X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=b8c6bf3da6d93c05a1e16ea6ef7ec94528ea3f60;hp=fe451a0bb556f6bf5e62287eb1a7433eaff0f85f;hb=ecea9a7ecaae77c1da240a683caca592b909e32c;hpb=815520c4c0adc3759fa8dd267ad82f79ed4ab89f diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index fe451a0b..b8c6bf3d 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -1,3 +1,5 @@ +[[!toc]] + # System F and recursive types In the simply-typed lambda calculus, we write types like σ @@ -24,9 +26,13 @@ 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): @@ -36,15 +42,19 @@ match up with usage in O'Caml, whose type system is based on 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.) +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 +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 +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 @@ -63,17 +73,22 @@ 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 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 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] 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 `'a`. Not surprisingly, the type of this type application is a function from Booleans to Booleans: @@ -87,7 +102,7 @@ of type `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 +of type `'a`. In general, then, the type of the uninstantiated (polymorphic) identity function is (Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a) @@ -96,10 +111,93 @@ 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. - +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, pp. 350--353, for `tail` for lists in System F.] +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. Types in OCaml