On this page:
31.1 Type Inference as Type Annotation Insertion
31.2 Understanding Inference
31.2.1 Constraint Generation
31.2.2 Constraint Solving Using Unification
31.3 Type Checking and Type Errors
31.4 Over- and Under-Constrained Solutions
31.5 Let-Polymorphism

31 Type Inference

    31.1 Type Inference as Type Annotation Insertion

    31.2 Understanding Inference

      31.2.1 Constraint Generation

      31.2.2 Constraint Solving Using Unification

    31.3 Type Checking and Type Errors

    31.4 Over- and Under-Constrained Solutions

    31.5 Let-Polymorphism

Until now, we have been studying programming languages that require a user to explicitly annotate their programs with types. In languages like Haskell and variants of ML, however, a user can leave out annotations and the language has the ability to automatically infer these annotations for them. For instance, a programmer can write the equivalent of

fun f(x, y): if x: y + 1 else: y - 1 end end

and the system will automatically infer that the header of f ought to have been

fun f(x :: Boolean, y :: Number): ...

Newer languages like Scala and Typed Racket have this in more limited measure: a feature called local type inference. Here, however, we will study the more traditional and powerful form.

31.1 Type Inference as Type Annotation Insertion

First, let’s understand what type inference is doing. Some people mistakenly think of languages with inference as having no type declarations, with inference taking their place. This is confused at multiple levels. For one thing, even in languages with inference, programmers are free (and for documentation purposes, are often encouraged) to annotate types. Furthermore, in the absence of such declarations, it is not quite clear what inference actually means.Sometimes, inference is also undecidable and programmers have no choice but to declare some of the types. Finally, writing explicit annotations can greatly reduce indecipherable error messages.

Instead, it is better to think of the underlying language as being fully, explicitly typed, like the languages we have studied [Reasoning about Programs: A First Look at Types]. It is as if the programmer had witten

fun f(x :: ___, y :: ___): ...

and some programming environment tool had filled in concrete annotations in place of the ___’s: an especially sophisticated kind of desugaring, as it were. This last remark helps us put inference in perspective: there are really two languages, one with optional type annotations and the other with required ones. Once these annotations are filled in, we are left with a traditional program that can be checked using the methods we have already studied, though in practice this is not necessary [Type Checking and Type Errors]. Thus, inference becomes simply a user convenience for alleviating the burden of writing type annotations, but the language underneath is explicitly typed.

31.2 Understanding Inference

For worked examples and more details, see Chapter 30 of Programming Languages: Application and Interpretation.

Suppose we have an expression (or program) e written in an explicitly typed language: i.e., e has type annotations everywhere they are required. Now suppose we erase all annotations in e, and use a procedure infer to deduce them back.

Do Now!

What property do we expect of infer?

We could demand many things of it. One might be that it produces precisely those annotations that e originally had. This is problematic for many reasons, not least that e might not even type-check, in which case how could infer possibly know what they were (and hence should be)? This might strike you as a pedantic trifle: after all, if e didn’t type-check, how can erasing its annotations and filling them back in make it do so? Since neither program type-checks, who cares?

Do Now!

Is this reasoning correct?

Suppose e is

lam(x :: Number) -> String: x end

This procedure obviously fails to type-check. But if we erase the type annotations—obtaining

lam(x): x end

we equally obviously obtain a typeable function! Therefore, a more reasonable demand might be that if the original e type-checks, then so must the version with annotations replaced by inferred types. This one-directional implication is useful in two ways:
  1. It does not say what must happen if e fails to type-check, i.e., it does not preclude a type inference algorithm that makes the faultily-typed identity function above typeable.

  2. More importantly, it assures us that we lose nothing by employing type inference: no program that was previously typeable will now cease to be so. That means we can focus on using explicit annotations where we want to, but will not be forced to do so.Of course, this only holds if inference is decidable.

We might also expect that both versions type to the same type, but that is not a given: the function

lam(x :: Number) -> Number: x end

types to Number -> Number, whereas applying inference to it after erasing types yields a much more general type, as we will see. Therefore, relating these types and giving a precise definition of type equality is not trivial [Relational Parametricity].

