext: Check that Next does not drive us outside of IndInv: $ apalache-mc 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 predicate Safety:

$apalache-mc 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
$apalache-mc check --length=0 --init=Init --inv=Inv y2k_override.tla$ apalache-mc 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.)

$cd test/tla apalache-mc 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 1. The current working directory. 2. The directory containing the main TLA+ file passed on the CLI. 3. A small Apalache standard library (bundled from $APALACHE_HOME/src/tla).
4. The Java package tla2sany.StandardModules (usually provided by the tla2tools.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 out-dir 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 yyyy-MM-ddTHH-mm-ss_<UNIQUEID>. For an example, consider using the default location of the run-dir for a run of Apalache on a spec named test.tla. This will create a directory structuring matching following pattern: ./_apalache-out/ └── test.tla └── 2021-11-05T22-54-55_810261790529975561  The default value for the out-dir is the _apalache-out 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 run-specific subdirectory 2021-11-05T22-54-55_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 run-specific subdirectory. The directory also includes a file run.txt, reporting the command line arguments used during the run. Additionally, if the parameter write-intermediate is set to true (see configuration instructions) each pass of the model checker produces intermediate TLA+ files in the run-specific subdirectory: • File out-parser.tla is produced as a result of parsing and importing into the intermediate representation, Apalache TLA IR. • File out-config.tla is produced as a result of substituting CONSTANTS, as described in Setting up specification parameters. • File out-inline.tla is produced as a result of inlining operator definitions and LET-IN definitions. • File out-priming.tla is produced as a result of replacing constants and variables in ConstInit and Init with their primed versions. • File out-vcgen.tla is produced as a result of extracting verification conditions, e.g., invariants to check. • File out-prepro.tla is produced as a result of running all preprocessing steps. • File out-transition.tla is produced as a result of finding assignments and symbolic transitions. • File out-opt.tla is produced as a result of expression optimizations. • File out-analysis.tla is produced as a result of analysis, e.g., marking Skolemizable expressions and expressions to be expanded. ## 5. Parsing and pretty-printing If you'd like to check that your TLA+ specification is syntactically correct, without running the model checker, you can run the following command: $ apalache-mc parse <myspec>.tla


In this case, Apalache performs the following steps:

1. It parses the specification with SANY.

2. It translates SANY semantic nodes into Apalache IR .

3. If the --write-intermediate flag is given, it pretty-prints 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.,

$apalache-mc 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  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 apalache-mc. For instance, for y2k_cinit, we would run the model checker as follows: $ cd $APALACHE_HOME/test/tla$ apalache-mc 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 non-empty 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

Let us go back to the example test/tla/y2k.tla and run apalache-mc against test/tla/y2k_override.tla while instructing Apalache to write intermediate output files:

$apalache-mc check --write-intermediate=true y2k_override.tla  We can check the detailed output of the TransitionFinderPass in the file _apalache-out/y2k_override.tla/<timestamp>/intermediate/<pass>_OutTransitionFinderPass.tla, where <timestamp> looks like 2021-12-01T12-07-41_1998641578103809179, and <pass> is a two-digit 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: 1. It has translated several expressions that look like x' = e into x' := e. For instance, you can see year' := 80 and hasLicense' := FALSE in Init_si_0000. We call these expressions assignments. 2. It has factored the operator Next into two operators Next_si_0000 and Next_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 breadth-first 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 with VARIABLE, there is exactly one assignment of the form x' := e in every symbolic transition A_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: apalache-mc 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: 1. What are the semantics of fold operators? 2. How do I use these operators in Apalache? 3. Should I use folding or recursion? 4. 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: 1. If C is empty, then ApaFoldSeqLeft( Op, b, <<>> ) = b, regardless of Op 2. If C is nonempty, we establish a recursive relation between folding over C and folding over Tail(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: 1. If C is empty, then ApaFoldSet( Op, b, {} ) = b, regardless of Op 2. If C is nonempty, we establish a recursive relation between folding over C and folding over some subset of C in the following way: ApaFoldSet( Op, b, C ) = ApaFoldSet( Op, Op(b, x), C \ {x} ), where x is some arbitrary member of C (e.g. x = CHOOSE y \in C: TRUE). Note that Apalache does not guarantee a deterministic choice of x, unlike what using CHOOSE 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: OrderApaFoldSet value 1 -> 2 -> 311 1 -> 3 -> 212 2 -> 1 -> 313 2 -> 3 -> 115 3 -> 1 -> 216 3 -> 2 -> 117 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/return-type 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 LET-IN operator Plus(p,q) == p + q instead X == ApaFoldSet( LAMBDA p,q: p + q, 0, {1,2,3} ) \* Built-in operators cannot be called by name in Apalache \* Define a LET-IN 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 re-entries, 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 non-primitively 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 fold-less 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 predicate-based 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 fold-based 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 ) \* Re-implementation of UNION setOfSets BigUnion(setOfSets) == LET Union(p,q) == p \union q IN ApaFoldSet( Union, {}, setOfSets ) \* Re-implementation 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 fold-less 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 fold-defined operators, is still very readable (most LET-IN 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 sub-call 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: S = Set(Int); \* @typeAlias: STATE = [ In: S, Done: S, Out: S ]; \* @type: S; In, \* @type: S; Done, \* @type: S; Out \* @type: <<S, S, S>>; 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 states Done \intersect In = {}, and • Predicate BuggyStateInv that states Done \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 --max-error=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 --max-error 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 --max-error 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 --max-error=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 model-based 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 number-like 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:

1. Command line arguments
2. Environment variables
3. A local configuration file
4. 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 the currently supported parameters, along with their default values:

# Directory in which to write all log files and records of each run

# 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 opt-in 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 opt-in 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 opt-in and opt-out To opt-in in the statistics collection, execute the following command: ./apalache-mc config --enable-stats=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 opt-out from the statistics collection, execute the following command:

./apalache-mc config --enable-stats=false


## What exactly is submitted to tlapl.us

You can check the daily log at exec-stats.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, or typecheck
• 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:11:5-13:18 4098,2020,1969,12000,powerset.tla:12:9-12:20 4098,2020,1969,12000,powerset.tla:12:14-12:20 ...  The meaning of the columns is as follows: • weight is the weight of the expression. Currently it is computed as nCells + 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 of line: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:

1. It automatically extracts symbolic transitions from the specification. This allows us to partition the action Next into a disjunction of simpler actions A_1, ..., A_n.

2. Apalache translates operators Init and A_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 of k steps s_0, s_1, ..., s_k, all of which execute action A_1, is encoded as a formula Run(k) that looks as follows:

[[Init(s_0)]] /\ [[A_1(s_0, s_1)]] /\ ... /\ [[A_1(s_(k-1), 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 non-empty 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

// 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 left-hand side
// with the identifier in the right-hand side.
replacement ::=
ident "<-" ident

// Replace the constant in the left-hand side
// with the constant expression in the right-hand side.
assignment ::=
ident "=" constExpr

// A constant expression that may appear in
// the right-hand 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 bit-width assumed)
integer ::=
<string matching regex [0-9]+>
| "-" <string matching regex [0-9]+>

// A string, starts and ends with quotes,
// a restricted set of characters is allowed (pre-UTF8 era, Paxon scripts?)
string ::=
'"' <string matching regex [a-zA-Z0-9_~!@#\$%^&*-+=|(){}[\],:;'<>.?/ ]*> '"' // 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/current-tools.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 [a-zA-Z_]([a-zA-Z0-9_])*>  ## 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 [--infer-poly=<bool>] <myspec>.tla


The arguments are as follows:

• General parameters:
• --infer-poly controls whether the type checker should infer polymorphic types. As many specs do not need polymorphism, you can set this option to false. The default value is true.

There is not much to explain about running the tool. When it successfully finds the types of all expressions, it reports:

 > Running Snowcat .::..
...
Type checker [OK]


When the type checker finds an error, it explains the error like that:

 > Running Snowcat .::.
[QueensTyped.tla:42:44-42:61]: Mismatch in argument types. Expected: (Seq(Int)) => Bool
[QueensTyped.tla:42:14-42:63]: 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 liveness-to-safety transformation.

## Language

### Module-Level constructs

ConstructSupported?MilestoneComment
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 LET-IN definitions

### The constant operators

#### Logic

OperatorSupported?MilestoneComment
/\, \/, ~, =>, <=>-
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.

OperatorSupported?MilestoneComment
=, /=, \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

OperatorSupported?MilestoneComment
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.

OperatorSupported?MilestoneComment
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.

OperatorSupported?MilestoneComment
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

ConstructSupported?MilestoneComment
"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

ConstructSupported?MilestoneComment
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 eAll applications of d1, ..., d_n are replaced with the expressions e1, ... e_n respectively. LET-definitions without arguments are kept in place.
multi-line /\ and \/-

### The Action Operators

ConstructSupported?MilestoneComment
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

ConstructSupported?MilestoneComment
[]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.

