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Combinators and Combinatory Logic
=================================

Combinatory logic is of interest here in part because it provides a
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
with a second parallel example later when we're thinking through
such topics as evaluation strategies and recursion.

Lambda expressions that have no free variables are known as **combinators**. Here are some common ones:

>   **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`.

>   **S** is defined to be `\f g x. f x (g x)`.  This is a more
complicated operation, but is extremely versatile and useful
(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.

>   **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. 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`.)

>   **C** is defined to be: `\f x y. f y x`. (So `C f` is a function like `f` except it expects its first two (curried) arguments in flipped order.)

>   **W** is defined to be: `\f x . f x x`. (So `W f` accepts one argument and gives it to `f` twice. What is the meaning of `W multiply`?) <!-- \x. multiply x x === \x. square x -->

>   **&omega;** (that is, lower-case omega) is defined to be: `\x. x x`. Sometimes this combinator is called **M**.

<!--
omitted: T, L

duplicate argument
W is \uv.uvv
M/omega is \x.xx; L is \uv.u(vv)

reorder arguments
C is \uvx.uxv (curried_flip)
T is \xy.yx

S, K, I, B also known
-->


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`.
(Though we'll see shortly that the behavior of `I` can be exactly
simulated by a combination of `S`'s and `K`'s.)  But it's possible to be
even more minimalistic, and get by with only a single combinator (see
links below for details). (And there are different single-combinator
bases you can choose.) <!-- Schoenfinkel already discovered one of them;
did Chris discover his? -->

There are some well-known linguistic applications of Combinatory
Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson.
They claim that natural language semantics is a combinatory system: that every
natural language denotation is a combinator.

For instance, Szabolcsi 1987 argues that reflexive pronouns are argument
duplicators.

<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        &forall;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)`:

<pre><code>S(CI) &equiv;
S((\fxy.fyx)(\x.x)) ~~>
S(\xy.(\x.x)yx) ~~>
S(\xy.yx) &equiv;
(\fgx.fx(gx))(\xy.yx) ~~>
\gx.(\xy.yx)x(gx) ~~>
\gx.(gx)x &equiv;
W</code></pre>

###A different set of reduction rules###

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) ~~> I(IX) ~~> IX ~~> X

The reduction rule for `K` is also straightforward:

    KXY ~~> X

That is, `K` throws away its second argument.  The reduction rule for `S` can be constructed by examining
the defining lambda term:

<pre><code>S &equiv; \fgx.fx(gx)</code></pre>

`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's behavior is simulated by a
certain crafty combination of `S`s and `K`s:

    SKKX ~~> KX(KX) ~~> X

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
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 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###

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
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
already seen that `K` deletes its second argument, so we have deletion covered.
`S` duplicates and reorders, so we have some reason to hope that `S` and `K` are
enough to define arbitrary functions.

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 `\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.

How about the other direction?  Here is a method for converting an arbitrary
lambda term into an equivalent Combinatory Logic term using only `S`, `K`, and `I`.
Besides the intrinsic beauty of this mapping, and the importance of what it
says about the nature of binding and computation, it is possible to hear an
echo of computing with continuations in this conversion strategy (though you
wouldn't be able to hear these echos until we've covered a considerable portion
of the rest of the course).  In addition, there is a direct linguistic
appliction of this mapping in chapter 17 of Barker and Shan 2014, where it is
used to establish a correpsondence between two natural language grammars, one
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:

     1. [a]               a
     2. [(M N)]           ([M][N])
     3. [\a.a]            I
     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]]

It's easy to understand these rules based on what `S`, `K` and `I` do.  The first rule says
that variables are mapped to themselves.
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 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.)

(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
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:

    KIXY ~~> IY ~~> Y

Throws away the first argument, returns the second argument---yep, it works.

Here's a more elaborate example of the translation.  The goal is to establish that combinators can reverse order, so we use the **T** combinator, where  <code>T &equiv; \x\y.yx</code>:

    [\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)

We can test this translation by seeing if it behaves like the original lambda term does.
The orginal lambda term lifts its first argument (think of it as reversing the order of its two arguments):

        S(K(SI))(S(KK)I) X Y ~~>
        (K(SI))X ((S(KK)I) X) Y ~~>
        SI ((KK)X (IX)) Y ~~>
        SI (KX) Y ~~>
        IY (KXY) ~~>
        Y X

Voil&agrave;: 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
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
enterprise Free Variable Free Semantics.

A philosophical connection: Quine went through a phase in which he developed a variable free logic.

  Quine, Willard. 1960. "Variables explained away" <cite>Proceedings of the American Philosophical Society</cite>.  Volume 104: 343--347.  Also in W. V. Quine.  1960. <cite>Selected Logical Papers</cite>.  Random House: New
  York.  227--235.

The reason this was important to Quine is similar to the worry that using
non-referring expressions such as Santa Claus might commit one to believing in
non-existant things.  Quine's slogan was that "to be is to be the value of a
variable."  What this was supposed to mean is that if and only if an object
could serve as the value of some variable, we are committed to recognizing the
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
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>.

Cresswell has also developed a variable-free approach of some philosophical and linguistic interest
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
a syntax/semantics interface using a small number of combinators, including `T` &equiv; `\xy.yx`, `B` &equiv; `\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!

The combinators `K` and `S` correspond to two well-known axioms of sentential logic:

###A connection between Combinatory Logic and Sentential Logic###

One way of getting a feel for the power of the SK basis is to note
that the following two axioms

    AK: A --> (B --> A)
    AS: (A --> (B --> C)) --> ((A --> B) --> (A --> C))

when combined with modus ponens (from `A` and `A --> B`, conclude `B`)
are complete for the implicational fragment of intuitionistic logic.
(To get a complete proof theory for *classical* sentential logic, you
need only add one more axiom, constraining the behavior of a new connective "not".)
The way we'll favor for viewing the relationship between these axioms
and the `S` and `K` combinators is that the axioms correspond to type
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.
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
*       [[!wikipedia SKI combinatory calculus]]
*       [[!wikipedia B,C,K,W system]]
*       [Chris Barker's Iota and Jot](http://semarch.linguistics.fas.nyu.edu/barker/Iota/)
*       Jeroen Fokker, "The Systematic Construction of a One-combinator Basis for Lambda-Terms" <cite>Formal Aspects of Computing</cite> 4 (1992), pp. 776-780.
<http://people.cs.uu.nl/jeroen/article/combinat/combinat.ps>