=================================
Combinatory logic is of interest here in part because it provides a
-useful computational system that is equivalent to the lambda calculus,
+useful computational system that is equivalent to the Lambda Calculus,
but different from it. In addition, Combinatory Logic has a number of
applications in natural language semantics. Exploring Combinatory
Logic will involve defining a difference notion of reduction from the
-one we have been using for the lambda calculus. This will provide us
+one we have been using for the Lambda Calculus. This will provide us
with a second parallel example later when we're thinking through
such topics as evaluation strategies and recursion.
(see below): it copies its third argument and distributes it
over the first two arguments.
-> **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.
+> **fst** was our function for extracting the first element of an ordered pair: `\a b. a`. Compare this to `K` and `true` as well.
-> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of `false`.
+> **snd** was our function for extracting the second element of an ordered pair: `\a b. b`. Compare this to our definition of `false`.
> **B** is defined to be: `\f g x. f (g x)`. (So `B f g` is the composition `\x. f (g x)` of `f` and `g`.)
-->
-It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all.
+It's possible to build a logical system equally powerful as the Lambda Calculus
+(and readily intertranslatable with it) using just combinators, considered as
+atomic operations. Such a language doesn't have any variables in it: not just
+no free variables, but no variables (or "bound positions") at all.
One can do that with a very spare set of basic combinators. These days
the standard base is just three combinators: `S`, `K`, and `I`.
For instance, Szabolcsi 1987 argues that reflexive pronouns are argument
duplicators.
- everyone hit himself
- S/(S!NP) (S!NP)/NP (S!NP)!((S!NP)/NP)
- \fAx[fx] \y\z[HIT y z] \h\u[huu]
- ---------------------------------
- S!NP \u[HIT u u]
- --------------------------------------------
- S Ax[HIT x x]
-
-Here, "A" is our crude markdown approximation of the universal quantifier.
-Notice that the semantic value of *himself* is exactly `W`.
-The reflexive pronoun in direct object position combines with the transitive verb. 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.
+<pre><code>
+everyone hit himself
+S/(S!NP) (S!NP)/NP (S!NP)!((S!NP)/NP)
+\fAx[fx] \y\z[HIT y z] \h\u[huu]
+ ---------------------------------
+ S!NP \u[HIT u u]
+--------------------------------------------
+ S ∀x[HIT x x]
+</code></pre>
+
+Notice that the semantic value of *himself* is exactly `W`. The reflexive
+pronoun in direct object position combines with the transitive verb. 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)`:
SKKX ~~> KX(KX) ~~> X
-So the combinator `SKK` is equivalent to the combinator `I`. (Really, it could be `SKy` for any `y`.)
+So the combinator `SKK` is equivalent to the combinator `I`. (Really, it could be `SKX` for any `X`.)
These reduction rule have the same status with respect to Combinatory
Logic as beta reduction and eta reduction, etc., have with respect to
-the lambda calculus: they are purely syntactic rules for transforming
+the Lambda Calculus: they are purely syntactic rules for transforming
one sequence of symbols (e.g., a redex) into another (a reduced
form). It's worth noting that the reduction rules for Combinatory
Logic are considerably more simple than, say, beta reduction. Also, since
there are no variables in Combiantory Logic, there is no need to worry
-about variable collision.
+about variables colliding when we substitute.
Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses `S`, `K`, and `I` as defined here.
-###The equivalence of the untyped lambda calculus and combinatory logic###
+###The equivalence of the untyped Lambda Calculus and Combinatory Logic###
-We've claimed that Combinatory Logic is equivalent to the lambda calculus. If
+We've claimed that Combinatory Logic is equivalent to the Lambda Calculus. If
that's so, then `S`, `K`, and `I` must be enough to accomplish any computational task
imaginable. Actually, `S` and `K` must suffice, since we've just seen that we can
simulate `I` using only `S` and `K`. In order to get an intuition about what it
-takes to be Turing complete, recall our discussion of the lambda calculus in
+takes to be Turing complete, recall our discussion of the Lambda Calculus in
terms of a text editor. A text editor has the power to transform any arbitrary
text into any other arbitrary text.
The way it does this is by deleting, copying, and reordering characters. We've
of which is based on lambda-like abstraction, the other of which is based on
Combinatory Logic like manipulations.
-Assume that for any lambda term T, [T] is the equivalent combinatory logic term. The we can define the [.] mapping as follows:
+Assume that for any lambda term T, [T] is the equivalent Combinatory Logic term. The we can define the [.] mapping as follows:
1. [a] a
2. [(M N)] ([M][N])
The second rule says that the way to translate an application is to translate the
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 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
+(Various, slightly differing translation schemes from Combinatory Logic to the
+Lambda Calculus are also possible. These generate different metatheoretical
correspondences between the two calculii. Consult Hindley and Seldin for
details. Also, note that the combinatorial proof theory needs to be
strengthened with axioms beyond anything we've here described in order to make
[M] convertible with [N] whenever the original lambda-terms M and N are
convertible. But then, we've been a bit cavalier about giving the full set of
-reduction rules for the lambda calculus in a similar way. For instance, one
-issue is whether reduction rules (in either the lambda calculus or Combinatory
+reduction rules for the Lambda Calculus in a similar way. For instance, one
+issue is whether reduction rules (in either the Lambda Calculus or Combinatory
Logic) apply to embedded expressions. Generally, we want that to happen, but
making it happen requires adding explicit axioms.)
Voilà: 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
+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.
+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
Szabolcsi's papers, see, for instance,
Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, "Towards a variable-free Semantics").
Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics
-express their meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their
+express their meanings using the Lambda Calculus rather than Combinatory Logic; perhaps they should call their
enterprise Free Variable Free Semantics.
A philosophical connection: Quine went through a phase in which he developed a variable free logic.
existence of that object in our ontology. Obviously, if there ARE no
variables, this slogan has to be rethought.
-Quine did not appear to appreciate that Shoenfinkel had already invented combinatory logic, though
+Quine did not appear to appreciate that Shoenfinkel had already invented Combinatory Logic, though
he later wrote an introduction to Shoenfinkel's key paper reprinted in Jean
van Heijenoort (ed) 1967 <cite>From Frege to Goedel, a source book in mathematical logic, 1879--1931</cite>.
in two books in the 1990's.
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
+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)`,
and our friend `S`. Steedman used Smullyan's fanciful bird
names for the combinators, Thrush, Bluebird, and Starling.
schemas for the combinators. This will become more clear once we have
a theory of types in view.
-Here's more to read about combinatory logic.
+Here's more to read about Combinatory Logic.
Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]].
Other sources include