From 38831ff5ac8222c8d44dce54d3c52c7fa09af3bf Mon Sep 17 00:00:00 2001 From: Chris Date: Tue, 24 Feb 2015 13:58:19 -0500 Subject: [PATCH] edits --- topics/_week5_system_F.mdwn | 95 +++++++++++++++++++++++++++++++-------------- 1 file changed, 66 insertions(+), 29 deletions(-) diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index cd1b6179..f6f7ab6b 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -24,9 +24,13 @@ 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 +40,19 @@ 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 +71,21 @@ 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. + +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 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 +99,7 @@ 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) @@ -96,11 +108,11 @@ 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; @@ -133,15 +145,40 @@ 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. +In fact, 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 [All X . X -> X] 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 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 (in this case, the identity function) to apply +to itself! - omega = lambda x:(All X. X->X) . x [All X . X->X] x in - omega; +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. -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... +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. (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 -- 2.11.0