X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=a80cc58e340154930fc4114cb61e1eb62d3028ee;hp=2c37ae3467e0f01203fb4ebb5dcdc203a3067ea5;hb=5cab83962241676d710c788561ac107a3563a3e8;hpb=cbf0c724914abae08e00292bdcbd3dea1f9460cd diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 2c37ae34..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 @@ -24,77 +26,265 @@ Reynolds, who called his version the *polymorphic lambda calculus*. continuations.) System F enhances the simply-typed lambda calculus with abstraction -over types. In order to state System F, we'll need to adopt the -notational convention that "x:α" represents a -expression whose type is α. +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 (choosing notation that will 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] - -In the definition of the types, "`c`" is a type constant (e.g., `e` or -`t`). "`'a`" is a type variable (the tick mark just indicates that -the variable ranges over types rather than values). "`τ1 -> τ2`" is -the type of a function from expressions of type `τ1` to expressions of -type `τ2`. And "`∀'a. τ`" is called a universal type, since it -universally quantifies over the type variable `'a`. - -In the definition of the expressions, we have variables "`x`". + System F: + --------- + 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`. "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 "`∀α. τ`" is called a +universal type, since it universally quantifies over the type variable +`'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 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 a type -abstraction, and "`e [τ]`" is a type application. The idea is that -Λ is a capital λ. Just like -the lower-case λ, Λ binds -variables in its body; unlike λ, -Λ binds type variables. So in the expression - -Λ 'a (λ x:'a . x) - -the Λ binds the type variable `'a` that occurs in -the λ abstract. This expression is a polymorphic -version of the identity function. It says that this one general -identity function can be adapted for use with expressions of any -type. In order to get it ready to apply to, say, a variable of 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 +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 'a (λ x:'a . x)) [t] +(Λ Î± (λ x:α . x)) [t] -The type application (where `t` is a type constant for Boolean truth -values) specifies the value of the type variable `α`, which is -the type of the variable bound in the `λ` expression. Not +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: -((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 unapplied +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)): (\forall 'a . 'a -> 'a) - - - - -## - +(Λ Î± (λ x: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 = All X . (X->X)->X->X; + Pair = (N -> N -> N) -> N; + let zero = lambda X . lambda s:X->X . lambda z:X. z in + let fst = lambda x:N . lambda y:N . x in + let snd = lambda x:N . lambda y:N . y in + let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in + let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in + let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in + let pre = lambda n: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 `All X . (X->X)->X->X`. 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, `(X -> X) -> X -> X`. And then we'd have to +replace each of these `X`'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 = lambda x:(All X. X->X) . x [All X . X->X] x + +In order to see how this works, we'll apply ω to the identity +function. + +ω id == + + (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x) + +Since the type of the identity function is `(All X . X->X)`, 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 `X` to the universal type `All X . X->X`. 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 = 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 -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.