# 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.

## Assignments and symbolic transitions

Let us go back to the example test/tla/y2k.tla and run apalache against test/tla/y2k_override.tla:

$apalache check y2k_override.tla  We can check the detailed output of the TransitionFinderPass in the file x/<timestamp>/out-transition.tla, where <timestamp> looks like 09.03-10.03.2020-508266549191958257: ----- MODULE y2k_override ----- VARIABLE year VARIABLE hasLicense ASSUME(80 \in 0 .. 99) ASSUME(18 \in 1 .. 99) Init$0 == year' := 80 /\ hasLicense' := FALSE
Next$0 == year' := ((year + 1) % 100) /\ (hasLicense' := hasLicense) Next$1 == 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$0. We call these expressions assignments. 2. It has factored the operator Next into two operators Next$0 and Next$1. 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 a \* this specification fails, as it has no expression \* that can be treated as an assignment Init == TRUE Next == a' = a Inv == FALSE ===============  Running the checker with apalache check Assignments20200309.tla  Apalache reports an error as follows: ... PASS #6: TransitionFinderPass I@09:39:33.527 To understand the error, check the manual: [https://apalache.informal.systems/docs/apalache/assignments.html] Assignment error: Failed to find assignments and symbolic transitions in InitPrimed E@09:39:33.676 It took me 0 days 0 hours 0 min 1 sec I@09:39:33.678 Total time: 1.88 sec I@09:39:33.678 EXITCODE: ERROR (99)  This error is cryptic. It does not indicate which parts of the specification have caused the problem. In the future, we will add better diagnostic in the assignment finder, see the open issue. Our current approach is to debug assignments by running TLC first. If running TLC takes too long, you may try to comment out parts of the specification to find the problematic action. Although this is tedious, it allows one to find missing assignments rather quickly. If you are interested in the technique for finding the assignments and symbolic transitions implemented in Apalache, check our paper at ABZ'18. The journal version is unfortunately behind the Elsevier paywall, which will be lifted after the two-year embargo period. ## Type annotations NOTE 1: Jure Kukovec is developing a completely automatic type inference engine. As soon as it is ready, type annotations will no longer be required. Until that happy day, refer to type annotations. NOTE 2: We are currently working on a better syntax for type annotations and a better type checker. Hence, the syntax will change in the future. Apalache requires two kinds of type annotations: • type annotations for empty sets and sequences, and • type annotations for records and sets of records. ### Empty sets and sequences Consider the following example NeedForTypes.tla: ------------------------ MODULE NeedForTypes ------------------------------ (** * This simple example transforms a set into a sequence. *) EXTENDS Integers, Sequences, FiniteSets CONSTANTS InSet \* an input set VARIABLES Left, \* a storage for the yet untransformed elements OutSeq \* the output sequence ConstInit == InSet = 1..4 Init == /\ OutSeq = << >> /\ Left = InSet Next == IF Left = {} THEN UNCHANGED <<Left, OutSeq>> ELSE \E x \in Left: /\ OutSeq' = Append(OutSeq, x) /\ Left' = Left \ {x} Inv == InSet = Left \union { OutSeq[i]: i \in DOMAIN OutSeq } ===========================================================================  While this example is perfectly fine for TLC, Apalache has to assign types to the variables, in order to construct SMT constraints. In some cases, Apalache can infer types completely automatically, e.g., as in the y2k example (see the example). However, if you run apalache check --cinit=ConstInit NeedForTypes.tla, the tool will complain: Step 0, level 0: checking if 1 transition(s) are enabled and violate the invariant I@15:17:14.313 Step 0, level 1: collecting 1 enabled transition(s) I@15:17:14.360 Step 1, level 1: checking if 2 transition(s) are enabled and violate the invariant I@15:17:14.374 NeedForTypes.tla:18:8-18:16, =(...), type error: Expected equal types: FinSet[Int] and FinSet[Unknown] E@15:17:14.379 The outcome is: Error I@15:17:14.388  In a somewhat obfuscated way, Apalache tells us the following. It has inferred that Left is a set of integers, that is, FinSet[Int]. First, it found that InSet is a set of integers, by applying ConstInit. Second, as Left = InSet in Init, it inferred that Left is also a set of integers. Third, when applying Next, it processed {}, which is an empty set of any kind of objects. Hence, {} was assigned the type FinSet[Unknown], that is, a set of some type. Finally, it found the expression Left = {}, and here the type checker has failed. To help the type checker, we have to introduce a few type annotations. But before doing that, we introduce the notation for type annotations in the specification. #### Syntax for type annotations Apalache reads any expression formed with the <: operator as an annotation of the value of the left hand side with the type on the right. E.g., v <: T  means "value v has type T". However, other tools (such as TLC and TLAPS) have no support for these annotations. To tell them to ignore type annotations, we maintain the convention that any file using Apalache type annotations begins with the following definition: v <: T == v  With this in place, Apalache can parse out the type annotations in the rest of the file, but other tools are told to simply read any occurrence of v <: T as v, effectively erasing the type ascription. Now we can help the type checker by rewriting the condition in Next as follows: #### Example of using type annotations Next == IF Left = {} <: {Int} THEN ... ELSE ...  Now the type checker treats the expression {} as a set of integers. However, it complains about another line: Step 0, level 0: checking if 1 transition(s) are enabled and violate the invariant I@15:43:35.932 Step 0, level 1: collecting 1 enabled transition(s) I@15:43:35.977 Step 1, level 1: checking if 2 transition(s) are enabled and violate the invariant I@15:43:35.992 NeedForTypes.tla:23:24-23:40, x$1, type error: Expected type Unknown, found Int E@15:43:36.012
NeedForTypes.tla:23:24-23:40, Append(...), type error: Expected a type, found: None E@15:43:36.018
NeedForTypes.tla:23:11-24:31, /\(...), type error: Expected a Boolean, found: None E@15:43:36.020
The outcome is: Error


