System F: the polymorphic lambda calculus

The simply-typed lambda calculus is beautifully simple, but it can't even express the predecessor function, let alone full recursion. And we'll see shortly that there is good reason to be unsatisfied with the simply-typed lambda calculus as a way of expressing natural language meaning. So we will need to get more sophisticated about types. The next step in that journey will be to consider System F.

System F was discovered by Girard (the same guy who invented Linear Logic), but it was independently proposed around the same time by Reynolds, who called his version the polymorphic lambda calculus. (Reynolds was also an early player in the development of continuations.)

System F enhances the simply-typed lambda calculus with abstraction over types. Normal lambda abstraction abstracts (binds) an expression (a term); type abstraction abstracts (binds) a type.

In order to state System F, we'll need to adopt the notational convention (which will last throughout the rest of the course) that "x:α" represents an expression x whose type is α.

Then System F can be specified as follows:

System F:
---------
types       τ ::= c | α | τ1 -> τ2 | ∀α.τ
expressions e ::= x | λx:τ.e | e1 e2 | Λα.e | e [τ]

In the definition of the types, "c" is a type constant. Type constants play the role in System F that base types play in the simply-typed lambda calculus. So in a lingusitics context, type constants might include e and t. "α" is a type variable. In various discussions, type variables are distinguished by using letters from the greek alphabet (α, β, etc.), as we do here, or by using capital roman letters (X, Y, etc.), or by adding a tick mark ('a, 'b, etc.), as in OCaml. "τ1 -> τ2" is the type of a function from expressions of type τ1 to expressions of type τ2. And "∀α.τ" is called a universal type, since it universally quantifies over the type variable &alpha;. You can expect that in ∀α.τ, the type τ will usually have at least one free occurrence of α somewhere inside of it.

In the definition of the expressions, we have variables "x" as usual. Abstracts "λx:τ.e" are similar to abstracts in the simply-typed lambda calculus, except that they have their shrug variable annotated with a type. Applications "e1 e2" are just like in the simply-typed lambda calculus.

In addition to variables, abstracts, and applications, we have two additional ways of forming expressions: "Λα.e" is called a type abstraction, and "e [τ]" is called a type application. The idea is that Λ is a capital λ: just like the lower-case λ, Λ binds variables in its body, except that unlike λ, Λ binds type variables instead of expression variables. So in the expression

Λ α (λ x:α. x)

the Λ binds the type variable α that occurs in the λ abstract.

This expression is a polymorphic version of the identity function. It defines one general identity function that can be adapted for use with expressions of any type. In order to get it ready to apply this identity function to, say, a variable of type boolean, just do this:

(Λ α (λ x:α. x)) [t]

This type application (where t is a type constant for Boolean truth values) specifies the value of the type variable α. Not surprisingly, the type of the expression that results from this type application is a function from Booleans to Booleans:

((Λα (λ x:α . x)) [t]): (b->b)

Likewise, if we had instantiated the type variable as an entity (base type e), the resulting identity function would have been a function of type e -> e:

((Λα (λ x:α. x)) [e]): (e->e)

Clearly, for any choice of a type α, the identity function can be instantiated as a function from expresions of type α to expressions of type α. In general, then, the type of the uninstantiated (polymorphic) identity function is

(Λα (λx:α . x)): (∀α. α->α)

Pred in System F

We saw that the predecessor function couldn't be expressed in the simply-typed lambda calculus. It can be expressed in System F, however. Here is one way:

let N = ∀α.(α->α)->α->α in
let Pair = (N->N->N)->N in

let zero = Λα. λs:α->α. λz:α. z in
let fst = λx:N. λy:N. x in
let snd = λx:N. λy:N. y in
let pair = λx:N. λy:N. λz:N->N->N. z x y in
let succ = λn:N. Λα. λs:α->α. λz:α. s (n [α] s z) in
let shift = λp:Pair. pair (succ (p fst)) (p fst) in
let pred = λn:N. n [Pair] shift (pair zero zero) snd in

pre (suc (suc (suc zero)));

[If you want to run this code in Benjamin Pierce's type-checker and evaluator for System F, the relevant evaluator is called "fullpoly", and you'll need to truncate the names of "suc(c)" and "pre(d)", since those are reserved words in Pierce's system.]

Exercise: convince yourself that zero has type N.

[By the way, in order to keep things as simple as possible here, the types used in this definition of the ancillary functions given here are not as general as they could be; see the discussion below of type inference and principal types in the OCaml type system.]

The key to the extra expressive power provided by System F is evident in the typing imposed by the definition of pred. The variable n is typed as a Church number, i.e., as N ≡ ∀α.(α->α)->α->α. The type application n [Pair] instantiates n in a way that allows it to manipulate ordered pairs: n [Pair]: (Pair->Pair)->Pair->Pair. In other words, the instantiation turns a Church number into a certain pair-manipulating function, which is the heart of the strategy for this version of computing the predecessor function.

