ADR-002: types and type annotations

Shon Feder, Igor Konnov, Jure Kukovec4

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.


  • 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. Comments inside types

When you introduce records that have dozens of fields, it is useful to explain those fields right in the type annotations. For that reason, the type lexer supports one-line comments right in the type definitions. The following text presents a type definition that contains comments:

// packets are stored in a set
  // unique sequence number
  seqno: Int,
  // payload hash
  payloadHash: Str

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.

1.4. 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}. However, 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.

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. Feel free to suggest Type System 2 :-)

Note: For the above example of a set of records, we are considering to introduce union types. So the type of the set

{[type |-> "1a", bal |-> 1],
 [type |-> "2a", bal |-> 2, val |-> 3]}

would be something like:

Set([type |-> "1a"], bal |-> 1]
  + [type |-> "2a", bal |-> 2, val |-> 3])

The value of the field type would serve as a type tag. However, we would have to fix a set of patterns that turn a union type into a precise record type. One such pattern is a set comprehension, e.g., { r \in S: r.type = "1a" }. If you have suggestions on this line of thought, please let us know.

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:

  \* @type: Int;
  \* @type: Set(ID);

  \* @type: ID;
  \* @type: Set(ID);

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) recursive 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: Set(a) -> Int;
Card[S \in T] ==
    IF S = {}
    THEN 0
    ELSE LET \* @type: a;
             \* we could also write: "() => a" instead of just "a"
             one_elem == (CHOOSE x \in S: TRUE)
         1 + Card[S \ {one_elem}]

In the definition of Card, we annotated the let-definition one_elem with its type, though any type checker should be able to compute the type of one_elem from its context. So the type of one_elem is there for clarification. According to our type grammar, the type of one_elem should be () => a, as one_elem 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 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 == {}
   \E x \in EmptyInts: x > 1
/\ LET \* @type: Set(Str);
       EmptyStrings == {}
   f = [x \in EmptyStrings |-> 2]
/\ LET \* @type: Seq(Int);
       EmptyIntSeq == {}
   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

    \* @type: Set(ENTRY);

\* @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.                                       *)
(*                 *)
(*                                                                         *)
(* 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:           *)
(* *)

EXTENDS Integers, FiniteSets

  \* @type: Set(INGREDIENT);
  \* @type: Set(Set(INGREDIENT));

  \* @type: INGREDIENT -> [smoking: Bool];
  \* @type: Set(INGREDIENT);

(* 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} \cup 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