Here the type checker stumbles upon the sequence operator Append(OutSeq, x) and complains about the type mismatch. Similar to {}, it has treated the expression << >> as a sequence of an unknown type. (In case of <<1, 2>> it would be even worse, as the type checker would not know, whether <<1, 2>> should be treated as a sequence or a tuple). Again, we help the type checker by modifying Init as follows:

Init ==
/\ OutSeq = << >> <: Seq(Int)
...


Having these two annotations, the type checker stops complaining. You can find the annotated specification in test/tla/NeedForTypesWithTypes.tla.

### Records and sets of records

Consider the following example in test/tla/Handshake.tla:

------------------------ MODULE Handshake ------------------------
(**
* A TCP-like handshake protocol:
* https://en.wikipedia.org/wiki/Transmission_Control_Protocol#Connection_establishment
*
* Igor Konnov, 2020
*)
EXTENDS Integers

VARIABLES msgs,     \* the set of all messages
iseqno,   \* Initiator's sequence number
istate,   \* Initiator's state

a <: b == a

Init ==
/\ msgs = {}
/\ iseqno = 0
/\ rseqno = 0
/\ istate = "INIT"
/\ rstate = "LISTEN"

SendSyn ==
/\ istate = "INIT"
/\ \E no \in Nat:
/\ msgs' = msgs \union {[syn |-> TRUE,
ack |-> FALSE, seqno |-> no]}
/\ iseqno' = no + 1
/\ istate' = "SYN-SENT"
/\ UNCHANGED <<rseqno, rstate>>

SendSynAck ==
/\ rstate = "LISTEN"
/\ \E seqno, ackno \in Nat:
/\ [syn |-> TRUE, ack |-> FALSE, seqno |-> seqno] \in msgs
/\ msgs' = msgs \union {[syn |-> TRUE, ack |-> TRUE,
seqno |-> seqno + 1,
ackno |-> ackno]}
/\ rseqno' = ackno + 1
/\ UNCHANGED <<iseqno, istate>>

