Type Constraints

Let's build up to the HM type inference algorithm by starting with this little language:

e ::= x | i | b | e1 bop e2                
    | if e1 then e2 else e3
    | fun x -> e
    | e1 e2

bop ::= + | * | <=

t ::= int | bool | t1 -> t2

That language is SimPL combined with the lambda calculus, but without let expressions. It turns out let expressions add a extra layer of complication, so we'll come back to them later.

Since anonymous functions in this language do not have type annotations, we have to infer the type of the argument x. For example,

  • In fun x -> x + 1, argument x must have type int hence the function has type int -> int.

  • In fun x -> if x then 1 else 0, argument x must have type bool hence the function has type bool -> int.

  • Function fun x -> if x then x else 0 is untypeable, because it would require x to have both type int and bool, which isn't allowed.

A syntactic simplification

We can treat e1 bop e2 as syntactic sugar for ( bop ) e1 e2. That is, we treat infix binary operators as prefix function application. Let's introduce a new syntactic class n for names, which generalize identifiers and operators. That changes the syntax to:

e ::= n | i | b
    | if e1 then e2 else e3
    | fun x -> e
    | e1 e2

n ::= x | bop

bop ::= ( + ) | ( * ) | ( <= )

t ::= int | bool | t1 -> t2

We already know the types of those built-in operators:

( + ) : int -> int -> int
( * ) : int -> int -> int
( <= ) : int -> int -> bool

Those types are given; we don't have to infer them. They are part of the initial static environment. In OCaml those operator names could later be shadowed by values with different types, but here we don't have to worry about that because we don't yet have let.

Constraint-based inference

How would you mentally infer the type of fun x -> 1 + x, or rather, fun x -> ( + ) 1 x? It's automatic by now, but we could break it down into pieces:

  • Start with x having some unknown type t.
  • Note that ( + ) is known to have type int -> (int -> int).
  • So its first argument must have type int. Which 1 does.
  • And its second argument must have type int, too. So t = int. That is a constraint on t.
  • Finally the body of the function must also have type int, since that's the return type of ( + ).
  • Therefore the type of the entire function must be t -> int.
  • Since t = int, that type is int -> int.

The type inference algorithm follows the same idea of generating unknown types, collecting constraints on them, and using the constraints to solve for the type of the expression.

Inference relation

Let's introduce a new 4-ary relation env |- e : t -| C, which should be read as follows: "in environment env, expression e is inferred to have type t and generates constraint set C." A constraint is an equation of the form t1 = t2 for any types t1 and t2.

If we think of the relation as a type-inference function, the colon in the middle separates the input from the output. The inputs are env and e: we want to know what the type of e is in environment env. The function returns as output a type t and constraints C.

The e : t in the middle of the relation is approximately what you see in the toplevel: you enter an expression, and it tells you the type. But around that is an environment and constraint set env |- ... -| C that is invisible to you. So, the turnstiles around the outside show the parts of type inference that the toplevel does not.

Inference of constants and names

The easiest parts of inference are constants:

env |- i : int -| {}

env |- b : bool -| {}

Any integer constant i, such as 42, is known to have type int, and there are no contraints generated. Likewise for Boolean constants.

Inferring the type of a name requires looking it up in the environment:

env |- n : env(n) -| {}

No constraints are generated.

If the name is not bound in the environment, the expression cannot be typed. It's an unbound name error.

Inference of if expressions

The remaining rules are at their core the same as the type-checking rules we saw previously, but they each generate a type variable and possibly some constraints on that type variable.

Here's the if rule. We'll explain it below.

env |- if e1 then e2 else e3 : 't -| C1, C2, C3, t1 = bool, 't = t2, 't = t3
  if fresh 't
  and env |- e1 : t1 -| C1
  and env |- e2 : t2 -| C2
  and env |- e3 : t3 -| C3

To infer the type of an if, we infer the types t1, t2, and t3 of each of its subexpressions, along with any constraints on them. We have no control over what those types might be; it depends on what the programmer wrote. But we do know that the type of the guard must be bool. So we generate a constraint that t1 = bool.

Furthermore, we know that both branches must have the same type—though, we don't know in advance what that type might be. So, we invent a fresh type variable 't to stand for that type. A type variable is fresh if it has never been used elsewhere during type inference. So, picking a fresh type variable just means picking a new name that can't possibly be confused with any other names in the program. We return 't as the type of the if, and we record two constraints 't = t2 and 't = t3 to say that both branches must have that type.

We therefore need to add type variables to the syntax of types:

t ::= 'x | int | bool | t1 -> t2

Some example type variables include 'a, 'foobar, and 't. In the last, t is an identifier, not a meta-variable.


{} |- if true then 1 else 0 : 't -| bool = bool, 't = int
  {} |- true : bool -| {}
  {} |- 1 : int -| {}
  {} |- 0 : int -| {}

The full constraint set generated is {}, {}, {}, bool = bool, 't = int, 't = int, but of course that simplifies to just bool = bool, 't = int. From that constraint set we can see that the type of if true then 1 else 0 must be int.

Inference of functions and applications

Anonymous functions. Since there is no type annotation on x, its type must be inferred:

env |- fun x -> e : 't1 -> t2 -| C
  if fresh 't1
  and env, x : 't1 |- e : t2 -| C

We introduce a fresh type variable 't1 to stand for the type of x, and infer the type of body e under the environment in which x : 't1. Wherever x is used in e, that can cause constraints to be generated involving 't1. Those constraints will become part of C.

Example. Here's a function where we can immediately see that x : bool, but let's work through the inference:

{} |- fun x -> if x then 1 else 0 : 't1 -> 't -| 't1 = bool, 't = int
  {}, x : 't1 |- if x then 1 else 0 : 't -| 't1 = bool, 't = int
    {}, x : 't1 |- x : 't1 -| {}
    {}, x : 't1 |- 1 : int -| {}
    {}, x : 't1 |- 0 : int -| {}

The inferred type of the function is 't1 -> 't, with constraints 't1 = bool and 't = int. Simplifying that, the function's type is bool -> int.

Function application. The type of the entire application must be inferred, because we don't yet know anything about the types of either subexpression:

env |- e1 e2 : 't -| C1, C2, t1 = t2 -> 't
  if fresh 't
  and env |- e1 : t1 -| C1
  and env |- e2 : t2 -| C2

We introduce a fresh type variable 't for the type of the application expression. We use inference to determine the types of the subexpressions and any constraints they happen to generate. We add one new constraint, t1 = t2 -> 't, which expresses that the type of the left-hand side e1 must be a function that takes in an argument of type t2 and returns a value of type 't.

Example. Let I be the initial environment that binds the boolean operators. Let's infer the type of a partial application of ( + ):

I |- ( + ) 1 : 't -| int -> int -> int = int -> 't
  I |- ( + ) : int -> int -> int -| {}
  I |- 1 : int -| {}

From the resulting constraint, we see that

int -> int -> int
int -> 't

Stripping the int -> off the left-hand side of each of those function types, we are left with

int -> int

Hence the type of ( + ) 1 is int -> int.

Solving constraint sets

We've now given an algorithm for generating types and constraint sets. But we've been waving our hands about how to solve the constraint set. The small examples we've seen so far have been easy to solve in our heads. For a large program, that won't be true. So let's turn our attention to that problem, next.

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