With these preliminaries out of the way, we are now ready to delve into the mechanics of type inference. The most important thing to note is that our simple, recursive-descent type-checking algorithm [A Type Checker for Expressions and Functions] will no longer work. That was possible because we already had annotations on all function boundaries, so we could descend into function bodies carrying information about those annotations in the type environment. Sans these annotations, it is not clear how to descend. In fact, it is not clear that there is any particular direction that makes more sense than another.

All this information is in the function. But how do we extract it systematically and in an algorithm that terminates and enjoys the property we have stated above? We do this in two steps. First we generate constraints, based on program terms, on what the types must be. Then we solve constraints to identify inconsistencies and join together constraints spread across the function body. Each step is relatively simple, but the combination creates magic.

31.2.1 Constraint Generation

Our goal, ultimately, is to find a type to fill into every type annotation position. It will prove to be just as well to find a type for every expression. A moment’s thought will show that this is likely necessary anyway: for instance, how can we determine the type to put on a function without knowing the type of its body? It is also sufficient, in that if every expression has had its type calculated, this will include the ones that need annotations.

First, we must generate constraints to (later) solve. Constraint generation walks the program source, emitting appropriate constraints on each expression, and returns this set of constraints. It works by recursive descent mainly for simplicity; it really computes a set of constraints, so the order of traversal and generation really does not matter in principle—so we may as well pick recursive descent, which is easy—though for simplicity we will use a list to represent this set.

What are constraints? They are simply statements about the types of expressions. In addition, though the binding instances of variables are not expressions, we must calculate their types too (because a function requires both argument and return types). In general, what can we say about the type of an expression?
  1. That it is related to the type of some identifier.

  2. That it is related to the type of some other expression.

  3. That it is a base type, such as numbers and Booleans.

  4. That it is a constructed type such as a function, whose domain and range types are presumably further constrained.

Thus, we define the following two datatypes:The name TyCHS is short for “type (Ty) constraint (C) left-or-right hand (H) side (S)”.

data TyCon: tyeq(l :: TyCHS, r :: TyCHS) end data TyCHS: | t-expr(e :: TyExprC) | t-con(name :: String, fields :: List<TyCHS>) end

Note that we have collapsed both base and constructed types into one representation, t-con: a base type will have an empty list of fields, while a constructed type will have a non-empty one corresponding to its arity. Concretely:

numeric-t-con = t-con("num", empty) boolean-t-con = t-con("bool", empty) fun mk-fun-t-con(a, r): t-con("fun", [list: a, r]) end

Now we can define the process of generating constraints:
<tyinf-generate> ::=

  fun generate(e :: TyExprC) -> List<TyCon>:

    cases (TyExprC) e:










When the expression is a number, all we can say is that we expect the type of the expression to be numeric:
<tyinf-generate-numC> ::=

  | numC(_) =>

    [list: tyeq(t-expr(e), numeric-t-con)]

This might sound trivial, but what we don’t know is what other expectations are being made of this expression by those containing it. Thus, there is the possibility that some outer expression will contradict the assertion that this expression’s type must be numeric, leading to a type error.

Identifiers do not constrain the program in any new way. The identifier will (if bound) have its type constrained at the point of binding. Therefore, there are no constraints:We are assuming that all bound identifiers in the program have distinct names, so there is no danger of confusion between two different identifiers.
<tyinf-generate-idC> ::=

  | idC(s) =>


If the context limits its type, then this expression’s type will automatically be limited, and must then be consistent with what its context expects of it.

