X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week2.mdwn;h=717b59432bb4759fdd64b41708eaab274a35b6ec;hp=77459bfda891dc264e5619162878263015ef5706;hb=64ccacaf20339168985b55aec3141bd9ba5c06fa;hpb=c359efaee71b34e8db4e7527c75766fde6c0e8fa diff --git a/week2.mdwn b/week2.mdwn index 77459bfd..717b5943 100644 --- a/week2.mdwn +++ b/week2.mdwn @@ -6,6 +6,13 @@ Define T to be `(\x. x y) z`. Then T and `(\x. x y) z` are syntactically equal,
T ≡ (\x. x y) z ≡ (\z. z y) z
 
+[Fussy note: the justification for counting `(\x. x y) z` as +equivalent to `(\z. z y) z` is that when a lambda binds a set of +occurrences, it doesn't matter which variable serves to carry out the +binding. Either way, the function does the same thing and means the +same thing. Look in the standard treatments for discussions of alpha +equivalence for more detail.] + This: T ~~> z y @@ -23,7 +30,10 @@ Lambda expressions that have no free variables are known as **combinators**. Her > **I** is defined to be `\x x` -> **K** is defined to be `\x y. x`, That is, it throws away its second argument. So `K x` is a constant function from any (further) argument to `x`. ("K" for "constant".) Compare K to our definition of **true**. +> **K** is defined to be `\x y. x`, That is, it throws away its + second argument. So `K x` is a constant function from any + (further) argument to `x`. ("K" for "constant".) Compare K + to our definition of **true**. > **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to **K** and **true** as well. @@ -35,9 +45,28 @@ It's possible to build a logical system equally powerful as the lambda calculus One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, **S**, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.) +There are some well-known linguistic applications of Combinatory +Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson. +Szabolcsi supposed that the meanings of certain expressions could be +insightfully expressed in the form of combinators. A couple more +combinators: + + **C** is defined to be: `\f x y. f y x` [swap arguments] + + **W** is defined to be: `\f x . f x x` [duplicate argument] + +For instance, Szabolcsi argues that reflexive pronouns are argument +duplicators. + + +![Szabolcsi's analysis of *himself* as the duplicator combinator](szabolcsi-reflexive.jpg) + + 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! -Here's more to read about combinatorial logic: +Here's more to read about combinatorial logic. +Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]]. +Other sources include * [[!wikipedia Combinatory logic]] at Wikipedia * [Combinatory logic](http://plato.stanford.edu/entries/logic-combinatory/) at the Stanford Encyclopedia of Philosophy