Could we try to accommodate the needs of the predecessor function by building a system for doing Church arithmetic in which the type for numbers always manipulated ordered pairs? The problem is that the ordered pairs we need here are pairs of numbers. If we tried to replace the type for Church numbers with a concrete (simple) type, we would have to replace each N with the type for Pairs, (N -> N -> N) -> N. But then we'd have to replace each of these N's with the type for Church numbers, which we're imagining is (Pair -> Pair) -> Pair -> Pair. And then we'd have to replace each of these Pairs's with... ad infinitum. If we had to choose a concrete type built entirely from explicit base types, we'd be unable to proceed.

[See Benjamin C. Pierce. 2002. Types and Programming Languages, MIT Press, chapter 23.]

Typing ω

In fact, unlike in the simply-typed lambda calculus, it is even possible to give a type for ω in System F.

ω = λx:(∀α.α->α). x [∀α.α->α] x

In order to see how this works, we'll apply ω to the identity function.

ω id ≡ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)

Since the type of the identity function is ∀α.α->α, it's the right type to serve as the argument to ω. The definition of ω instantiates the identity function by binding the type variable α to the universal type ∀α.α->α. Instantiating the identity function in this way results in an identity function whose type is (in some sense, only accidentally) the same as the original fully polymorphic identity function.

So in System F, unlike in the simply-typed lambda calculus, it is possible for a function to apply to itself!

Does this mean that we can implement recursion in System F? Not at all. In fact, despite its differences with the simply-typed lambda calculus, one important property that System F shares with the simply-typed lambda calculus is that they are both strongly normalizing: every expression in either system reduces to a normal form in a finite number of steps.

Not only does a fixed-point combinator remain out of reach, we can't even construct an infinite loop. This means that although we found a type for ω, there is no general type for Ω ≡ ω ω. In fact, it turns out that no Turing complete system can be strongly normalizing, from which it follows that System F is not Turing complete.

Polymorphism in natural language

Is the simply-typed lambda calclus enough for analyzing natural language, or do we need polymorphic types? Or something even more expressive?

The classic case study motivating polymorphism in natural language comes from coordination. (The locus classicus is Partee and Rooth 1983.)

Type of the arguments of "and":
Ann left and Bill left.           t
Ann left and slept.               e->t
Ann read and reviewed the book.   e->e->t
Ann and Bill left.                (e->t)-t  (i.e, generalize quantifiers)

In English (likewise, many other languages), and can coordinate clauses, verb phrases, determiner phrases, transitive verbs, and many other phrase types. In a garden-variety simply-typed grammar, each kind of conjunct has a different semantic type, and so we would need an independent rule for each one. Yet there is a strong intuition that the contribution of and remains constant across all of these uses.

Can we capture this using polymorphic types?

Ann, Bill      e
left, slept    e -> t
read, reviewed e -> e -> t

With these basic types, we want to say something like this:

and:t->t->t = λl:t. λr:t. l r false
gen_and = Λα.Λβ.λf:(β->t).λl:α->β.λr:α->β.λx:α. f (l x) (r x)

The idea is that the basic and (the one defined in the first line) conjoins expressions of type t. But when and conjoins functional types (the definition in the second line), it builds a function that distributes its argument across the two conjuncts and then applies the appropriate lower-order instance of and.

and (Ann left) (Bill left)
gen_and [e] [t] and left slept
gen_and [e] [e->t] (gen_and [e] [t] and) read reviewed

Following the terminology of Partee and Rooth, this strategy of defining the coordination of expressions with complex types in terms of the coordination of expressions with less complex types is known as Generalized Coordination, which is why we call the polymorphic part of the definition gen_and.

In the first line, the basic and is ready to conjoin two truth values. In the second line, the polymorphic definition of gen_and makes explicit exactly how the meaning of and when it coordinates verb phrases depends on the meaning of the basic truth connective. Likewise, when and coordinates transitive verbs of type e->e->t, the generalized and depends on the e->t version constructed for dealing with coordinated verb phrases.

On the one hand, this definition accurately expresses the way in which the meaning of the conjunction of more complex types relates to the meaning of the conjunction of simpler types. On the other hand, it's awkward to have to explicitly supply an expression each time that builds up the meaning of the and that coordinates the expressions of the simpler types. We'd like to have that automatically handled by the polymorphic definition; but that would require writing code that behaved differently depending on the types of its type arguments, which goes beyond the expressive power of System F.

And in fact, discussions of generalized coordination in the linguistics literature are almost always left as a meta-level generalizations over a basic simply-typed grammar. For instance, in Hendriks' 1992:74 dissertation, generalized coordination is implemented as a method for generating a suitable set of translation rules, which are in turn expressed in a simply-typed grammar.

