Let-Polymorphic Type Inferencer (Damas-Hindley-Milner)

In this lesson we discuss a type inferencer based on what we call today the Damas-Hindley-Milner type system, which is at the core of many modern functional programming languages. The first variant of it was proposed by Hindley in 1969, then, interestingly, Milner rediscovered it in 1978 in the context of the ML language. Damas formalized it as a type system in his PhD thesis in 1985. More specifically, our type inferencer here, like many others as well as many implementations of it, follows more closely the syntax-driven variant proposed by Clement in 1987.

In terms of K, we will see how easily we can turn one definition which is considered naive (our previous type inferencer in Lesson 8) into a definition which is considered advanced. All we have to do is to change one existing rule (the rule of the let binder) and to add a new one. We will also learn some new predefined features of K, which make the above possible.

The main idea is to replace the rule

rule let X = E in E' => E'[E/X]

which creates potentially many copies of E within E' with a rule which types E once and then reuses that type in each place where X occurs free in E'. The simplest K way to type E is to declare the let construct strict(2). Now we cannot simply bind X to the type of E, because we would obtain a variant of the naive type inferencer we already discussed, together with its limitations, in Lesson 5 of this tutorial. The trick here is to parameterize the type of E in all its unconstrained fresh types, and then create fresh copies of those parameters in each free occurrence of X in E'.

Let us discuss some examples, before we go into the technical details. Consider the first let-polymorphic example which failed to be typed with our first naive type-inferencer:

let id = lambda x . x
in if (id true) then (id 1) else (id 2)

When typing lambda x . x, we get a type of the form t -> t, for some fresh type t. Instead of assigning this type to id as we did in the naive type inferencers, we now first parametrize this type in its fresh variable t, written

(forall t) t -> t

and then bind id to this parametric type. The intuition for the parameter is that it can be instantiated with any other type, so this parametric type stands, in fact, for infinitely many non-parametric types. This is similar to what happens in formal logic proof systems, where rule schemas stand for infinitely many concrete instances of them. For this reason, parametric types are also called type schemas.

Now each time id is looked up within the let-body, we create a fresh copy of the parameter t, which can this way be independently constrained by each local context. Let's suppose that the three id lookups yield the types t1 -> t1, t2 -> t2, and respectively t3 -> t3. Then t1 will be constrained to be bool, and t2 and t3 to be int, so we can now safely type the program above to int.

Therefore, a type schema comprises a summary of all the typing work that has been done for typing the corresponding expression, and an instantiation of its parameters with fresh copies represents an elegant way to reuse all that typing work.

There are some subtleties regarding what fresh types can be made parameters. Let us consider another example, discussed as part of Lesson 7 on naive let-polymorphism:

lambda x . (
  let y = lambda z . x
  in if (y true) then (y 1) else (y (lambda x . x))

This program should type to bool -> bool, as explained in Lesson 7. The lambda construct will bind x to some fresh type tx. Then the let-bound expression lambda z . x types to tz -> tx for some additional fresh type tz. The question now is what should the parameters of this type be when we generate the type schema? If we naively parameterize in all fresh variables, that is in both tz and tx obtaining the type schema (forall tz,tx) tz -> tx, then there will be no way to infer that the type of x, tx, must be a bool! The inferred type of this expression would then wrongly be tx -> t for some fresh types tx and t. That's because the parameters are replaced with fresh copies in each occurrence of y, and thus their relationship to the original x is completely lost. This tells us that we cannot parameterize in all fresh types that appear in the type of the let-bound expression. In particular, we cannot parameterize in those which some variables are already bound to in the current type environment (like x is bound to tx in our example above). In our example, the correct type schema is (forall tz) tz -> tx, which now allows us to correctly infer that tx is bool.

Let us now discuss another example, which should fail to type:

lambda x .
  let f = lambda y . x y
  in if (f true) then (f 1) else (f 2)

This should fail to type because lambda y . x y is equivalent to x, so the conditional imposes the conflicting constraints that x should be a function whose argument is either a bool or an int. Let us try to type it using our currently informal procedure. Like in the previous example, x will be bound to a fresh type tx. Then the let-bound expression types to ty -> tz with ty and tz fresh types, adding also the constraint tx = ty -> tz. What should the parameters of this type be? If we ignore the type constraint and simply make both ty and tz parameters because no variable is bound to them in the type environment (indeed, the only variable x in the type environment is bound to tx), then we can wrongly type this program to tx -> tz following a reasoning similar to the one in the example above. In fact, in this example, none of ty and tz can be parameters, because they are constrained by tx.

The examples above tell us two things: first, that we have to take the type constraints into account when deciding the parameters of the schema; second, that after applying the most-general-unifier solution given by the type constraints everywhere, the remaining fresh types appearing anywhere in the type environment are consequently constrained and cannot be turned into parameters. Since the type environment can in fact also hold type schemas, which already bind some types, we only need to ensure that none of the fresh types appearing free anywhere in the type environment are turned into parameters of type schemas.

Thanks to generic support offered by the K tool, we can easily achieve all the above as follows.

First, add syntax for type schemas:

syntax TypeSchema ::= "(" "forall" Set ")" Type  [binder]

The definition below will be given in such a way that the Set argument of a type schema will always be a set of fresh types. We also declare this construct to be a binder, so that we can make use of the generic free variable function provided by the K tool.

We now replace the old rule for let

rule let X = E in E' => E'[E/X]

with the following rule:

rule <k> let X = T:Type in E => E ~> tenv(TEnv) ...</k>
     <mgu> Theta:Mgu </mgu>
     <tenv> TEnv
      => TEnv[(forall freeVariables(applyMgu(Theta, T)) -Set
                      freeVariables(applyMgu(Theta, values TEnv))
              ) applyMgu(Theta, T) / X]

So the type T of E is being parameterized and then bound to X in the type environment. The current mgu Theta, which comprises all the type constraints accumulated so far, is applied to both T and the types in the type environment. The remaining fresh types in T which do not appear free in the type environment are then turned into type parameters. The function freeVariables returns, as expected, the free variables of its argument as a Set; this is why we declared the type schema to be a binder above.

Now a LAMBDA variable in the type environment can be bound to either a type or a type schema. In the first case, the previous rule we had for variable lookup can be reused, but we have to make sure we check that T there is of sort Type (adding a sort membership, for example). In the second case, as explained above, we have to create fresh copies of the parameters. This can be easily achieved with another predefined K function, as follows:

rule <k> X:Id => freshVariables(Tvs,T) ...</k>
     <tenv>... X |-> (forall Tvs) T ...</tenv>

Indeed, freshVariables takes a set of variables and a term, and returns the same term but with each of the given variables replaced by a fresh copy.

The operations freeVariables and freshVariables are useful in many K definitions, so they are predefined in module substitution.k.

Our definition of this let-polymorphic type inferencer is now complete. To test it, kompile it and then krun all the LAMBDA programs discussed since Lesson 4. They should all work as expected.