Addition gives us our first look at a contextual constraint. For an addition expression, we must first make sure we generate (and return) constraints in the two sub-expressions, which might be complex. That done, what do we expect? That each of the sub-expressions be of numeric type. (If the form of one of the sub-expressions demands a type that is not numeric, this will lead to a type error.) Finally, we assert that the entire expression’s type is itself numeric. Because the treatment of multiplication is identical, we abstract over both:
<tyinf-generate-plusC/multC> ::=

  | plusC(l, r) => generate-arith-binop(e, l, r)

  | multC(l, r) => generate-arith-binop(e, l, r)

where the interesting work is done by the abstraction:

fun generate-arith-binop(e :: TyExprC, l :: TyExprC, r :: TyExprC) -> List<TyCon>: [list: tyeq(t-expr(e), numeric-t-con), tyeq(t-expr(l), numeric-t-con), tyeq(t-expr(r), numeric-t-con)] + generate(l) + generate(r) end

Like numbers, Boolean values constrain the current expression to be of Boolean type:
<tyinf-generate-trueC/falseC> ::=

  | trueC =>

    [list: tyeq(t-expr(e), boolean-t-con)]

  | falseC =>

    [list: tyeq(t-expr(e), boolean-t-con)]

The case for the conditional is again interesting. We must make sure the conditional expression is of Boolean type, and that the two brances have the same type:
<tyinf-generate-ifC> ::=

  | ifC(cnd, thn, els) =>

    [list: tyeq(t-expr(cnd), boolean-t-con),

      tyeq(t-expr(thn), t-expr(els)),

      tyeq(t-expr(thn), t-expr(e))] +

    generate(cnd) + generate(thn) + generate(els)


What happens if you leave out the

tyeq(t-expr(thn), t-expr(e))



What if we instead wrote

tyeq(t-expr(els), t-expr(e))

? Would it make a difference?

Now we get to the other two interesting cases, function declaration and application. In both cases, we must remember to generate and return constraints of the sub-expressions.

In a function definition, the type of the function is a function type, whose argument type is that of the formal parameter, and whose return type is that of the body:
<tyinf-generate-fdC> ::=

  | fdC(a, b) =>



        mk-fun-t-con(t-expr(idC(a)), t-expr(b)))] +


Do Now!

Do you see why we have idC(a) instead of just a?

This is necessary to make the types work out: a on its own is just bound to a string, which is not a TyExprC. Of course, the binding positions of functions are not truly identifiers, so we’re playing fast-and-loose here. In this particular context, we can get away with it, and it saves us having to come up with a whole new representation of programs.

Finally, we have applications. We cannot directly state a constraint on the type of the application. Rather, we can say that the function in the application position must consume arguments of the actual parameter expression’s type, and return types of the application expression’s type:
<tyinf-generate-appC> ::=

  | appC(f, a) =>



        mk-fun-t-con(t-expr(a), t-expr(e)))] +

    generate(f) +


And that’s it! We have finished generating constraints; now we just have to solve them.

31.2.2 Constraint Solving Using Unification

The process used to solve constraints is known as unification. A unifier is given a set of equations. Each equation maps a variable to a term, whose datatype is above.

For our purposes, the goal of unification is to generate a substitution, or mapping from variables to terms that do not contain any variables. This should sound familiar: we have a set of simultaneous equations in which each variable is used linearly; such equations are solved using Gaussian elimination. In that context, we know that we can end up with systems that are both under- and over-constrained. The same thing can happen here, as we will soon see.

The unification algorithm works iteratively over the set of constraints. Because each constraint equation has two terms and each term can be one of two kinds, there are four cases to cover.

The algorithm begins with the set of all constraints, and the empty substitution. Each constraint is considered once and removed from the set, so in principle the termination argument should be utterly simple, but it will prove to be slightly more tricky. As constraints are disposed, the substitution set tends to grow. When all constraints have been disposed, unification returns the final substitution set.

For a given constraint, the unifier examines the left-hand-side of the equation. If it is a variable, it is now ripe for elimination. The unifier adds the variable’s right-hand-side to the substitution and, to truly eliminate it, replaces all occurrences of the variable in the substitution with the this right-hand-side.It is worth noting that because the constraints are equalities, eliminating a variable is tantamount to associating it with the same set as whatever replaces it. In other words, we can use union-find [Disjoint Sets Redux] to implement this process efficiently, though if we need to backtrack during unification (as we do for logic programming [REF]), this becomes much more tricky.

Do Now!

Did you notice the subtle error above?