OperatorSupported?MilestoneComment
+, -, *, <=, >=, <, >-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

OperatorSupported?MilestoneComment
<<...>>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

OperatorSupported?MilestoneComment
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

OperatorSupported?MilestoneComment
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 operatorsDummy 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 user-defined 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:

1. A recursive operator may be non-terminating (although a non-terminating operator is useless in TLA+);

2. 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:

1. OpInternal was non-recursive
2. If computing Op(x) required a recursion stack of depth at most UNROLL_TIMES_Op, then OpInternal(x) = Op(x)
3. Otherwise, OpInternal(x) would return the value, which would have been produced by the computation of Op(x), if all applications of Op while the recursion stack height was UNROLL_TIMES_Op returned UNROLL_DEFAULT_Op instead of the value produced by another recursive call to Op

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. non-primitive or non-tail), 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 empty-set 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 or ApaFoldSeqLeft
• 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 than UNROLL_TIMES_Op steps
• The use of recursive operators produces unpredictable results, particularly when used in invariants

# Known issues

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 non-constant 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:

1. Incremental EXCEPT
G ==
LET F(g, x) == [g EXCEPT ![x] = A(x)]
IN ApaFoldSet(F, f, S)

1. Incremental \union
R ==
LET F(T, e) == T \union {A(e)}
IN ApaFoldSet(F, S0, S)

1. 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 programming-language objects in its own implementation. This makes it easy to construct temporary mutable objects, manipulate them (e.g. via for-loops) and garbage-collect them after they stop being useful. For constraint-based 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 state-space exploration, but on in-state value computation, which is not its strong suit. Below we show how to rewrite these APs.

1. 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]
]

1. 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 }

1. 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 user-defined operator with the operator's body
• replaces every call to a let-in defined operator of arity at least 1 with the operator's body
• PrimingPass: adds primes to variables in Init and ConstInit (required by TransitionPass)
• VCGen: extracts verification conditions from the invariant candidate.
• Desugarer: removes syntactic sugar like short-hand expressions in EXCEPT.
• Normalizer: rewrites all expressions in negation-normal 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. Addison-Wesley 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 command-line option --tuning-options-file=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 command-line by passing the option --tuning-options that has the following format:  --tuning-options=key1=val1 --tuning-options=key1=val1:key2=val2 ...   The following options are supported: 1. Randomization: smt.randomSeed=<int> passes the random seed to z3 (via z3's parameters sat.random_seed and smt.random_seed). 2. 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. 3. Invariant mode: search.invariant.mode=(before|after) defines the moment when the invariant is checked. In the after mode, all transitions are first translated, one of them is picked non-deterministically 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. 4. Guided search: search.transitionFilter=<regex>. Restrict the choice of symbolic transitions at every step with a regular expression. The regular expression should recognize words over of the form 's->t', where s is a regular expression over step numbers and t is a regular expression over transition numbers. For instance, search.transitionFilter=(0->0|1->5|2->(2|3)|[3-9]->.*|[1-9][0-9]+->.*) 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 --write-intermediate=true and check the transition numbers in _apalache-out/<MySpec>.tla/*/intermediate/XX_OutTransitionFinderPass.tla: the 0th transition is called Next_si_0000, 1st transition is called Next_si_0001, etc. 5. Invariant checking at certain steps: search.invariantFilter=regex. Check the invariant only at the steps that satisfy the regular expression. For instance, search.invariantFilter=10|15|20 tells the model checker to check the invariant only after exactly 10, 15, or 20 step were made. Step 0 corresponds to the initialization with Init, step 1 is the first step with Next, etc. This option is useful 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. 6. Translation to SMT: 1. Short circuiting: rewriter.shortCircuit=(false|true). When rewriter.shortCircuit=true, A \/ B and A /\ B are translated to SMT as if-then-else 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 Boolean-valued 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 sub-requirements: 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 closed-form 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, or • x' := 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 non-trivial 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 control-flow operators in this way, the or (\/) operator and the if-then-else 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 action-or, otherwise it is a value-or. Similarly, if both the THEN _ and ELSE _ subexpressions of if-then-else are actions, it is an action-ITE, otherwise it is a value-ITE (in particular, a value-ITE can be non-Boolean). Using these two operators we can define the following terms: A minimal action is an action which contains no action-or and no action-ITE. Conversely, a compound action is an action which contains at least one action-or or at least one action-ITE. ### 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 if-then-else, 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, or • x' := e (note that := is the operator defined in Apalache.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 left-to-right syntax order of and-operator (/\) 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. ### Assignment-before-use 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:10:16-10:17: x' is used before it is assigned.  notifying the user of any variables used before assignment. In particular, right-hand-sides 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 or-operator (\/), 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:10:15-10:19: 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 if-then-else: both the THEN _ and ELSE _ branch must assign the same variables, however, the IF _ condition is ignored when determining assignments. ### Assignment-free 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 shape A_1 /\ ... /\ A_n, then assignments are selected from A_1, ... , A_n sequentially, subject to the syntax-order rule. • If A has the shape A_1 \/ ... \/ A_n, then assignments are selected in all A_1, ... , A_n independently, subject to the balance rule. • If A has the shape IF p THEN A_1 ELSE A_2, then: • p may not contain assignments. Any assignment candidates in p are subject to the assignment-before-use rule. • Assignments are selected in both A_1 and A_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 in S are subject to the assignment-before-use rule. • Assignments are selected in A_1 • In any other case, A may not contain assignments, however, any assignment candidates in A are subject to the assignment-before-use 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 assignment-before-use 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 assignment-before-use. 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 if-then-else 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, if-conditions are assignment-free, 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 assignment-before-use. ## Manual Assignments 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 assignment-free 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:10:12-10:18: Manual assignment is spurious, x is already assigned!  or Assignment error: test.tla:10:15-10:21: Illegal assignment inside an assignment-free 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 # Overview # How to write type annotations Warning: This HOWTO discusses how to write type annotations for the new type checker Snowcat, which is used in Apalache since version 0.15.0. Note that the example specification uses recursive operators, which were removed in version 0.23.1. 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: Recipe 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: $ apalache-mc 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 .::.
> 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:

$apalache-mc 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. Run the type checker again. You should see the following message: > 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: $ apalache-mc typecheck CarTalkPuzzle.tla


Now you should see the following error:

[CarTalkPuzzle.tla:52:32-52:35]: Cannot apply f to the argument x() in f[x()].
[CarTalkPuzzle.tla:50:1-52:53]: 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:160:23-160:26]: Cannot apply B to the argument x in B[x].
[CarTalkPuzzle.tla:160:7-160:37]: 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 .::.
> All expressions are typed
...


## Recipe 4: Annotating records

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


By looking at the spec, it is easy to guess the types of the variables rmState, tmState, and tmPrepared:

VARIABLES
\* @type: RM -> Str;
rmState,
\* @type: Str;
tmState,
\* @type: Set(RM);
tmPrepared,


The type of the variable msgs is less obvious. We can check the definitions of TPTypeOK and Message to get the 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 definitions, you can see that msgs is a set that contains records of two types: [type: Str] and [type: Str, rm: RM]. When you have a set of heterogeneous records, you have to choose the type of a super-record that contains the fields of all records that could be put in the set. That is:

  \* @type: Set([type: Str, rm: RM]);
msgs


A downside of this approach is that Snowcat will not help you in finding an incorrect field access. We probably will introduce more precise types for records later. See Issue 401.

## 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
...
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:35:44-35:61]: 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:35:1-35:63]: 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 funny: IsSolution is expecting a sequence, whereas Solutions is clearly producing a set of functions. Of course, it is not a problem in the untyped TLA+. In fact, it is a well-known idiom: Construct a function by using function operators 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 .::.
> 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: ALIAS = <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>TypeAliases, where the <PREFIX> is replaced with a unique prefix to prevent name clashes. In the MissionariesAndCannibals.tla example, we have

\* @typeAlias: PERSONS = Set(PERSON);
MCTypeAliases = 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. When 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 ADR002.

## Recipe 7: Multi-line annotations

A type annotation may span over multiple lines. You may use both the (* ... *) syntax as well as the single-line 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 multi-line 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 n-th 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.

## 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 non-polymorphic <,>,>=,<= 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 [A-Z_][A-Z0-9_]*, and is not a type alias defined elsewhere.

## How to introduce values belonging to an uninterpreted type

Apalache uses the following convention-based naming scheme for values of uninterpreted types:

"identifier_OF_TYPENAME"


where:

• TYPENAME is the uninterpreted type to which this value belongs, matching the pattern [A-Z_][A-Z0-9_]*, and
• identifier is a unique identifier within the uninterpreted type, matching the pattern [a-zA-Z0-9_]+.