There is some work using System F to express generalizations about natural language: Ponvert, Elias. 2005. Polymorphism in English Logical Grammar. In Lambda Calculus Type Theory and Natural Language: 47--60.

Not incidentally, we're not aware of any programming language that makes generalized coordination available, despite is naturalness and ubiquity in natural language. That is, coordination in programming languages is always at the sentential level. You might be able to evaluate (delete file1) and (delete file2), but never delete (file1 and file2).

We'll return to thinking about generalized coordination as we get deeper into types. There will be an analysis in term of continuations that will be particularly satisfying.

Types in OCaml

OCaml has type inference: the system can often infer what the type of an expression must be, based on the type of other known expressions.

For instance, if we type

# let f x = x + 3;;

The system replies with

val f : int -> int = <fun>

Since + is only defined on integers, it has type

# (+);;
- : int -> int -> int = <fun>

The parentheses are there to turn off the trick that allows the two arguments of + to surround it in infix (for linguists, SOV) argument order. That is,

# 3 + 4 = (+) 3 4;;
- : bool = true

In general, tuples with one element are identical to their one element:

# (3) = 3;;
- : bool = true

though OCaml, like many systems, refuses to try to prove whether two functional objects may be identical:

# (f) = f;;
Exception: Invalid_argument "equal: functional value".

Oh well.

[Note: There is a limited way you can compare functions, using the == operator instead of the = operator. Later when we discuss mutation, we'll discuss the difference between these two equality operations. Scheme has a similar pair, which they name eq? and equal?. In Python, these are is and == respectively. It's unfortunate that OCaml uses == for the opposite operation that Python and many other languages use it for. In any case, OCaml will accept (f) == f even though it doesn't accept (f) = f. However, don't expect it to figure out in general when two functions are equivalent. (That question is not Turing computable.)

# (f) == (fun x -> x + 3);;
- : bool = false

Here OCaml says (correctly) that the two functions don't stand in the == relation, which basically means they're not represented in the same chunk of memory. However as the programmer can see, the functions are extensionally equivalent. The meaning of == is rather weird.]

Booleans in OCaml, and simple pattern matching

Where we would write true 1 2 in our pure lambda calculus and expect it to evaluate to 1, in OCaml boolean types are not functions (equivalently, they're functions that take zero arguments). Instead, selection is accomplished as follows:

# if true then 1 else 2;;
- : int = 1

The types of the then clause and of the else clause must be the same.

The if construction can be re-expressed by means of the following pattern-matching expression:

match <bool expression> with true -> <expression1> | false -> <expression2>

That is,

# match true with true -> 1 | false -> 2;;
- : int = 1

Compare with

# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9

Unit and thunks

All functions in OCaml take exactly one argument. Even this one:

# let f x y = x + y;;
# f 2 3;;
- : int = 5

Here's how to tell that f has been curry'd:

# f 2;;
- : int -> int = <fun>

After we've given our f one argument, it returns a function that is still waiting for another argument.

There is a special type in OCaml called unit. There is exactly one object in this type, written (). So

# ();;
- : unit = ()

Just as you can define functions that take constants for arguments

# let f 2 = 3;;
# f 2;;
- : int = 3;;

you can also define functions that take the unit as its argument, thus

# let f () = 3;;
val f : unit -> int = <fun>

Then the only argument you can possibly apply f to that is of the correct type is the unit:

# f ();;
- : int = 3

Now why would that be useful?

Let's have some fun: think of rec as our Y combinator. Then

# let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
val f : int -> int = <fun>
# f 5;;
- : int = 120

We can't define a function that is exactly analogous to our ω. We could try let rec omega x = x x;; what happens?

[Note: if you want to learn more OCaml, you might come back here someday and try:

# let id x = x;;
val id : 'a -> 'a = <fun>
# let unwrap (Wrap a) = a;;
val unwrap : [< Wrap of 'a ] -> 'a = <fun>
# let omega ((Wrap x) as y) = x y;;
val omega : [< Wrap of [> Wrap of 'a ] -> 'b as 'a ] -> 'b = <fun>
# unwrap (omega (Wrap id)) == id;;
- : bool = true
# unwrap (omega (`Wrap omega));;
<Infinite loop, need to control-c to interrupt>

But we won't try to explain this now.]

Even if we can't (easily) express omega in OCaml, we can do this:

# let rec blackhole x = blackhole x;;

By the way, what's the type of this function?

If you then apply this blackhole function to an argument,

# blackhole 3;;

the interpreter goes into an infinite loop, and you have to type control-c to break the loop.

Oh, one more thing: lambda expressions look like this:

# (fun x -> x);;
- : 'a -> 'a = <fun>
# (fun x -> x) true;;
- : bool = true

(But (fun x -> x x) still won't work.)