The subtle error is this. We said that the unifier eliminates the variable by replacing all instances of it in the substitution. However, that assumes that the right-hand-side does not contain any instances of the same variable. Otherwise we have a circular definition, and it becomes impossible to perform this particular substitution. For this reason, unifiers include a occurs check: a check for whether the same variable occurs on both sides and, if it does, decline to unify. For simplicity we will ignore this here.

Do Now!

Construct a term whose constraints would trigger the occurs check.

Do you remember ω [Recursion and Non-Termination]?

Let us now implement unification. For simplicity, we will use a list of type constraints as the representation of the subtitution.As you read this, keep in mind that unification is a generic procedure, completely independent of type-inference: indeed, the unification algorithm was invented before and spurred the creation of the type-inference process.


If we use type constraints to represent the substitution, what invariant would we expect the computed set of constraints to have?

It will be convenient to have a helper function that takes the current substitution as an accumulated parameter. Let’s therefore include it, and get the easy case out of the way:

<tyinf-unify> ::=

  fun unify(cs :: List<TyCon>) -> List<TyCon>:

    fun help(shadow cs :: List<TyCon>, sub :: List<TyCon>) -> List<TyCon>:

      cases (List) cs:

        | empty => sub

        | link(f, r) =>




    help(cs, empty)


There are four cases we need to consider, because either side can be a t-expr or t-con:
  • If both sides are t-expr’s, then we simply replace one with the other (this is the “variable elimination” case of the Gaussian procedure). We must perform this replacement everywhere: in the remaining terms but also in the substitution already performed.


    What happens if we miss doing this replacement in one or the other?

  • If one side is a t-expr and the other a t-con, then we have resolved that expression’s type to a concrete type. Record this and substitute.

  • There are two cases of a t-expr and t-con: for simplicity, we handle one case and in the other case, rewrite the problem to the former case and recur.This swapping of sides is legal because these are equational constraints.

  • If we have to unify two constructors, then they had better be the same constructor! If they are not, we have a type error. If they are, then we recur on their parameters.

Here it is in code:

<tyinf-unify-link> ::=

  lhs = f.l

  rhs = f.r


    | is-t-expr(lhs) and is-t-expr(rhs) then:

      help(subst(lhs, rhs, r), link(f, subst(lhs, rhs, sub)))

    | is-t-expr(lhs) and is-t-con(rhs) then:

      help(subst(lhs, rhs, r), link(f, subst(lhs, rhs, sub)))

    | is-t-con(lhs) and is-t-expr(rhs) then:

      help(link(tyeq(rhs, lhs), r), sub)

    | is-t-con(lhs) and is-t-con(rhs) then:

      if lhs.name == rhs.name:

        help(map2(tyeq, lhs.fields, rhs.fields) + r, sub)


        raise('type error: ' + lhs.name + ' vs. ' + rhs.name)



In terms of proving termination, note that the last two cases do not shrink the input: the third keeps it the same, while the fourth in some cases grows it.

The unifier depends on:

fun subst(to-rep :: TyCHS%(is-t-expr), rep-with :: TyCHS, in :: List<TyCon>) -> List<TyCon>: cases (List) in: | empty => empty | link(f, r) => lhs = f.l rhs = f.r link( tyeq( if lhs == to-rep: rep-with else: lhs end, if rhs == to-rep: rep-with else: rhs end), subst(to-rep, rep-with, r)) end end


There is a subtle bug in the above implementation of unification. It assumes that two textually identical expressions must have the same type. Construct a counter-example to show that this is not true. Then fix the implementation (consider using reference rather than structural equality [Re-Examining Equality]).


The algorithm above is rather naive. Given a choice, we would rather see the types of identifiers rather than those of expressions. Modify the algorithm to bias in this direction.


The output of the above algorithm is unsatisfying: a set of (solved) constraints rather than an “answer”. Extract the type of the top-level expression, and “pretty-print” it in terms of only type constants, referring to expressions only when necessary (Over- and Under-Constrained Solutions).


Prove the termination of this algorithm. Make an argument based on the size of the constraint set and on the size of the substitution.


