From 95124aa2037f01f4d4aeaa55ecb2dacf785b6cd6 Mon Sep 17 00:00:00 2001 From: Jim Date: Thu, 12 Feb 2015 12:26:54 -0500 Subject: [PATCH] combinatory tweaks and formatting --- topics/week3_combinatory_logic.mdwn | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/topics/week3_combinatory_logic.mdwn b/topics/week3_combinatory_logic.mdwn index 53c2cd67..0b3390ee 100644 --- a/topics/week3_combinatory_logic.mdwn +++ b/topics/week3_combinatory_logic.mdwn @@ -156,7 +156,7 @@ enough to define arbitrary functions. We've already established that the behavior of combinatory terms can be perfectly mimicked by lambda terms: just replace each combinator with its -equivalent lambda term, i.e., replace `I` with `\x.x`, replace `K` with `\fxy.x`, +equivalent lambda term, i.e., replace `I` with `\x.x`, replace `K` with `\xy.x`, and replace `S` with `\fgx.fx(gx)`. So the behavior of any combination of combinators in Combinatory Logic can be exactly reproduced by a lambda term. @@ -177,7 +177,7 @@ Assume that for any lambda term T, [T] is the equivalent combinatory logic term. 1. [a] a 2. [(M N)] ([M][N]) 3. [\a.a] I - 4. [\a.M] K[M] assumption: a does not occur free in M + 4. [\a.M] K[M] when a does not occur free in M 5. [\a.(M N)] S[\a.M][\a.N] 6. [\a\b.M] [\a[\b.M]] @@ -189,7 +189,7 @@ The third rule should be obvious. The fourth rule should also be fairly self-evident: since what a lambda term such as `\x.y` does it throw away its first argument and return `y`, that's exactly what the combinatory logic translation should do. And indeed, `Ky` is a function that throws away its argument and returns `y`. The fifth rule deals with an abstract whose body is an application: the `S` combinator takes its next argument (which will fill the role of the original variable a) and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. This ensures that any free occurrences of a inside M or N will end up taking on the appropriate value. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambdas in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.) -[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of `S`, `K`, and `I` (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.] +(*Fussy notes:* if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of `S`, `K`, and `I` (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.) (Various, slightly differing translation schemes from combinatory logic to the lambda calculus are also possible. These generate different metatheoretical @@ -203,7 +203,7 @@ issue is whether reduction rules (in either the lambda calculus or Combinatory Logic) apply to embedded expressions. Generally, we want that to happen, but making it happen requires adding explicit axioms.) -Let's check that the translation of the false boolean behaves as expected by feeding it two arbitrary arguments: +Let's check that the translation of the `false` boolean behaves as expected by feeding it two arbitrary arguments: KIXY ~~> IY ~~> Y @@ -263,7 +263,7 @@ in two books in the 1990's. A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is from combinatory logic (see especially his 2012 book, Taking Scope). Steedman attempts to build -a syntax/semantics interface using a small number of combinators, including T ≡ `\xy.yx`, B ≡ `\fxy.f(xy)`, +a syntax/semantics interface using a small number of combinators, including `T` ≡ `\xy.yx`, `B` ≡ `\fxy.f(xy)`, and our friend `S`. Steedman used Smullyan's fanciful bird names for the combinators, Thrush, Bluebird, and Starling. -- 2.11.0