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.

## 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 type-incorrect:
"abc" = "bc_OF_A" \* Cannot compare values of types Str and A

• The following expression is type-incorrect:
"1_OF_A" = "1_OF_B" \* Cannot compare values of types A and B

• The following expressions are type-correct:
\* 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


# 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 higher-level language on top of TLA+. If you are interested in learning Pluscal, check LearnTla.com by Hillel Wayne.

## Contents

1. The standard operators of TLA+ 🔌
2. User-defined operators 💡
3. 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 non-deterministic, so they require the oracle to resolve non-determinism, see Control Flow and Non-determinism. 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+

### Strings 🔡

String constants. You learned it!

• String literals, e.g., "hello" and "TLA+ is awesome".
• In Apalache, the literals have the type Str.
• Set of all finite strings: STRING.
• In Apalache, the set STRING has the type Set(Str).

### Tuples 📐

• Tuple constructor: << e_1, ..., e_n >>
• Cartesian product: S_1 \X ... \X S_n (also S_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,...

• 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 minus -a and Int

## 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)
• Leads-to: F ~> G
• Guarantee: F -+-> G
• Temporal hiding: \EE x: F
• Temporal universal quantification: \AA x: F

# Booleans

[Back to all operators]

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 set-theory 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 Non-determinism.

### 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 both F and G evaluate to TRUE.

• FALSE, if F evaluates to FALSE, or F evaluates to TRUE and G evaluates to FALSE.

The general case F_1 /\ ... /\ F_n can be understood by evaluating the expression F_1 /\ (F_2 /\ ... /\ (F_{n-1} /\ F_n)...).

Determinism: Deterministic, if the arguments are deterministic. Otherwise, the possible effects of non-determinism of each argument are combined. See Control Flow and Non-determinism.

Errors: In pure TLA+, the result is undefined if either conjunct evaluates to a non-Boolean 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 both F and G evaluate to FALSE.

• TRUE, if F evaluates to TRUE, or F evaluates to FALSE and G evaluates to TRUE.

The general case F_1 \/ ... \/ F_n can be understood by evaluating the expression F_1 \/ (F_2 \/ ... \/ (F_{n-1} \/ F_n)...).

Determinism: deterministic, if the arguments may not update primed variables. If the arguments may update primed variables, disjunctions may result in non-determinism, see Control Flow and Non-determinism.

Errors: In pure TLA+, the result is undefined, if a non-Boolean 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 to FALSE, then ~F is evaluated to TRUE,
• if F is evaluated to TRUE, then ~F is evaluated to FALSE.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined, if the argument evaluates to a non-Boolean 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, if F evaluates to FALSE, or F evaluates to TRUE and G evaluates to TRUE.

• FALSE, if F evaluates to TRUE and G evaluates to FALSE.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined, if one of the arguments evaluates to a non-Boolean 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 both F and G evaluate to TRUE, or both F and G evaluate to FALSE.

• FALSE, if one of the arguments evaluates to TRUE, while the other argument evaluates to FALSE.

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 non-Boolean 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 Non-determinism in TLA+

Back to all operators

Author: Igor Konnov

Peer review: Shon Feder, Jure Kukovec

Non-determinism 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 non-determinism. 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 non-determinism means in the mathematical framework of TLA+, and then proceed with the explanation that is friendly to computers and software engineers.

## Explaining non-determinism 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 well-defined 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 to TRUE on state s_0, and
• The operator Next evaluates to TRUE on every pair of states <<s_i, s_j>> for 0 <= i < k and j = 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 non-determinism. 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 non-determinism. 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 non-determinism in the operator Init. Specification coord3 has non-determinism 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 non-determinism to a computer? It turns out that we can. In fact, many model checkers support non-determinism 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 non-determinism that is close to the approach in Apalache.

## Explaining non-determinism 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 left-to-right. The left-hand 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 right-hand 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 non-primed and primed variables, that is, over x, x', y, y'. Non-primed 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 left-hand 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:

1. For a non-empty set S, a call GUESS S returns some value v \in S.
2. A call GUESS {} halts the evaluation.
3. 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 non-determinism in computer science. So we can think of GUESS S as being one of the following implementations:

1. 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 non-deterministic.

2. GUESS S can be simply the user input. In this case, the user resolves non-determinism.

3. GUESS S can be controlled by an adversary, who is trying to break the system.

4. GUESS S can pick an element by calling a pseudo-random number generator. However, note that RNG is a very special way of resolving non-determinism: 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 non-determinism. With GUESS S we can model all of them. As TLA+ does not introduce special primitives for different kinds of non-determinism, 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.

### Non-determinism 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:

1. The predicate P refers to y', that is, P has to assign a value to y'.
2. The value of y' is not defined yet, that is, binding s does not have a value for the name y'.

If the above assumptions do not hold true, the expression \E x \in S: P does not have non-determinism 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:

1. Predicate P evaluates to TRUE when using the provided value of x. In this case, P assigns the value of an expression e to y' as soon as the evaluator meets the expression y' = e. The evaluation may continue.
2. Predicate P evaluates to FALSE when using the provided value of x. Well, that was a wrong guess. According to our semantics, the evaluation halts. See the above discussion on "halting".
3. The set S is empty, and GUESS 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 non-determinism 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.


### Non-determinism 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:

1. All sets Use_i agree on which variables are to be assigned. Formally, Use_i \intersect Unassigned(s) = Use_j \intersect Unassigned(s) /= {} for i, j \in 1..n. This is the case that we consider below.
2. 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 ill-structured. TLC would raise an error when it found a binding like this. Apalache would detect this problem when preprocessing the specification.
3. The clauses do not assign values to the primed variables. Formally, Use_i \intersect Unassigned(s) = {} for i \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 non-deterministic 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 short-circuiting of disjunction in the deterministic case, we have non-deterministic choice here. Hence, short-circuiting does not apply to non-deterministic disjunctions.

### Non-determinism in Boolean IF-THEN-ELSE

For the deterministic use of IF-THEN-ELSE, see Deterministic conditionals.

Consider an IF-THEN-ELSE 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, IF-THEN-ELSE can be evaluated as the equivalent expression:

  \/  A /\ B
\/ ~A /\ C


We do not recommend you to use IF-THEN-ELSE with non-determinism. The structure of the disjunction provides a clear indication that the expression may assign to variables as a side effect. IF-THEN-ELSE 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.

### Non-determinism 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 IF-THEN-ELSE, we do not recommend using CASE for expressing non-determinism. 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 non-determinism 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.

Back to all operators

# Deterministic conditionals

[Back to all operators]

In this section, we consider the instances of IF-THEN-ELSE and CASE that may not update primed variables. For the case, when the operators inside IF-THEN-ELSE or CASE can be used to do non-deterministic assignments, see Control Flow and Non-determinism.

Warning: Because frequent use of IF-THEN-ELSE 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 IF-THEN-ELSE 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 IF-THEN-ELSE to compute predicate-dependent 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 IF-THEN-ELSE instead.

## Deterministic IF-THEN-ELSE

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, if A evaluates to TRUE.
• The value of C, if A evaluates to FALSE.

Determinism: This is a deterministic version. For the non-deterministic version, see Control Flow and Non-determinism.

Errors: If A evaluates to a non-Boolean 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 if-else 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 some i \in I, if I 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 implementation-agnostic.

Owing to the flexible semantics of simultaneously enabled predicates, TLC interprets the above CASE operator as a chain of IF-THEN-ELSE 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 ITE-series 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 CASE-OTHER expressions (see below), which it treats as a chain of IF-THEN-ELSE expressions.

Determinism. The result of CASE is deterministic, if there are no primes inside. For the non-deterministic version, see [Control Flow and Non-determinism]. 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 non-Boolean 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 non-exclusive 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 CASE-OTHER

Better use IF-THEN-ELSE.

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 IF-THEN-ELSE 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 CASE-OTHER. Hence, we recommend using IF-THEN-ELSE instead of CASE-OTHER. Although IF-THEN-ELSE is a bit more verbose, its semantics are precisely defined.

Determinism. The result of CASE-OTHER is deterministic, if e_0, e_1, ..., e_n may not update primed variables. For the non-deterministic version, see [Control Flow and Non-determinism]. 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 non-Boolean 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

[Back to all operators]

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 non-linear 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 set Int has the type Set(Int). A bit confusing, right? 😎
• The set Nat that consists of all natural numbers, that is, Nat contains every integer x that has the property x >= 0. This set is infinite. In Apalache, the set Nat 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 non-integer 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 non-integer 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



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 non-integer 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 non-integer 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 non-integer 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:

1. When a >= 0 and b > 0, then the result of a \div b is the integer c that has the property: a = b * c + d for some d in 0..(b-1).
2. When a < 0 and b > 0, then the result of a \div b is the integer c that has the property: a = b * c + d for some d in 0..(b-1).
3. When a >= 0 and b < 0, then the result of a \div b is the integer c that has the property: a = b * c + d for some d in 0..(-b-1).
4. When a < 0 and b < 0, then the result of a \div b is the integer c that has the property: a = b * c + d for some d in 0..(-b-1).