Augment this implementation with the occurs check.


Use union-find to optimize this implementation. Measure the performance gain.

With this, we are done. Unification produces a substitution. We can now traverse the substitution and find the types of all the expressions in the program, then insert the type annotations accordingly.

31.3 Type Checking and Type Errors

A theorem, which we will not prove here, dictates that the success of the above process implies that the program would have typed-checked, so we need not explicitly run the type-checker over this program.

Observe, however, that the nature of a type error has now changed dramatically. Previously, we had a recursive-descent algorithm that walked a expressions using a type environment. The bindings in the type environment were programmer-declared types, and could hence be taken as (intended) authoritative specifications of types. As a result, any mismatch was blamed on the expressions, and reporting type errors was simple (and easy to understand). Here, however, a type error is a failure to notify. The unification failure is based on events that occur at the confluence of two smart algorithms—constraint generation and unification—and hence are not necessarily comprehensible to the programmer. In particular, the equational nature of these constraints means that the location reported for the error, and the location of the “true” error, could be quite far apart. As a result, producing better error messages remains an active research area.In practice the algorithm will maintain metadata on which program source terms were involved and probably on the history of unification, to be able to trace errors back to the source program.

31.4 Over- and Under-Constrained Solutions

Remember that the constraints may not precisely dictate the type of all variables. If the system of equations is over-constrained, then we get clashes, resulting in type errors. If instead the system is under-constrained, that means we don’t have enough information to make definitive statements about all expressions. For instance, in the expression (fun (x) x) we do not have enough constraints to indicate what the type of x, and hence of the entire expression, must be. This is not an error; it simply means that x is free to be any type at all. In other words, its type is “the type of x -> the type of x” with no other constraints. The types of these underconstrained identifiers are presented as type variables, so the above expression’s type might be reported as (A -> A).

The unification algorithm actually has a wonderful property: it automatically computes the most general types for an expression, also known as principal types. That is, any actual type the expression can have can be obtained by instantiating the inferred type variables with actual types. This is a remarkable result: in another example of computers beating humans, it says that no human can generate a more general type than the above algorithm can!

31.5 Let-Polymorphism

Unfortunately, though these type variables are superficially similar to the polymorphism we had earlier [Parametric Polymorphism], they are not. Consider the following program:

(let ([id (fun (x) x)])

  (if (id true)

      (id 5)

      (id 6)))

If we write it with explicit type annotations, it type-checks:

(if (id<Boolean> true)

    (id<Number> 5)

    (id<Number> 6))

However, if we use type inference, it does not! That is because the A’s in the type of id unify either with Boolean or with Number, depending on the order in which the constraints are processed. At that point id effectively becomes either a (Boolean -> Boolean) or (Number -> Number) function. At the use of id of the other type, then, we get a type error!

The reason for this is because the types we have inferred through unification are not actually polymorphic. This is important to remember: just because you type variables, you don’t necessarily have polymorphism! The type variables could be unified at the next use, at which point you end up with a mere monomorphic function. Rather, true polymorphism only obtains when you can instantiate type variables.

In languages with true polymorphism, then, constraint generation and unification are not enough. Instead, languages like ML and Haskell implement something colloquially called let-polymorphism. In this strategy, when a term with type variables is bound in a lexical context, the type is automatically promoted to be a quantified one. At each use, the term is effectively automatically instantiated.

There are many implementation strategies that will accomplish this. The most naive (and unsatisfying) is to merely copy the code of the bound identifier; thus, each use of id above gets its own copy of (fun (x) x), so each gets its own type variables. The first might get the type (A -> A), the second (B -> B), the third (C -> C), and so on. None of these type variables clash, so we get the effect of polymorphism. Obviously, this not only increases program size, it also does not work in the presence of recursion. However, it gives us insight into a better solution: instead of copying the code, why not just copy the type? Thus at each use, we create a renamed copy of the inferred type: id’s (A -> A) becomes (B -> B) at the first use, and so on, thus achieving the same effect as copying code but without its burdens. Because all these strategies effectively mimic copying code, however, they only work within a lexical context.