X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=a80cc58e340154930fc4114cb61e1eb62d3028ee;hb=5cab83962241676d710c788561ac107a3563a3e8;hp=d7a2cf12e68708f389b99b3d55cb60d3dd956fc4;hpb=8a2e3fac5b5faf8eebef6b4c24db35dfc101c07c;p=lambda.git diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index d7a2cf12..a80cc58e 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -1,3 +1,5 @@ +[[!toc levels=2]] + # System F and recursive types In the simply-typed lambda calculus, we write types like σ @@ -37,22 +39,22 @@ match up with usage in O'Caml, whose type system is based on System F): System F: --------- - types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ - expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ] + types τ ::= c | α | τ1 -> τ2 | ∀'a. 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`. "`'a`" is a type variable. The +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 variable can be distinguished by using letters from the greek alphabet (α, β, etc.), or by using capital roman letters (X, Y, etc.). "`τ1 -> τ2`" is the type of a function from expressions of -type `τ1` to expressions of type `τ2`. And "`∀'a. τ`" is called a +type `τ1` to expressions of type `τ2`. And "`∀α. τ`" is called a universal type, since it universally quantifies over the type variable -`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually -have at least one free occurrence of `'a` somewhere inside of it. +`'a`. 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 @@ -60,7 +62,7 @@ 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: "`Λ'a. e`" is called a *type +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 @@ -68,9 +70,9 @@ variables in its body, except that unlike λ, Λ binds type variables instead of expression variables. So in the expression -Λ 'a (λ x:'a . x) +Λ Î± (λ x:α . x) -the Λ binds the type variable `'a` that occurs in +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 @@ -83,27 +85,27 @@ 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 'a (λ x:'a . x)) [t] +(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 `'a`. Not +values) specifies the value of the type variable `α`. Not surprisingly, the type of this type application is a function from Booleans to Booleans: -((Λ 'a (λ x:'a . x)) [t]): (b -> b) +((Λ Î± (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`: -((Λ 'a (λ x:'a . x)) [e]): (e -> e) +((Λ Î± (λ x:α . x)) [e]): (e -> e) -Clearly, for any choice of a type `'a`, the identity function can be -instantiated as a function from expresions of type `'a` to expressions -of type `'a`. In general, then, the type of the uninstantiated +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 -(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a) +(Λ Î± (λ x:α . x)): (∀ α . α -> α) Pred in System F ---------------- @@ -198,8 +200,91 @@ be strongly normalizing, from which it follows that System F is not Turing complete. -Types in OCaml --------------- +## 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 = lambda l:t . lambda r:t . l r false + and = lambda 'a . lambda 'b . + lambda l:'a->'b . lambda r:'a->'b . + lambda x:'a . and:'b (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 'b 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.