From: barker Date: Sat, 18 Sep 2010 19:54:59 +0000 (-0400) Subject: (no commit message) X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=7743c45c8440825886fa216aadf1f788b1b2783e --- diff --git a/week2.mdwn b/week2.mdwn index a42ec6bf..47a46667 100644 --- a/week2.mdwn +++ b/week2.mdwn @@ -124,7 +124,7 @@ 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.b] Kb assumption: a and b are different + 4. [\a.M] KM assumption: a does not occur free in M 5. [\a.(M N)] S[\a.M][\a.N] 6. [\a\b.M] [\a[\b.M]] @@ -134,7 +134,13 @@ The second rule says that the way to translate an application is to translate th first element and the second element separately. 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 breaks down an abstract whose body is an application. The S combinator takes its next argument and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. 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 lambda in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.) +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. 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.) + +Here's an example of the translation: + + [\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I) + + [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 `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In that intermediate stage, we have `\x.I`. It's possible to avoid this, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]