SendAck ==
/\ istate = "SYN-SENT"
/\ \E ackno \in Nat:
/\ [syn |-> TRUE, ack |-> TRUE,
seqno |-> iseqno, ackno |-> ackno] \in msgs
/\ istate' = "ESTABLISHED"
/\ msgs' = msgs \union {[syn |-> FALSE, ack |-> TRUE,
seqno |-> iseqno,
ackno |-> ackno + 1]}
/\ UNCHANGED <<iseqno, rseqno, rstate>>

RcvAck ==
/\ \E seqno \in Nat:
/\ ([syn |-> FALSE, ack |-> TRUE,
seqno |-> seqno, ackno |-> rseqno]) \in msgs
/\ rstate' = "ESTABLISHED"
/\ UNCHANGED <<msgs, iseqno, rseqno, istate>>

Next == SendSyn \/ SendSynAck \/ SendAck \/ RcvAck

Inv == (rstate = "ESTABLISHED" => istate = "ESTABLISHED")
======================================================================


As we have seen before, the type checker complains about the set msgs, which is initialized as {}. So we have to specify the type of {}. But which type shall we use for the empty set?

In our example, the set msgs may contain records of three kinds:

• a SYN request that is modeled as a record [ack |-> FALSE, syn |-> TRUE, seqno |-> i] for some number i,
• a SYN-ACK reply that is modeled as a record [ack |-> TRUE, syn |-> TRUE, seqno |-> i, ackno |-> j] for some numbers i and j,
• an ACK reply that is modeled as a record [ack |-> TRUE, syn |-> FALSE, seqno |-> i, ackno |-> j] for some numbers i and j.

From the perspective of the type checker, the three records shown above have three different types. Although we would love to reject this example as an ill-typed one, mixing records of different types is a widely-accepted idiom in TLA+, for instance, see Lamport's specification of Paxos. Think of records as of C unions, rather than C structs!

To help the type checker, we first introduce a handy operator for the type that contains the fields of the three records:

MT == [syn |-> BOOLEAN, ack |-> BOOLEAN, seqno |-> Int, ackno |-> Int]


Then we add annotations as follows:

Init ==
/\ msgs = {} <: {MT}
...

SendSyn ==
...
/\ \E no \in Nat:
/\ msgs' = msgs \union {[syn |-> TRUE, ack |-> FALSE, seqno |-> no] <: MT}
...

SendSynAck ==
...
/\ \E seqno, ackno \in Nat:
/\ ([syn |-> TRUE, ack |-> FALSE, seqno |-> seqno] <: MT) \in msgs
...

SendAck ==
...
/\ \E ackno \in Nat:
...


As you can see, we have to annotate only those records that do not have all four fields of MT. As soon as we have added the annotations, the type checker stopped complaining and let the model checker to run. The annotated code can be found in test/tla/HandshakeWithTypes.tla.

Type annotations can be also applied to sets of records. For example:

[syn |-> BOOLEAN, ack |-> BOOLEAN, seqno |-> Int] <: {MT}


You can find more details on the simple type inference algorithm and the type annotations in type annotations.

### Naturals

If you look carefully at the type 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 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. We are currently working on a different kind of type annotations, which would not require Int.

## Recursive operators and functions

### Recursive operators

In the preprocessing phase, Apalache replaces every application of a user operator with its body. We call this process "operator inlining". This cannot be done for recursive operators, for two reasons:

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 unpredicted number of iterations.

However, in practice, when one fixes specification parameters (that is, CONSTANTS), it is usually easy to find a bound on the number of operator iterations. For instance, consider the following specification:

--------- MODULE Rec6 -----------------
CONSTANTS N
VARIABLES set, count

RECURSIVE Sum(_)

Sum(S) ==
IF S = {}
THEN 0
ELSE LET x == CHOOSE x \in S: TRUE IN
x + Sum(S \ {x})

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 the number of iterations that is equal to Cardinality(S) + 1. Moreover, the expression set \subseteq 1..N is an invariant, and thus every call Sum(set) requires up to N+1 iterations.

