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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!


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: tuple

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: set-prod

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.