From c43f7e5448e344d33bc9271d34e801e1feb108f7 Mon Sep 17 00:00:00 2001 From: barker Date: Sun, 19 Sep 2010 11:37:20 -0400 Subject: [PATCH] --- week2.mdwn | 12 +++++++++++- 1 file changed, 11 insertions(+), 1 deletion(-) diff --git a/week2.mdwn b/week2.mdwn index 53e86eeb..a22bf2a6 100644 --- a/week2.mdwn +++ b/week2.mdwn @@ -138,7 +138,17 @@ The fifth rule deals with an abstract whose body is an application: the S combin [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.] -Here's an example of the translation: +Here's a simple example of the translation. We already have a combinator for our true boolean (K is true: it returns its first argument and discards its second argument). What about false? + + [\x\y.y] = [\x[\y.y]] = [\xI] = KI + +We can test this translation by feeding it two arbitrary arguments: + + KIXY ~~> IY ~~> Y + +Yep, it works. + +Here's a more elaborat example of the translation. The goal is to establish that combinators can reverse order, so we use the T combinator, where `T = \x\y.yx`: [\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) -- 2.11.0