When we can find an upper bound on the number of iterations, Apalache can unroll the recursive operator up to this bound. To this end, we define two additional operators. For instance:

--------- MC_Rec6 ----------
VARIABLES set, count

INSTANCE Rec6 WITH N <- 3

UNROLL_TIMES_Sum == 4
UNROLL_DEFAULT_Sum == 0
============================


In this case, Apalache unrolls every call to Sum exactly UNROLL_TIMES_Sum times, that is, four times. On the default branch, Apalache places UNROLL_DEFAULT_Sum, that is, 0.

All recursively defined operators should follow this convention where, for every such operator Oper, the user defines both UNROLL_TIMES_Oper, which expands to a positive integer value, and UNROLL_DEFAULT_Oper, which expands to some default value Oper(args*) should take, if the computation would require more than UNROLL_TIMES_Oper recursive calls. At present, we only support literals (e.g. 4) or primitive arithmetic expressions (e.g. 2 + 2) in the body of UNROLL_TIMES_Oper.

#### Recursive functions

Apalache offers limited support for recursive functions. However, read the warning below on why you should not use recursive functions. The restrictions are as follows:

1. Apalache supports recursive functions that return an integer or a Boolean.

2. As Apalache's simple type checker is not able to find the type of a recursive function, all uses of a recursive function should come with a type annotation.

3. As in TLC, the function domain must be a finite set.

The example below shows a recursive function that computes the factorial of n.

------------------------------ MODULE Rec8 ------------------------------------
EXTENDS Integers

VARIABLES n, factSpec, factComp

\* the syntax for type annotations
a <: b == a

\* the type of the factorial function
FactT == [Int -> Int]

(*
Defining a recursive function on a finite domain. Although it is rather
unnatural to define factorial on a finite set, both Apalache and TLC
require finite domains. As is usual for function application, the result
of the application is not defined on the elements outside of the function
domain.
*)
Fact[k \in 1..20] ==
IF k <= 1
THEN 1
ELSE k * (Fact <: FactT)[k - 1]

Init ==
/\ n = 1
/\ factSpec = Fact[n]
/\ factComp = 1

Next ==
/\ n' = n + 1
/\ factSpec' = Fact[n']
/\ factComp' = n' * factComp

Inv ==
factComp = factSpec
===============================================================================


Check other examples in test/tla that start with the prefix Rec.

Why you should avoid recursive functions. Sometimes, recursive functions concisely describe the function that you need. The nice examples are the factorial function (see above) and Fibonacci numbers (see Rec3). However, when you define a recursive function over sets, the complexity gets ugly really fast.

Consider the example Rec9, which computes set cardinality. Here is a fragment of the spec:

 The set cardinality function. It needs an upper bound on the set size.
Although this function looks nice, be warned that this definition requires us
to construct the powerset SUBSET NUMS and then write down the constraints
for the function Card. This encoding is (at least) double-exponential.
*)
Card[S \in SUBSET NUMS] ==
IF S = {}
THEN 0
ELSE LET i == CHOOSE j \in S: TRUE IN
1 + Card[S \ {i}]

Init ==
/\ set = {}
/\ size = 0


Since we cannot fix the order, in which the set elements are evaluated, we define function Card over SUBSET NUMS, that is, all possible subsets of NUMS. Apalache translates the function in a quantifier-free theory of SMT. Hence, in this case, Apalache expands SUBSET NUMS, so it introduces 2^|NUMS| sets! Further, Apalache writes down the SMT constraints for the domain of Card. As a result, it produces NUMS * 2^|NUMS| constraints. As you can see, recursive functions over sets explode quite fast.

It is usually a good idea to use recursive operators over sets rather than recursive functions. The downside is that you have to provide an upper bound on the number of the operator iterations. The upside is that recursive operators are usually unrolled more efficiently. (If they contain only constant expressions, they are even computed by the translator!) For instance, set cardinality does not require 2^|NUMS| constraints, when using a recursive operator.