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 2.13RustPython 3.8.6TLA+ (TLC)SMT (z3 4.8.8)
100 / 3 == 33100 / 3 == 33100 / 3 == 33100 // 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) == -33100 / (-3) == -33100 / (-3) == -33100 // (-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 1-2 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 non-integer 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 non-integer 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:

1. b > 0,
2. b = 0 and a /= 0.

Apalache type: (Int, Int) => Int.

Effect: a^b evaluates to a raised to the b-th power:

• If b = 1, then a^b is defined as a.
• If a = 0 and b > 0, then a^b is defined as 0.
• If a /= 0 and b > 1, then a^b is defined as a * a^(b-1).
• 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 non-integer 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 less-than

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, if a is less than b,
• FALSE, otherwise.

Determinism: Deterministic.

Errors: In pure TLA+, the result is undefined if one of the arguments evaluates to a non-integer 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 less-than-or-equal

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, if a < b or a = 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 non-integer 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 greater-than

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, if a is greater than 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 non-integer 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 greater-than-or-equal

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, if a > b or a = 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 non-integer 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

[Back to all operators]

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:

1. Booleans,
2. integers,
3. strings,
4. sets,
5. 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, if S is a set that contains an element that is equal to the value of e; and
• FALSE, if S is a set and all of its elements are not equal to the value of e.

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 Non-determinism.

Determinism: Deterministic, unless you are using the special form x' \in S to assign a value to x', see Control Flow and Non-determinism.

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 non-membership

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, if S is a set that contains an element that is equal to the value of e; and
• TRUE, if S is a set and all of its elements are not equal to the value of e.

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, if S and T are sets, and every element of S is a member of T;
• FALSE, if S and T are sets, and there is an element of S that is not a member of T.

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 type a, for some type a,
• S has the type Set(a),
• P has the type Bool,
• the expression { x \in S: P } has the type Set(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):

1. Bind the element e to variable x,
2. Evaluate the predicate P,
3. If P evaluates to TRUE under the binding [x |-> e], then insert the element of e into set F.

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 one-argument case:

• x has the type a, for some type a,
• S has the type Set(a),
• e has the type b, for some type b,
• the expression { e: x \in S } has the type Set(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:

1. Bind the element e_1 to variable x,
2. Bind the element e_2 to variable y,
3. Compute the value of e under the binding [x |-> e_1, y |-> e_2],
4. Insert the value e into the set M.

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, then X \subseteq S.
• If X \subseteq S, then X \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 similar-looking 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, then X \subseteq T.
• If y \in T, then there is a set Y \in S that contains y, 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, when S is a finite set,
• FALSE, when S 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

[Back to all operators]

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 first-order logic, as opposed to propositional logic (see Booleans).

Note that the special form \E y \in S: x' = y is often used to express non-determinism in TLA+. See Control Flow and Non-determinism. 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 type a, for some type a,
• S has the type Set(a),
• P has the type Bool,
• the expression \A x \in S: P has the type Bool.

Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:

• \A x \in S: P evaluates to TRUE, if for every element e of S, the expression P evaluates to TRUE against the binding [x |-> e].
• Conversely, \A x \in S: P evaluates to FALSE, if there exists an element e of S that makes the expression P evaluate to FALSE 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 type a, for some type a,
• S has the type Set(a),
• P has the type Bool,
• the expression \E x \in S: P has the type Bool.

Effect: This operator evaluates to a Boolean value. We explain semantics only for a single variable:

• \E x \in S: P evaluates to TRUE, if there is an element e of S that makes the expression P evaluate to TRUE against the binding [x |-> e].
• Conversely, \E x \in S: P evaluates to FALSE, if for all elements e of S, the expression P evaluate to FALSE 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 non-deterministic assignment, duality does not work anymore! See Control Flow and Non-determinism.

Determinism: Deterministic when P contains no action operators (including the prime operator '). For the non-deterministic case, see Control Flow and Non-determinism.

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 and v_2 are Booleans, then e_1 = e_2 evaluates to v_1 <=> v_2.

• If v_1 and v_2 are integers, then e_1 = e_2 evaluates to TRUE if and only if v_1 and v_2 are exactly the same integers.

• If v_1 and v_2 are strings, then e_1 = e_2 evaluates to TRUE if and only if v_1 and v_2 are exactly the same strings.

• If v_1 and v_2 are sets, then e_1 = e_2 evaluates to TRUE if and only if the following expression evaluates to TRUE:

v_1 \subseteq v_2 /\ v_2 \subseteq v_1

• If v_1 and v_2 are functions, tuples, records, or sequences, then e_1 = e_2 evaluates to TRUE if and only if the following expression evaluates to TRUE:

  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 to FALSE 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 non-deterministic case, see Control Flow and Non-determinism.

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 type a, for some type a,
• S has the type Set(a),
• P has the type Bool,
• the expression CHOOSE x \in S: P has the type a.

Effect: This operator implements a black-box 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, then CHOOSE 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 non-determinism with CHOOSE. For non-determinism, see: Control Flow and Non-determinism.

Apalache picks a set element that satisfies the predicate P, but it does not guarantee the repeatability property of CHOOSE. It does not guarantee non-determinism 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 first-order 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 first-order 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

[Back to all operators]

Contributors: @konnov, @shonfeder, @Kukovec, @Alexander-N

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 Two-phase commit, the function rmState stores the transaction state for each process:

argumentrmState[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 non-deterministically 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 built-in 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 type a, for some type a,
• S has the type Set(a),
• e has the type b, for some type b,
• the expression [ x \in S |-> e ] has the type a -> 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:

1. Bind the element elem to variable x,
2. Compute the value of e under the binding [x |-> elem] and store it in a temporary variable called result.
3. Set f[elem] to result.

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 well-typed (as a set), and it requires the mapping expression e to be well-typed. 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 single-index case, the type is ((a -> b), a) => b, for some types a and b. In the multi-index 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, then f[e] evaluates to the value that function f associates with the value of e.
• 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 type-compatible 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 two-dimensional 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 single-point 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 to DOMAIN f.
• For every element b \in DOMAIN f, do:
• If b = a_i for some i \in 1..n, then set g[b] to e_i.
• If b \notin { a_1, ..., a_n }, then set g[b] to f[b].

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 type-compatible 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

[Back to all operators]

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 duck-typing 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 C-style struct-member 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.

It is quite common to mix records of different shapes into sets. For instance, see how the variable msgs is updated in Paxos. To address this pattern, Apalache treats records that do not disagree on field types to be type-compatible. For instance, the records [type |-> "A", a |-> 3] and [type |-> "B", b |-> TRUE] have the joint type:

  [type: Str, a: Int, b: Bool]


If your spec tries to access a field on a record without that field, Apalache will fail with the following error:

Access to non-existent record field ...


## 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 of e_i for i \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.

>>> { "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_1: a_1, ..., field_i: a_i, ..., field_n: a_n]) => a_i, for some types a_1, ..., a_n. Due to the record unification rule, we usually write this type simply as: [field_i: a_i] => a_i.

Note that r.field_i is just a syntax sugar for r["field_i"] in TLA+. Hence, if the type of r cannot be inferred, you can see an error message about Apalache not knowing, whether r is a record, or a function.

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

[Back to all operators]

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 some n > 0.

That is right. You can construct the empty tuple <<>> in TLA+ as well as a single-element 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 well-known 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 fine-grained 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:

1. A tuple constructor: (a_1, ..., a_n) => <<a_1, ..., a_n>>, for some types a_1, ..., a_n.
2. A sequence constructor: (a, ..., a) => Seq(a), for some type a.

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 to 1..n,
• set r[i] to the value of e_i for i \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 "tuples-are-functions" 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

[Back to all operators]

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 duck-typing 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) and Tail(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:

1. A tuple constructor: (a_1, ..., a_n) => <<a_1, ..., a_n>>, for some types a_1, ..., a_n.
2. A sequence constructor: (a, ..., a) => Seq(a), for some type a.

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 to 1..n,
• set r[i] to the value of e_i for i \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 two-element list
['Printer', 631]
>>> { 1: "Printer", 2: 631 }  # the "sequences-are-functions" 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] to seq[i] for i \in 1..Len(seq).
• set newSeq[Len(seq) + 1] to e.

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.

