+++ /dev/null
-[[!toc levels=2]]
-
-# 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 "<code>x:α</code>" represents an expression `x`
-whose type is <code>α</code>.
-
-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 `α`. 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 <code>Λ</code> is a capital <code>λ</code>: just
-like the lower-case <code>λ</code>, <code>Λ</code> binds
-variables in its body, except that unlike <code>λ</code>,
-<code>Λ</code> binds type variables instead of expression
-variables. So in the expression
-
-<code>Λ α (λ x:α. x)</code>
-
-the <code>Λ</code> binds the type variable `α` that occurs in
-the <code>λ</code> 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:
-
-<code>(Λ α (λ x:α. x)) [t]</code>
-
-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:
-
-<code>((Λα (λ x:α . x)) [t]): (b->b)</code>
-
-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`:
-
-<code>((Λα (λ x:α. x)) [e]): (e->e)</code>
-
-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
-
-<code>(Λα (λx:α . x)): (∀α. α->α)</code>
-
-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|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]], 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`.
-
-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.
-
-<code>ω = λx:(∀α.α->α). x [∀α.α->α] x</code>
-
-In order to see how this works, we'll apply ω to the identity
-function.
-
-<code>ω id ≡ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)</code>
-
-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?
-
-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.