From: Chris Date: Thu, 26 Feb 2015 18:24:58 +0000 (-0500) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=293b63f453994f8c3c458ca264ef0fb04b79e1fb;ds=sidebyside edits --- diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 4bde11e0..4517509b 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -51,16 +51,16 @@ Then System F can be specified as follows: 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`. "Î±" 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.). "`Ï1 -> Ï2`" is the type of a function from expressions of -type `Ï1` to expressions of type `Ï2`. And "`âÎ±.Ï`" is called a -universal type, since it universally quantifies over the type variable -`'a`. You can expect that in `âÎ±.Ï`, the type `Ï` will usually -have at least one free occurrence of `Î±` somewhere inside of it. +constants might include `e` and `t`. "Î±" is a type variable. In +various discussions, type variables are distinguished by using letters +from the greek alphabet (α, β, etc.), as we do here, or by +using capital roman letters (X, Y, etc.), or by adding a tick mark +(`'a`, `'b`, etc.), as in OCaml. "`Ï1 -> Ï2`" is the type of a +function from expressions of type `Ï1` to expressions of type `Ï2`. +And "`âÎ±.Ï`" is called a universal type, since it universally +quantifies over the type variable `α`. You can expect that in +`âÎ±.Ï`, the type `Ï` will usually have at least one free occurrence of +`Î±` somewhere inside of it. In the definition of the expressions, we have variables "`x`" as usual. Abstracts "`Î»x:Ï.e`" are similar to abstracts in the simply-typed lambda @@ -79,24 +79,19 @@ variables. So in the expression `Λ Î± (λ x:Î±. x)` the `Λ` binds the type variable `Î±` 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 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: +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 this +identity function to, say, a variable of type boolean, just do this: `(Λ Î± (λ x:Î±. x)) [t]` This type application (where `t` is a type constant for Boolean truth values) specifies the value of the type variable `Î±`. Not -surprisingly, the type of this type application is a function from -Booleans to Booleans: +surprisingly, the type of the expression that results from this type +application is a function from Booleans to Booleans: `((ΛÎ± (λ x:Î± . x)) [t]): (b->b)` @@ -111,20 +106,17 @@ instantiated as a function from expresions of type `Î±` to expressions of type `Î±`. In general, then, the type of the uninstantiated (polymorphic) identity function is -`(ΛÎ± (λx:Î± . x)): (∀Î±. Î±-Î±)` +`(ΛÎ± (λx:Î± . x)): (∀Î±. Î±->Î±)` 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 -relevant evaluator is called "fullpoly"): +however. Here is one way: - N = âÎ±.(Î±->Î±)->Î±->Î±; - Pair = (N->N->N)->N; + let N = âÎ±.(Î±->Î±)->Î±->Î± in + let Pair = (N->N->N)->N in let zero = ÎÎ±. Î»s:Î±->Î±. Î»z:Î±. z in let fst = Î»x:N. Î»y:N. x in @@ -136,12 +128,14 @@ relevant evaluator is called "fullpoly"): 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). +[If you want to run this code 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", and you'll need to +truncate the names of "suc(c)" and "pre(d)", since those are +reserved words in Pierce's system.] + +Exercise: convince yourself that `zero` has type `N`. 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