everyone hit himself
S/(S!NP) (S!NP)/NP (S!NP)!((S!NP)/NP)
- \fAx[fx] \y\z[HIT y z] \h\u[huu]
+ \fAx[fx] \y\z[HIT y z] \h\u[huu]
---------------------------------
S!NP \u[HIT u u]
--------------------------------------------
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.
+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)`:
###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,
+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
KXY ~~> X
-That is, K throws away its second argument. The reduction rule for S can be constructed by examining
+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 ≡ \fgx.fx(gx)</code></pre>
SFGX ~~> FX(GX)
-If the meaning of a function is nothing more than how it behaves with respect to its arguments,
+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.
+We can use these rules to compute without resorting to beta reduction.
For instance, we can show how the I combinator is equivalent to a
certain crafty combination of Ss and Ks:
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 variable collision.
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.
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.
+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 `\fxy.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.
+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
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
+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
+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.]
+[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 combinatorial
logic to the lambda calculus are also possible. These generate
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
-variable collision---since there are no (bound) variables, there is no possibility of accidental variable capture,
+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
+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
+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
+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
+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.
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
+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)`,
-and our friend S. Steedman used Smullyan's fanciful bird
+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,
+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: