We've already established that the behavior of combinatory terms can be
perfectly mimicked by lambda terms: just replace each combinator with its
-equivalent lambda term, i.e., replace `I` with `\x.x`, replace `K` with `\fxy.x`,
+equivalent lambda term, i.e., replace `I` with `\x.x`, replace `K` with `\xy.x`,
and replace `S` with `\fgx.fx(gx)`. So the behavior of any combination of
combinators in Combinatory Logic can be exactly reproduced by a lambda term.
1. [a] a
2. [(M N)] ([M][N])
3. [\a.a] I
- 4. [\a.M] K[M] assumption: a does not occur free in M
+ 4. [\a.M] K[M] when a does not occur free in M
5. [\a.(M N)] S[\a.M][\a.N]
6. [\a\b.M] [\a[\b.M]]
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. 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 our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]
+(*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 our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.)
(Various, slightly differing translation schemes from combinatory logic to the
lambda calculus are also possible. These generate different metatheoretical
Logic) apply to embedded expressions. Generally, we want that to happen, but
making it happen requires adding explicit axioms.)
-Let's check that the translation of the false boolean behaves as expected by feeding it two arbitrary arguments:
+Let's check that the translation of the `false` boolean behaves as expected by feeding it two arbitrary arguments:
KIXY ~~> IY ~~> Y
A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is
from combinatory logic (see especially his 2012 book, <cite>Taking Scope</cite>). Steedman attempts to build
-a syntax/semantics interface using a small number of combinators, including T ≡ `\xy.yx`, B ≡ `\fxy.f(xy)`,
+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.