You may also see this:

# (function x -> x);;
- : 'a -> 'a = <fun>

This works the same as fun in simple cases like this, and slightly differently in more complex cases. If you learn more OCaml, you'll read about the difference between them.

We can try our usual tricks:

# (fun x -> true) blackhole;;
- : bool = true

OCaml declined to try to fully reduce the argument before applying the lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language?

Remember that blackhole is a function too, so we can reverse the order of the arguments:

# blackhole (fun x -> true);;

Infinite loop.

Now consider the following variations in behavior:

# let test = blackhole blackhole;;
<Infinite loop, need to control-c to interrupt>

# let test () = blackhole blackhole;;
val test : unit -> 'a = <fun>

# test;;
- : unit -> 'a = <fun>

# test ();;
<Infinite loop, need to control-c to interrupt>

We can use functions that take arguments of type unit to control execution. In Scheme parlance, functions on the unit type are called thunks (which I've always assumed was a blend of "think" and "chunk").

Question: why do thunks work? We know that blackhole () doesn't terminate, so why do expressions like:

let f = fun () -> blackhole ()
in true

terminate?

Type inference and principal types

As we mentioned, the types given to some of the functions defined above in the System F definition of pred are not as general as they might be.

let pair = λx:N. λy:N. λz:N->N->N. z x y in
...

For instance, in the definition of pair, repeated here, we assumed that the function z would return something of type N, i.e., a Church number. But we can give a more general treatment.

let general_pair = Λα. Λβ. λx:α. λy:β. Λρ. λz:α->β->ρ. z x y in
...

In this more general version, the pair function accepts any kind of objects as its first and second element. The resulting pair will expect a handler function (z) that is ready to handle arguments of the same types the pair was built from, but there is no restriction on the type (ρ) of the result returned by the handler function.

The number specific type we gave the pair function above (i.e., N->N->(N->N->N)->N) is a specific instance of the more general type, with α, β, and ρ all set to N. Many practical type systems guarantee that under reasonably general conditions, there will be a principal type: a type such that every other possible type for that expression is a more specific version of the principal type.

As we have seen, it is often possible to infer a type for an expression based on its internal structure, as well as by the way in which it is used. In the simply-typed lambda calculus, types for a well-typed expression can always be inferred; this is what enables programs to be written "Curry-style", i.e., without explicit types.

Unfortunately, reliably inferring types is not always possible. For unrestricted System F, type inference is undecidable, so a certain amount of explicit type specification is required. OCaml places restrictions on its type system that makes type inference feasible.

When programming language interpreters and compilers infer types, they often (but not always) aim for the principal type (if one is guaranteed to exist).

# let pair a b z = z a b;;
val pair : 'a -> 'b -> ('a -> 'b -> 'c) -> 'c = <fun>

For instance, when we define the same pair function given above in the OCaml interpreter, it correctly infers the principal type we gave above (remember that OCaml doesn't bother giving the explicit universal type quantifiers required by System F).

Computing principal types involves unification algorithms. Inferring types is a subtle and complicated business, and all sorts of extensions to a programming language can interfere with it.

Bottom type, divergence

Expressions that don't terminate all belong to the bottom type. This is a subtype of every other type. That is, anything of bottom type belongs to every other type as well. More advanced type systems have more examples of subtyping: for example, they might make int a subtype of real. But the core type system of OCaml doesn't have any general subtyping relations. (Neither does System F.) Just this one: that expressions of the bottom type also belong to every other type. It's as if every type definition in OCaml, even the built in ones, had an implicit extra clause:

type 'a option = None | Some of 'a;;
type 'a option = None | Some of 'a | bottom;;

Here are some exercises that may help better understand this. Figure out what is the type of each of the following:

fun x y -> y;;

fun x (y:int) -> y;;

fun x y : int -> y;;

let rec blackhole x = blackhole x in blackhole;;

let rec blackhole x = blackhole x in blackhole 1;;

let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;

let rec blackhole x = blackhole x in (blackhole 1) + 2;;

let rec blackhole x = blackhole x in (blackhole 1) || false;;

let rec blackhole x = blackhole x in 2 :: (blackhole 1);;

By the way, what's the type of this:

let rec blackhole (x:'a) : 'a = blackhole x in blackhole

Back to thunks: the reason you'd want to control evaluation with thunks is to manipulate when "effects" happen. In a strongly normalizing system, like the simply-typed lambda calculus or System F, there are no "effects." In Scheme and OCaml, on the other hand, we can write programs that have effects. One sort of effect is printing. Another sort of effect is mutation, which we'll be looking at soon. Continuations are yet another sort of effect. None of these are yet on the table though. The only sort of effect we've got so far is divergence or non-termination. So the only thing thunks are useful for yet is controlling whether an expression that would diverge if we tried to fully evaluate it does diverge. As we consider richer languages, thunks will become more useful.