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.]
+same thing.
+Linguistic trivia: some linguistic discussions suppose that alphabetic variance
+has important linguistic consequences (notably Ivan Sag's dissertation).
+Look in the standard treatments for discussions of alpha
+equivalence for more detail. Also, as mentioned below, one of the intriguing
+properties of Combinatory Logic is that alpha equivalence is not an issue.]
This:
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 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.)
+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 `\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)
Viola: the combinator takes any X and Y as arguments, and returns Y applied to X.
+One very nice property of combinatory logic is that there is no need to worry about alphabetic variance, or
+variable collision---since there are no (bound) variables, there is no possibility of accidental variable capture,
+and so reduction can be performed without any fear of variable collision. We haven't mentioned the intricacies of
+alpha equivalence or safe variable substitution, but they are in fact quite intricate. (The best way to gain
+an appreciation of that intricacy is to write a program that performs lambda reduction.)
+
Back to linguistic applications: one consequence of the equivalence between the lambda calculus and combinatory
logic is that anything that can be done by binding variables can just as well be done with combinators.
This has given rise to a style of semantic analysis called Variable Free Semantics (in addition to
Cresswell has also developed a variable-free approach of some philosophical and linguistic interest
in two books in the 1990's.
-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!
+A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is
+from combinatory logic (see especially his 2000 book, *The Syntactic Process*). Steedman attempts to build
+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.
+
+Many of these combinatory logics, in particular, the SKI system,
+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.
Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]].