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-# System F and recursive types
+[[!toc levels=2]]
-In the simply-typed lambda calculus, we write types like σ
--> τ
. This looks like logical implication. We'll take
-that resemblance seriously when we discuss the Curry-Howard
-correspondence. In the meantime, note that types respect modus
-ponens:
+# System F: the polymorphic lambda calculus
-
-Expression Type Implication ----------------------------------- -fn α -> β α ⊃ β -arg α α ------- ------ -------- -fn arg β β -+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. -The implication in the right-hand column is modus ponens, of course. +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 is usually attributed to Girard, but was independently -proposed around the same time by Reynolds. It enhances the -simply-typed lambda calculus with quantification over types. In -System F, you can say things like +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. -
Λ α (\x.x):(α -> α)
+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
.
-This says that the identity function maps arguments of type α to
-results of type α, for any choice of α. So the Λ is
-a universal quantifier over types.
+Then System F can be specified as follows:
+ System F:
+ ---------
+ types Ï ::= c | α | Ï1 -> Ï2 | âα.Ï
+ expressions e ::= x | λx:Ï.e | e1 e2 | Îα.e | e [Ï]
+
+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. The
+tick mark just indicates that the variable ranges over types rather
+than over values; in various discussion below and later, type variables
+can be distinguished by using letters from the greek alphabet
+(α, β, etc.), or by using capital roman letters (X, Y,
+etc.). "`Ï1 -> Ï2`" is the type of a function from expressions of
+type `Ï1` to expressions of type `Ï2`. And "`âα.Ï`" is called a
+universal type, since it universally quantifies over the type variable
+`'a`. You can expect that in `âα.Ï`, the type `Ï` 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:Ï.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: "`Îα.e`" is called a *type
+abstraction*, and "`e [Ï]`" 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 Î± (λ x:α. x)
+
+the Λ
binds the type variable `α` that occurs in
+the λ
abstract. Of course, as long as type
+variables are carefully distinguished from expression variables (by
+tick marks, Grecification, or capitalization), there is no need to
+distinguish expression abstraction from type abstraction by also
+changing the shape of the lambda.
+
+The expression immediately below 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 Î± (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 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, coded 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"):
+
+ N = âα.(α->α)->α->α;
+ Pair = (N->N->N)->N;
+
+ let zero = Îα. 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 suc = λn:N. Îα. λs:α->α. λz:α. s (n [α] s z) in
+ let shift = λp:Pair. pair (suc (p fst)) (p fst) in
+ let pre = λn:N. n [Pair] shift (pair zero zero) snd in
+
+ pre (suc (suc (suc zero)));
+
+We've truncated the names of "suc(c)" and "pre(d)", since those are
+reserved words in Pierce's system. Note that in this code, there is
+no typographic distinction between ordinary lambda and type-level
+lambda, though the difference is encoded in whether the variables are
+lower case (for ordinary lambda) or upper case (for type-level
+lambda).
+
+The key to the extra expressive power provided by System F is evident
+in the typing imposed by the definition of `pre`. The variable `n` is
+typed as a Church number, i.e., as `âα.(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
+pair-manipulating function, which is the heart of the strategy for
+this version of predecessor.
+
+Could we try to build 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 `X` 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, `(α -> α) -> α -> α`. And then we'd have to
+replace each of these `α`'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:(âα.α->α). x [âα.α->α] x
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+ω id ==
+
+ (λx:(âα.α->α). x [âα.α->α] x) (Îα.λx:α.x)
+
+Since the type of the identity function is `âα.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 `âα.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. Furthermore, 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.)
+
+ Ann left and Bill left.
+ Ann left and slept.
+ Ann and Bill left.
+ Ann read and reviewed the book.
+
+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
+ and = Îα.Îβ.λl:α->β.λr:α->β.λx:α. and [β] (l x) (r x)
+
+The idea is that the basic *and* conjoins expressions of type `t`, and
+when *and* conjoins functional types, it builds a function that
+distributes its argument across the two conjuncts and conjoins the two
+results. So `Ann left and slept` will evaluate to `(\x.and(left
+x)(slept x)) ann`. Following the terminology of Partee and Rooth, the
+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.
+
+But the definitions just given are not well-formed expressions in
+System F. There are three problems. The first is that we have two
+definitions of the same word. The intention is for one of the
+definitions to be operative when the type of its arguments is type
+`t`, but we have no way of conditioning evaluation on the *type* of an
+argument. The second is that for the polymorphic definition, the term
+*and* occurs inside of the definition. System F does not have
+recursion.
+
+The third problem is more subtle. The defintion as given takes two
+types as parameters: the type of the first argument expected by each
+conjunct, and the type of the result of applying each conjunct to an
+argument of that type. We would like to instantiate the recursive use
+of *and* in the definition by using the result type. But fully
+instantiating the definition as given requires type application to a
+pair of types, not to just a single type. We want to somehow
+guarantee that β will always itself be a complex type.
+
+So conjunction and disjunction provide a compelling motivation for
+polymorphism in natural language, but we don't yet have the ability to
+build the polymorphism into a formal system.
+
+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.
+
+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 =