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] to seq[i + 1] for i \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 be 1..(Len(s) + Len(t)).
• set newSeq[i] to s[i] for i \in 1..Len(s).
• set newSeq[Len(s) + i] to t[i] for i \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 be 1..(n - m).
• set newSeq[i] to s[m + i - 1] for i \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 one-argument 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.

# User-defined operators

Like macros, to do a lot of things in one system step...

User-defined 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 user-defined operators in versions 1 and 2.

Short digression. The most important thing to understand about user-defined 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 user-defined 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 user-defined operators that you would probably find unexpected:

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

2. Operators may accept other operators as parameters. Such operators are called Higher-order operator definitions.

3. Although operators may be passed as parameters, they are not first-class 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).

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

5. Operators can be nested. However, nested operators require a slightly different syntax. They are defined with LET-IN definitions.

Details about operators. We go in detail about different kinds of operators and recursive functions below:

# Top-level operator definitions

[Back to user operators]

## Quick example

Here is a quick example of a top-level 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 Higher-order 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:

1. 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+.

2. The one-argument operator Fahrenheit2Celcius is applied to t, which is a parameter of the operator UpdateCelcius.

3. The two-argument operator Max is applied to ABSOLUTE_ZERO_IN_CELCIUS and Fahrenheit2Celcius(t).

4. The one-argument operator UpdateCelcius is applied to fahrenheit', which is the value of state variable fahrenheit in the next state of the state machine. TLA+ has no problem applying the operator to fahrenheit' or to fahrenheit.

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 non-recursive 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:

1. Change the names in the definition of A in such a way such they do not clash with the names in B (as well as with other names that may be used in B). This is the well-known technique of Alpha conversion in programming languages. This may also require renaming of the parameters p_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

1. Substitute the expression A(e_1, ..., e_n) in the definition of B with body_of_uniq_A.

2. Substitute the names uniq_p_1, ..., uniq_p_n with the expressions e_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 user-defined operators. For instance, Apalache applies Alpha conversion and Beta reduction to remove user-defined operator and then translates the TLA+ expression to SMT.

# LET-IN operator definitions

work in progress...

[Back to user operators]

# Higher-order operator definitions

work in progress...

[Back to user operators]

# Anonymous operator definitions

work in progress...

[Back to user operators]

# Local operator definitions

work in progress...

[Back to user operators]

# 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 state-variable 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")


## Non-deterministically 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: Non-deterministically pick a value out of the set S, if S is non-empty. If S is empty, return some value of the proper type.

Determinism: Non-deterministic if S is non-empty, 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 non-determinism.

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 ill-typed, 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 non-deterministic
\/ 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 property 1..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 on len, 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 side-outputs 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 side-outputs 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 side-outputs 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


# 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:

1. understood by distributed system engineers,
2. understood by verification engineers, and
3. 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 book-keeping 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 2: Apply primes only to state variables 📌

Idiom 3: Isolate updates to VARIABLES 👻

Idiom 4: Isolate non-determinism in actions 🔮

Idiom 5: Introduce pure operators 🙈

Idiom 6: Introduce a naming convention for operator parameters 🛂

Idiom 8: CHOOSE smart, prefer \E 💂‍♂

Idiom 9: Do not over-structure 🔬

Idiom 10: Do not over-modularize 🦆

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 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:

1. To avoid repetitive computations of the same expression,
2. To break down a large expression into a series of smaller expressions,
3. To make the code concise.

Point 1 is a non-issue 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 LET-definitions 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.

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.

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 model-based 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 let-definitions:

        ...
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 let-definitions 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 let-definition. 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 non-deterministic 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, non-deterministic choice of the next value of x. When combined with the operator :=, the effect of non-deterministic 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


• Non-determinism 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 and x' \in S as assignments, as long as x' 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

• 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
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 := {}
/\ deliveredSeqNo := -1

(* Subsequent assignments are all to primed variables, designating changed values
after state transition. *)
Send ==
/\ sent' := sent \union {sentSeqNo}
/\ sentSeqNo' := sentSeqNo + 1

(* We make the nonderministic assignment explicit, by use of existential quantification *)
/\ \E msgs \in SUBSET (sent \ received):
/\ UNCHANGED <<sentSeqNo, sent, deliveredSeqNo>>

Deliver ==
/\ (deliveredSeqNo + 1) \in received
/\ deliveredSeqNo' := deliveredSeqNo + 1
\* deliver the message with the sequence number deliveredSeqNo'

Next ==
\/ Send
\/ 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+1-th 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 round-robin 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 312-313 of Specifying Systems.

• 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.

• 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 writing Inv'. However, you have to be careful about propagation of primes in Inv.

## 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 round-robin to decide who makes the next step
/\ turn' := 1 + (turn + 1) % 3
===============================================================================


# Idiom 7: Use Boolean operators in actions, not IF-THEN-ELSE

author: Gabriela Moreira

## Description

TLA+ provides an IF-THEN-ELSE 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 transition-defining action declared with IF-THEN-ELSE 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 IF-THEN-ELSE definitions.

The IF-THEN-ELSE operator can be used either to define a value, or to branch some action as a sort of flow control. Defining values with IF-THEN-ELSE 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 IF-THEN-ELSE for flow control unnecessary. This idiom aims to clarify different usages of IF-THEN-ELSE expressions, keeping in mind the TLA+ essence of declaring actions to define transitions.

## When to use IF-THEN-ELSE

### When the result is not Boolean

When the IF-THEN-ELSE 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 IF-THEN-ELSE 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 IF-THEN-ELSE 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 IF-THEN-ELSE 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 IF-THEN-ELSE 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_{n-1} /\ cn /\ an
\/ ~c1 /\ ... /\ ~c_{n-1} /\ ~cn /\ a


## When (and how) not to use IF-THEN-ELSE

Mixing IF-THEN-ELSE expressions with action operators introduces unnecessary branching to definitions that could be self-contained 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 IF-THEN-ELSE block:

WithdrawSuccess(amount) == /\ balance >= amount
/\ balance' = balance - amount
/\ response' = "SUCCESS"

WithdrawFailure(amount) == /\ balance < amount
/\ response' = "FAILURE"
/\ UNCHANGED balance

Withdraw(amount) == WithdrawSuccess(amount) \/ WithdrawFailure(amount)


• 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 IF-THEN-ELSE expression is
• Nested IF-THEN-ELSE expressions are an extrapolation of these problems and can over-constrain some branches if not done carefully. Using different actions defining its conditions explicitly leaves less room for implicit wrong constraints that an ELSE 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 IF-THEN-ELSE is used without attention.

A disjunction in TLA+ may or may not represent non-determinism, while an IF-THEN-ELSE is incapable of introducing non-determinism. If it's important that readers can easily differentiate deterministic and non-deterministic definitions, using IF-THEN-ELSE 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 type-checking. 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 backward-compatibility with specifications of message-based 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 type-system for TLA+, as well as introduces unsoundness, in the sense that it makes it impossible, at the type-checking level, to detect record-field 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 model-checking process, instead of at the level of static analysis provided by the type-checker.

## 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 type-checker, 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 integer-valued field "x" and "t2" with a string-valued 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}

/\ \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

/\ \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 ==
\/ CheckIntMsg
\/ 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}]

/\ \E v \in Ints:
/\ SendInt( [x |-> v] )
/\ UNCHANGED <<found3, foundC>>

CheckIntMsg ==
/\ \E m \in msgs.int:
/\ found3' = ( m.x = 3 )
/\ RmInt(m)
/\ UNCHANGED foundC

/\ \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 ==
\/ CheckIntMsg
\/ 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 type-unification for record sets is generally expensive for the solver.

# ADR-002: types and type annotations

authorsrevisionrevision date
Shon Feder, Igor Konnov, Jure Kukovec5April 08, 2022

This is an architectural decision record. For user documentation, check the Snowcat tutorial and Snowcat HOWTO.

This is a follow up of RFC-001, which discusses plenty of alternative solutions. In this ADR-002, 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.

1. How to write types in TLA+ (Type System 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.

In contrast, the type inference algorithm by @Kukovec is fully automatic and thus it eliminates the need for type annotations. Jure's algorithm is using Type System 1 too. The type inference algorithm is still in the prototype phase.

