X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=cd1b6179a31949736fa76ab3473846f434e60b74;hp=2c37ae3467e0f01203fb4ebb5dcdc203a3067ea5;hb=de74e6bac683afd7a0d6c64716814ac6c4942c6b;hpb=422bf1f49096fad4e372d80b7bab3432e4165d60 diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 2c37ae34..cd1b6179 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -25,49 +25,55 @@ 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. +notational convention 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: + 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`". +`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.) + +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 +additional ways of forming expressions: "`Λ'a. 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 '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 -boolean, just do this: +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: (Λ 'a (λ x:'a . 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, which is +the type of the variable bound in the λ expression. Not surprisingly, the type of this type application is a function from Booleans to Booleans: @@ -84,13 +90,58 @@ instantiated as a function from expresions of type `'a` to expressions of type `'a`. In general, then, the type of the unapplied (polymorphic) identity function is -(Λ 'a (λ x:'a . x)): (\forall 'a . 'a -> 'a) - - +(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a) + +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 +part you want 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 + + 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 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. + +[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT +Press, pp. 350--353, for `tail` for lists in System F.] + +Typing ω +-------------- +In fact, it is even possible to give a type for &omeage; in System F. -## + 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