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=4afb43bae5fd783c4007e9158ab3a15bb44858fd;hb=c98d4e2d9d0c85b16ac707323031114f2a3db1aa;hpb=c09bd7005b179db8ab4c09c4c60be32d1a1c8881 diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 4afb43ba..a7b4bb91 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -34,35 +34,34 @@ 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] + 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`. "`'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 +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 "`∀'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 +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 @@ -70,9 +69,9 @@ 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) +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 @@ -85,27 +84,27 @@ 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) +((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 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) +(Λα (λx:α . x)): (∀α. α-α) Pred in System F ---------------- @@ -117,15 +116,16 @@ however. Here is one way, coded in 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 + 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))); @@ -138,7 +138,7 @@ 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 +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 @@ -151,8 +151,8 @@ 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 +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. @@ -165,19 +165,19 @@ 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 +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 == - (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x) + (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x) -Since the type of the identity function is `(All X . X->X)`, it's the +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 `X` to the universal type `All X . X->X`. Instantiating the +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. @@ -228,10 +228,8 @@ uses. Can we capture this using polymorphic types? 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) + 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 @@ -258,7 +256,7 @@ 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. +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