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)

We simply write types as strings that follow the type grammar:

T ::=   'Bool' | 'Int' | 'Str'
| T '->' T
| 'Set' '(' T ')'
| 'Seq' '(' T ')'
| '<<' T ',' ...',' T '>>'
| '[' field ':' T ',' ...',' field ':' T ']'
| '(' T ',' ...',' T ')' '=>' T
| typeConst
| typeVar
| '(' T ')'

field     ::= <an identifier that matches [a-zA-Z_][a-zA-Z0-9_]*>

typeConst ::= <an identifier that matches [A-Z_][A-Z0-9_]*>

typeVar   ::= <a single letter from [a-z]>


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 type T.
• The rule Seq(T) produces a sequence type over elements of type T.
• The rule <<T, ..., T>> produces a tuple type over types that are produced by T. Types at different positions may differ.
• The rule [field: T, ..., field: T] produces a record type over types that are produced by T. Types at different positions may differ.
• The rule (T, ..., T) => T defines an operator whose result type and parameter types are produced by T.
• 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 multi-argument function always receives a tuple, e.g., <<Int, Bool>> -> Int, whereas a single-argument function receives the type of its argument, e.g., Int -> Int. The arrow -> is right-associative, 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 variable-free 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 alias-free form.

Examples.

• x is an integer. Its type is Int.
• f is a function from an integer to an integer. Its type is Int -> Int.
• f is a function from a set of integers to a set of integers. Its type is Set(Int) -> Set(Int).
• r is a record that has the fields a and b, where a is an integer and b is a string. Its type is [a: Int, b: Str].
• F is a set of functions from a pair of integers to an integer. Its type is Set(<<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 higher-order 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 and Faulty are sets of the same type. Their type is Set(PID).

### 1.2. Type aliases

The grammar of T includes one more rule for defining a type alias:

A ::= typeConst "=" T


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. For examples, see Section 2.4.

### 1.3. Type System 1.2, including precise records, variants, and rows

This is work in progress. You can track the progress of this work in Issue 401. Once this work is complete, we will switch to Type System 1.2.

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 ADR-002. To this end, we extend the grammar with new records, variants, and rows as follows:

// Type System 1.2
T2 ::=
// all types of Type System 1
T
// A new record type with a fully defined structure.
// The set of fields may be empty.
| '{' field ':' T2 ',' ...',' field ':' T2 '}'
// A new record type with a partially defined structure
// (the type variable should be a 'row').
// The set of fields may be empty.
| '{' field ':' T2 ',' ...',' field ':' T2 ',' typeVar '}'
// A variant that contains several options.
| variantOption '|' ... '|' variantOption
// A variant of undefined structure (the type variable should be a 'row')
| 'Variant' '(' typeVar ')'
// An empty variant
| 'Variant' '(' ')'

variantOption ::=
// A variant option with a fully defined structure.
| { tag: stringLiteral, field: T2, ..., field: T2 }
// a variant option with a partially defined structure
//   (a variant option over a row).
| { tag: stringLiteral, field: T2, ..., field: T2, typeVar }

// Special syntax for the rows, which is internal to the type checker.
row ::=
// A row with a fully defined structure.
| '(|' field ':' T2 '|' ...'|' field ':' T2 '|)'
// A row with a partially defined structure (ending with a row).
| '(|' field ':' T2 '|' ...'|' field ':' T2 '|' typeVar '|)'


Examples.

• r1 is a record that has the fields a and b, where a is an integer and b is a string. Its type is { a: Int, b: Str }.

• r2 is a record that has the fields a of type Int and b of type Str and other fields, whose precise structure is captured with a type variable c. The type of r2 is { a: Int, b: Str, c }. More precisely, the variable c should 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:

• It has the fields tag of type Str and a of type Int (if the field tag is equal to "A").

• It has the fields tag of type Str and b of type Bool (if the field tag is equal to "B").

• v2 is an empty variant, which admits no options. It has the type Variant().

• v3 is a variant whose structure is defined by the type variable b. The type of v3 is Variant(b). Note that b type variable should be a row. For instance, it can be equal to the type (| A: { tag: Str, f: Int } | B: { tag: Str, g: 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.

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 one-line 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,
])


The parser only supports one-line 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 ill-typed some perfectly legal TLA+ values. For instance, we cannot assign a reasonable type to {1, TRUE}.

Sets of records in Type System 1. We can assign a reasonable type to {[type |-> "1a", bal |-> 1], [type |-> "2a", bal |-> 2, val |-> 3]}, a pattern that 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 better tests the field type carefully.

Sets of records in Type System 1.2. Consider the following set:

{[tag |-> "1a", bal |-> 1],
[tag |-> "2a", 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({ tag: "1a", bal: Int } | { tag: "2a", bal: Int, val: Int })


The value of the field tag serves as a type tag. However, we have to fix a set of patterns that turn a variant type into a precise record type. In untyped TLA+, such pattern is a set comprehension, e.g., { r \in S: r.tag = "1a" }. In the typed version, we define a minimal set of operators over variants in the module Variants.tla. For instance, instead of writing the set comprehension, we have to use a filter over a set of variants: FilterByTag(S, "1a").

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 user-defined operators, on top of their types for TLA+ expressions that do not contain user-defined 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})


Higher-order 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 let-definition 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 LET-definitions 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 LET-definition 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>; on a dummy operator called <PREFIX>TypeAliases. For example:

\* @typeAlias: ENTRY = [a: Int, b: Bool];
EXTypeAliases = 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> convention protects against name clashes when the module is extended or instantiated.

The actual rules around the placement of the @typeAlias annotation allows more flexibility:

1. You can define a type alias with @typeAlias anywhere you can define a @type.

2. The names of type aliases must be unique in a module.

3. There is no scoping for aliases within a module. Even if an alias is defined deep in a tree of LET-IN definitions, it can be references at any level in the module.

## 3. Example

As an example that contains non-trivial 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
=============================================================================


authorrevision
Igor Konnov1

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)
• 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 non-incremental (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 first-order 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.

authorrevision
Igor Konnov2

This ADR documents our decision on using Java-like annotations in comments. Our main motivation to have annotations is to simplify type annotations, as presented in ADR-002. 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:

1. Constant declarations, e.g., CONSTANT N.
2. Variable declarations, e.g., VARIABLE x.
3. Operator declarations, including:
4. Top-level operator declarations, e.g., Foo(x) == e.
5. Operators defined via LET-IN, e.g., Foo(x) == LET Inner(y) == e IN f.
6. Recursive operators, e.g., RECURSIVE Fact(_) Fact(n) == ...
7. Recursive and non-recursive functions including:
8. Top-level functions, e.g., foo[i \in Int] == e.
9. Functions defined via LET-IN, 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     ::= '-'? [0-9]+
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 or t.

Examples. The following strings are examples of syntactically correct annotations:

1. @tailrec
2. @type("(Int, Int) => Int")
3. @require(Init)
4. @type: (Int, Int) => Int ;
5. @random(true)
6. @deprecated("Use operator Foo instead")
7. @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 one-argument 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, LET-definitions, 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 single-argument 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 tla-io, 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.

authorrevision
Jure Kukovec1.1

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 third-party tools in the future.

## 1. Serializable classes

The following classes are serializable:

1. 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
• Operator expressions OperEx
• LET-IN expressions LetInEx
2. TLA+ declarations (see TlaDecl) and subclasses thereof:

• Variable declarations TlaVarDecl
• Constant declarations TlaConstDecl
• Operator declarations TlaOperDecl
• Assumption declarations TlaAssumeDecl
• Theorem declarations TlaTheoremDecl
3. 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 ADR-002, ADR-004 for a description of our type system and the syntax for types-as-string-annotations 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}, and
• modules, the value of which is an array containing the JSON encodings of zero or more TlaModule 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 sub-expressions, 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:

1. For TlaStr: a JSON string
2. For TlaBool: a JSON Boolean
3. For TlaInt(bigIntValue):
1. If bigIntValue.isValidInt: a JSON number
2. Otherwise: { "bigInt": bigIntValue.toString() }
4. For TlaDecimal(decValue): a JSON string decValue.toString

The reason for the non-uniform 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 tla-import, see TlaToJson.

# RFC-006: Unit testing and property-based testing of TLA+ specifications

authorsrevision
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 unit-testing 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 built-in 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 built-in 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 Property-based 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 well-known 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 non-deterministic 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 Apalache-specific 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 property-based 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 fine-grained 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 that TypeOK should act as an initialization predicate for testing TestAction_n0.
• The annotation @testAction indicates that TestAction_n0 should be tested as an action that is an operator over unprimed and primed variable.
• The annotation @ensure(Assert_n0) tells the framework that Assert_n0 should hold after TestAction_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 and TypeOK (similar to Init),
• executing the action by evaluating the action predicate TestAction_n0 (like a single instance of Next),
• 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:

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

2. 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 error-prone approach. Let's do it step-by-step.

First of all, there is no simple way to initialize constants in TLA+, as we did with ConstInit (this is an Apalache-specific 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 state-of-the-art 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 constraint-based 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 Design-by-contract languages. In particular, they are used as pre- and post-conditions 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 self-contained, 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 Property-based testing, e.g., we use generators similar to Scalacheck. In contrast to property-based 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 Property-based 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 Property-based 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 command-line 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 tool-specific option search.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 tool-specific option checker.algo to offline, meaning that the model checker should use the offline solver instead of the incremental one.

