X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=a7b4bb912333577b0afea0cb4f5ec855da96d566;hp=cd1b6179a31949736fa76ab3473846f434e60b74;hb=c98d4e2d9d0c85b16ac707323031114f2a3db1aa;hpb=de74e6bac683afd7a0d6c64716814ac6c4942c6b diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index cd1b6179..a7b4bb91 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 @@ -24,35 +26,42 @@ 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 an -expression `x` 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): +Then System F can be specified as follows: 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`, or in arithmetic contexts, `N` or `Int`). "`'a`" is a type -variable (the tick mark just indicates that the variable ranges over -types rather than over 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`. (You can expect that in -`∀'a. τ`, the type `τ` will usually have at least one free occurrence -of `'a` somewhere inside of it.) + 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. 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.). "`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 +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 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 @@ -60,56 +69,63 @@ 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 `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 Λ binds the type variable `'a` 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 to, say, a variable of type boolean, just do -this: +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] This 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 +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) +((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`: -((Λ 'a (λ x:'a . x)) [e]): (e -> e) +((Λα (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)): (∀ 'a . 'a -> 'a) +(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, +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 -part you want is called "fullpoly"): +relevant evaluator is called "fullpoly"): - N = All X . (X->X)->X->X; - Pair = All X . (N -> N -> X) -> X; - let zero = lambda X . lambda s:X->X . lambda z:X. z in - let snd = lambda x:N . lambda y:N . y in - let pair = lambda x:N . lambda y:N . lambda X . lambda z:N->N->X . 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 . p [Pair] (lambda a:N . lambda b:N . pair (suc a) a) in - let pre = lambda n:N . n [Pair] shift (pair zero zero) [N] snd in + N = ∀α.(α->α)->α->α; + Pair = (N->N->N)->N; + + 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 suc = λn:N. Λα. 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))); @@ -120,32 +136,153 @@ 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 flexibility provided by System F is that we can -instantiate the `pair` function to return a number, as in the -definition of `pre`, or we can instantiate it to return an ordered -pair, as in the definition of the `shift` function. Because we don't -have to choose a single type for all uses of the pair-building -function, we aren't forced into a infinite regress of types. - +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 `∀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 +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, pp. 350--353, for `tail` for lists in System F.] +Press, chapter 23.] Typing ω -------------- -In fact, it is even possible to give a type for &omeage; in System F. +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. 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 = Λα.Λβ.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 - omega = lambda x:(All X. X->X) . x [All X . X->X] x in - omega; - -Each time the internal application is performed, the type of the head -is chosen anew. And each time, we choose the same type as before, the -type of a function that takes an argument of any type and returns a -result of the same type... - - -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.