From d2fa64a9e75f27c2d8fdaeae2f25eb29da82e760 Mon Sep 17 00:00:00 2001
From: barker
Date: Sat, 18 Sep 2010 10:06:31 -0400
Subject: [PATCH]
---
week2.mdwn | 39 +++++++++++++++++++++++++++++++++++++++
1 file changed, 39 insertions(+)
diff --git a/week2.mdwn b/week2.mdwn
index 31eef33e..8de3aab9 100644
--- a/week2.mdwn
+++ b/week2.mdwn
@@ -63,7 +63,46 @@ duplicators.
Notice that the semantic value of *himself* is exactly W.
The reflexive pronoun in direct object position combines first with the transitive verb (through compositional magic we won't go into here). The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
+Note that W = S(CI):
+ S(CI) =
+ S((\fxy.fyx)(\x.x)) =
+ S(\xy.(\x.x)yx) =
+ (\fgx.fx(gx))(\xy.yx) =
+ \gx.[\xy.yx]x(gx) =
+ \gx.(gx)x =
+ W
+
+Ok, here comes a shift in thinking. Instead of defining combinators as equivalent to certain lambda terms,
+we can define combinators by what they do. If we have the I combinator followed by any expression X,
+I will take that expression as its argument and return that same expression as the result. In pictures,
+
+ IX ~~> X
+
+Thinking of this as a reduction rule, we can perform the following computation
+
+ II(IX) ~~> IIX ~~> IX ~~> X
+
+The reduction rule for K is also straigtforward:
+
+ KXY ~~> X
+
+That is, K throws away its second argument. The reduction rule for S can be constructed by examining
+the defining lambda term:
+
+ S = \fgx.fx(gx)
+
+S takes three arguments, duplicates the third argument, and feeds one copy to the first argument and the second copy to the second argument. So:
+
+ SFGX ~~> FX(GX)
+
+If the meaning of a function is nothing more than how it behaves with respect to its arguments,
+these reduction rules capture the behavior of the combinators S,K, and I completely.
+We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of S's and K's:
+
+ SKKX ~~> KX(KX) ~~> X
+
+So the combinator SKK is equivalent to the combinator I.
These systems are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation!
--
2.11.0