• The annotation testOption("checker.nworkers", 2) sets the tool-specific option checker.nworkers to 2, 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 re-design 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.

# ADR-007: Apalache Package Structure Guidelines

authorrevision
Jure Kukovec1

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:

1. Interface core: Classes within the interface core relate to general TLA+ concepts, and are intended to be usable by 3rd party developers.
2. Apalalache core: Classes in the Apalache core relate to specific Apalache- or model-checking- functionality

Staking the "core" strata together gives us the complete stack implementing Apalache.

## 2. Interface core

Together, L0-L6 make up the interface core. Notably, these levels are Apalache-agnostic. The concepts of these levels are packaged as follows:

1. tla-ir : L0-L3
2. tla-types : L4
3. tla-aux : L5-L6

# L0: TLA+ IR

In L0, we have implementations of the following concepts:

• The internal representation of TLA+, with the exception of:
• Apalache-specific operators (ApalacheOper)
• Any utility required to define the IR, such as:
• UIDs
• TypeTags & TT1

# L1: Auxiliary structures

In L1, we have implementations of the following concepts:

• Meta-information 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 TLA-IR
• Reading from or writing to files (.tla, .json, TLC-formats)

# L3: Basic IR manipulation

In L3, we have implementations of the following concepts:

• TLA+ transformations that:
• do not introduce operators excluded from L0
• have type-correctness 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 type-correctness. 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 type-related static analysis, such as:
• Type checking
• Type inference

# L5: Self-typing Builder:

In L5, we have implementations of the following concepts:

• The TLA-IR builder

# L6: Type-guaranteed 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 type-tags, but use the self-typing builder instead

## 3. Apalache core

L7+ make up the Apalache core. The concepts of these levels are packaged as follows:

1. apa-base : L7-L9
2. apa-pass : L10
3. apa-tool : L11+

If individual passes in L10 are deemed complex enough, they may be placed in their own package (e.g. apa-bmc)

# L7: Extensions of TLA+

In L7, we have implementations of the following concepts:

• Any Apalache-specific operators, introduced into the IR (e.g. ApalacheOper)
• Extensions of any printer/reader/writer classes with support for the above specific operators

# L8: Apalache-specific 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 type-correctness.

# 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.

authorrevision
Jure Kukovec1

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. Parser-level

Since we depend on Sany for parsing, Apalache rejects any syntax which Sany cannot parse. If Sany produces an exception Apalache catches it and re-throws a class extending

ParserException


The exception should report the following details:

• Location of at least one parser issue

### 1.2. Apalache-specific 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. Type-related exceptions

This category of exceptions deals with problems arising from the type-system 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 built-in 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 user-defined 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. Static-analysis 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/Transformation-specific 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 --quiet-exceptions .

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

## authors: Jure Kukovec, Shon Feder last revised: 2021-12-14

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:

1. Counterexamples
2. Log files
3. Intermediate state outputs
4. 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: --write-intermediate 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 out-dir 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>/apalache-out/.

Each run looks for a subdirectory inside of the out-dir 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 out-dir. 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 YYYY-MM-DD format, <TIME> is the local time in HH-MM-SS format.

Example file structure for a run executed on a file test.tla:

_apalache-out/
└── test.tla
├── 2021-11-05T22-54-55_810261790529975561


### Custom run directories

The --run-dir flag can be used to specify an output directory into which outputs are written directly. When the --run-dir 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 pre-filled bug report file BugReport.md, if the tool exited with a FailureMessage
• if --write-intermediate is set, a subdirectory intermediate, 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. profile-rules.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 out-dir. 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:

1. If --<flag>=<value> is given, use <value>, otherwise
2. if a local .apalache.cfg file is found (or is specified with the --config-file argument) containing <flag>: <value>, then use <value>, otherwise
3. if the global apalache.cfg specifies <flag>: <value> use <value>, otherwise
4. Use the defaults specified in the ApalacheConfig class.

# ADR-011: alternative SMT encoding using arrays

authorrevision
Rodrigo Otoni1.6

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 SMT-LIB Standard (Version 2.6) in conjunction with Z3-specific 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.

## 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.

--smt-encoding : the SMT encoding: oopsla19, arrays (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 to SolverConfig.
• Add new class SymbStateRewriterImplWithArrays, which extends class SymbStateRewriterImpl.
• Use the new option to set the SolverConfig encoding field and select between different SymbStateRewriter implementations in classes BoundedCheckerPassImpl and SymbStateRewriterAuto.

## 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 tla-bmcmt/src/test to enable unit testing with different configurations of SolverConfig and implementations of SymbStateRewriter.
• Add unit tests for the new encoding, which should be similar to existing tests, but use a different solver configuration and SymbStateRewriterImplWithArrays instead of SymbStateRewriterImpl.
• Add integration tests for the new encoding by tagging existing tests with array-encoding, which will be run by the CI with envvar SMT_ENCODING set to arrays.

## 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.

(declare-sort some_sort 0)
(declare-const elem1 some_sort)
(declare-const elem2 some_sort)
(declare-const 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.

(declare-sort set_of_some_Sort 0)
(declare-const set1 set_of_some_Sort)

(declare-const elem1_in_set1 Bool)
(declare-const elem2_in_set1 Bool)
(declare-const 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 quantifier-free fragment of first-order 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.

(declare-const set2_0 (Array some_sort Bool))
(declare-const set2_1 (Array some_sort Bool))
(declare-const set2_2 (Array some_sort Bool))
(declare-const 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.
• In class SymbStateRewriterImplWithArrays, add the new rules to ruleLookupTable by overriding the entries to their older versions.
• Add four new Apalache IR operators in ApalacheOper, Builder, ConstSimplifierForSmt, and PreproSolverContext, to represent the array operations.
• The selectInSet IR operator represents the SMT select.
• The storeInSet IR operator represents the SMT store.
• 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.
• In class Z3SolverContext, add/change appropriate methods to handle SMT constraints over arrays.
• The main changes will de done in declareCell and the new mkSelect, mkStore, and mkUnchangedSet methods, as these methods are directly responsible for creating the SMT constraints representing sets and set membership.
• With the new IR operators, the "in-relation" concept, which underpins declareInPredIfNeeded and getInPred, will not be applied to the new encoding. Cases for the new IR operators will be added to toExpr, which will default to TlaSetOper.in and TlaSetOper.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 and PowSetT will be added to getOrMkCellSort, 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 of push and pop 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


## 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.

(declare-sort Tuple_Int 0) ; Sort of <Int>
(declare-const tuple_with_1 Tuple_Int) ; <1>
(declare-const tuple_with_2 Tuple_Int) ; <2>
(declare-const tuple_with_3 Tuple_Int) ; <3>

(declare-const finSucc_domain_0 (Array Tuple_Int Bool))
(declare-const finSucc_domain_1 (Array Tuple_Int Bool))
(declare-const finSucc_domain_2 (Array Tuple_Int Bool))
(declare-const 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.

(declare-const finSucc_0 (Array Tuple_Int Int))
(declare-const finSucc_1 (Array Tuple_Int Int))
(declare-const finSucc_2 (Array Tuple_Int Int))
(declare-const 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

(declare-const 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.

(declare-const finSucc_4 (Array Tuple_Int Int))
(declare-const 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 counter-example generation in SymbStateDecoder. In order to continue having counter-examples, these sets will keep being produced, but will not be present in the SMT constraints. They will be carried only as metadata in the Arena.
• 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, and unchangedSet IR operators will be used when handling them. A future refactoring may rename these operators.
• Update class Z3SolverContext to handle the new SMT constraints over arrays.
• A case for FunT will be added to getOrMkCellSort.
• 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 is true, 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 first funAppRes = f[s \in S], followed by funAppRes \in T.

TODO

## 6. Encoding control operators and TLA+ quantifiers

TODO

authors: Igor Konnov

proposed by: Vitor Enes, Andrey Kupriyanov

last revised: 2022-06-01

# ADR-015: Informal Trace Format in JSON

## 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 ADR-015 contains a very simple format that does not require any knowledge of TLA+ and can be easily integrated into other tools.

## 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:

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

2. 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 finite-state system can be represented by a lasso. (In general, executions of infinite-state 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:

1. They are simply sequences of states.

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

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

4. Integer and string literals.

5. Set constructor, sequence/tuple constructor, record constructor.

6. 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

1. 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.
2. Use the JSON serialization as in ADR005:

• Pros:

• easy to parse automatically.
• Cons:

• too detailed and too verbose.
• requires the knowledge of Apalache IR and of TLA+.
3. 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 machine-readable). 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:

1. A JSON Boolean: either false or true.

2. A JSON string literal, e.g., "hello". TLA+ strings are written as strings in this format.

3. A JSON integer, e.g., 123. According to RFC7159, JSON integers must be in the range: [-(2**53)+1, (2**53)-1]. Integers in this range may be written as JSON integers.

4. A big integer of the following form: { "#bigint": "[-][0-9]+" }. We are using this format, as many JSON parsers impose limits on integer values, see RFC7159. Big and small integers may be written in this format.

5. A list of the form [ <expr>, ..., <expr> ]. A list is just a JSON array. TLA+ sequences are written as lists in this format.

6. 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.

7. 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.

8. 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.

9. A map of the form { "#map": [ [ <expr>, <expr> ], ..., [ <expr>, <expr> ] ] }. That is, a map holds a JSON array of two-element arrays. Each two-element array p is interpreted as follows: p[0] is the map key and p[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.

10. 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.

### Example

The counterexample to NoSolution may be written in the ITF format as follows:

{
"#meta": {
"source": "MC_MissionariesAndCannibalsTyped.tla"
},
"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 machine-readability and tool automation in mind, whereas TLA+ counterexamples are not. However, the example in the ITF format is also self-explanatory 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 heavy-weight solution with no obvious value. However, we should keep it in mind and use schemas, when the need arises.

## Consequences

Reserved for the future.

# ADR-16: ReTLA - Relational TLA

authorrevision
Jure Kukovec1

## Summary

We propose introducing support for a severely restricted fragment of TLA+, named Relational TLA (reTLA for short), which covers uninterpreted first-order 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 model-checking 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 message-passing system:

CONSTANT Values
VARIABLE messages

SendT1(v) == messages \union { [ type |-> "t1", x |-> v ] }

\E msg \in messages:
/\ msg.type = "t1"
/\ F(msg.x) \* some action

Next ==
\/ \E v \in Values: SendT1(v)


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]

\E v \in Values:
/\ T1messages[v]
/\ F(v) \* some action

Next ==
\/ \E v \in Values: SendT1(v)


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, sets-of-sets, 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

1. 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
2. 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 or BOOLEAN, or
• CONSTANT-declared and has a type Set(T), for some uninterpreted type T, or
• Boolean operators (/\, \/, =>, <=>, ~)
• Quantified expressions (\E x \in S: P, \A x \in S: P), on the condition that P is in reTLA and S is a restricted set.
• Functions:
• Definitions ([x1 \in S1, ..., xn \in Sn |-> e]), on the condition that:
• e is in reTLA and has an Int, Bool or uninterpreted type
• All Si are restricted sets.
• Updates ([f EXCEPT ![x] = y]), if y is in reTLA
• Applications (f[x])
• (In)equality and assignments:
• a = b and a /= b if both a and b are in reTLA
• x' = v if x is a VARIABLE and v is in reTLA
• Control flow:
• IF p THEN a ELSE b if p,a,b are all in reTLA

In potential future versions we are likely to also support:

• Standard integer operators (+, -, u-, *, %, <, >, <=, >=)
• ranges a..b, where both a and b are in reTLA.
• Tuples

## Consequences

Reserved for the future.

# PDR-017: Checking temporal properties

This is a preliminary design document. It will be refined and it will mature into an ADR later.

## 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 16.2.3-4). 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 formula A and an expression e (usually a tuple of variables). This predicate is equivalent to A \/ e' = e.

• No-stutter A: <A>_e, for an action formula A and an expression e, which is equivalent to A /\ e' /= e.

• ENABLED A, for an action formula A, is true in a state s if there is a state t such that s is transformed to t via action A.

• Temporal formulas:

• Eventually: <>F, for a temporal formula F.

• Always: []F, for a temporal formula F.

• Weak fairness: WF_e(A), for an expression e and a formula F, which is equivalent to []<>~(ENABLED <A>_e) \/ []<><A>_e.

• Strong fairness: SF_e(A), for an expression e and a formula F, which is equivalent to <>[]~(ENABLED <A>_e) \/ []<><A>_e.

• Leads-to: 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 16.2.3-4).

## 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:

1. 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.

2. Specification preprocessing.

In this approach, given a specification S and a temporal property P, we would produce another specification S_P and an invariant I_P that has the following property: the temporal property P holds on specification S if and only if the invariant I_P holds on specification S_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 ADR-010 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), and SF_e(A) via preprocessing.

The task on Temporal operators is well-understood 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] satisfies Init,
• every pair of states s[i] and s[i+1] satisfies Next, for i \in 0..k-1, 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 finite-state systems:

Whenever a finite-state 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 finite-state systems. For example, if all variable domains are finite (and bounded), then the specification is finite-state. However, if a specification contains integer variables, it may produce infinitely many states. That is, an infinite-state system may still contain lassos as counterexamples but it does not have to, which makes this technique incomplete. An extension to infinite-state 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.

• 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).

• 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.

• 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

• 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.

• 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

• 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

authorrevisionlast revised
Jure Kukovec1Apr 21, 2022

## 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 LET-IN, 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 non-recursive 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:

1. Standard inlining: the instantiation described above
1. Non-nullary inlining: the instantiation described above, except the inlining skips nullary LET-defined operators
2. Pass-by-name inlining: replacing an operator name A used as an argument to a higher-order operator with a local LET-definition: 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 1-to-1 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 non-nullary operators is strictly necessary, inlining nullary operators is not, because nullary operators, by definition, do not have formal parameters. Therefore, in a well-constructed expression, all variables appearing in a nullary operator are scoped, i.e. they are either specification-level 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 non-nullary 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 non-nullary variant, because it is strictly more efficient in our cell-arena framework fo rewriting rules.

The reason for doing (2) is more pragmatic; in order to rewrite expressions which feature any of the higher-order (HO) built-in 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 on-demand as part of rewriting, or whether to isolate it to an independent inlining-pass (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).

1. 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 simplify IF TRUE THEN a ELSE b, but not IF p THEN a ELSE b)
• Rewriting rules for different encodings have to deal with operators in their generality.
2. Independently perform only standard (non-nullary) inlining (1), but no pass-by-name inlining (2)

• Pros: Allows for additional optimizations after inlining (simplification, normalization, keramelization, etc.)
• Cons: Rewriting rules still need scope, to resolve higher-order operator arguments in certain built-in operators (e.g. folds)
• Recall that the non-nullary variant of (1) is strictly better than the simple one (while being trivial to implement), because nullary inlining is prone to repetition.
3. Independently perform non-nullary inlining and pass-by-name inlining

• Pros:
• Enables further optimizations (simplification, normalization, keramelization, etc.)
• Using non-nullary inlining has all of the benefits of standard inlining, while additionally being able to avoid repetition (e.g. not inlining A in A + A)
• Pass-by-name inlining allows us to keep rewriting rules local
• Cons: Implementation is more complex

## 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 non-nullary 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

authors: Igor Konnov

last revised: 2022-06-01

# ADR-020: Introduce static membership in arenas

## 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 cell c_p potentially contains an element c_e (e.g., if c_p represents a set). These edges encode many-to-many relations.

• dom. A domain edge (c_f, c_d) represents that a function cell c_f has the domain represented with a cell c_d. These edges encode many-to-one relations.

Likewise, a domain edge (c_F, c_c) represents that a function set cell c_F has the domain represented with a cell c_c.

• cdm. A co-domain edge (c_F, c_c) represents that a function set cell c_F has the co-domain represented with a cell c_c. These edges encode many-to-one 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 co-domain. We would prefer to call label the relation edge with rel, not cdm. As these edges are many-to-one, 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 has-edges. 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 constant in_${c_e.id}_${c_S.id} in SMT. Hence, every time we introduce a copy c_T of a set c_S, we introduce a new edge (c_T, c_e) in the arena, and thus we have to introduce another Boolean constant in_${c_e.id}_${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 has-edges. 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 has-edge directly refers to its parent in the edge name (that is, in_${c_e.id}_${c_S.id}), we cannot share edges when encoding SUBSET 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 and c_r to represent the sets S and T. These cells are backed with two SMT constants of an uninterpreted sort, which corresponds to the common type of S and T.

• Five SMT constants of the Boolean sort that express set membership of a, b, c and d, e in S and T, respectively. The sets S and T are backed with SMT constants of the sort of S and T.

• One cell c_u to represent the set S \union T.

• Five Boolean constants of the Boolean sort that express set membership of a, b, c, d, e in S \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):

Having an abstract edge, we introduce various case classes. The simplest case is the FixedElemPtr, which always evaluates to a fixed Boolean value:

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:

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:

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 in S and the value of P for every x.

### 3.2. Optimization 1: constant propagation via membership pointers

One immediate application of using the new representation is completely SMT-free 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 arena-based 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.