XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek3_combinatory_logic.mdwn;h=a176603cfc57571957c0a5b16f407a83b16ed2b5;hp=f0b28cd7d9e394ee84229f0a0b8a8d2ffbd623fb;hb=d765cd3f1ae243cedf9ae80fcbf9f1051ff72c52;hpb=cdd2eadc0a38d4d42cf19648f3cb0f6fb8ff9995
diff git a/topics/week3_combinatory_logic.mdwn b/topics/week3_combinatory_logic.mdwn
index f0b28cd7..a176603c 100644
 a/topics/week3_combinatory_logic.mdwn
+++ b/topics/week3_combinatory_logic.mdwn
@@ 30,11 +30,27 @@ Lambda expressions that have no free variables are known as **combinators**. Her
> **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 arguments in swapped order.)
+> **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 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`?)
> **ω** (that is, lowercase omega) is defined to be: `\x. x x`
+> **ω** (that is, lowercase omega) is defined to be: `\x. x x`
+
+
+
+
+
+
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.
@@ 126,33 +142,35 @@ Combinatory Logic is what you have when you choose a set of combinators and regu
###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 `\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.

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 lambdalike abstraction,
the other of which is based on Combinatory Logic like manipulations.
+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 `\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.
+
+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 lambdalike 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:
@@ 173,18 +191,17 @@ The fifth rule deals with an abstract whose body is an application: the S combin
[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 variablesi.e., is a combinatorthen 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 lambdaterms 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.]
+(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 lambdaterms 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:
@@ 229,14 +246,13 @@ A philosophical connection: Quine went through a phase in which he developed a v
Quine, Willard. 1960. "Variables explained away" Proceedings of the American Philosophical Society. Volume 104: 343347. Also in W. V. Quine. 1960. Selected Logical Papers. Random House: New
York. 227235.
The reason this was important to Quine is similar to the worry that
using nonreferring expressions such as Santa Claus might commit one
to believing in nonexistant 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.
+The reason this was important to Quine is similar to the worry that using
+nonreferring expressions such as Santa Claus might commit one to believing in
+nonexistant 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
@@ 256,7 +272,7 @@ are Turing complete. In other words: every computation we know how to describe c
The combinators K and S correspond to two wellknown axioms of sentential logic:
###A connection between Combinatory Logic and logic 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
@@ 266,9 +282,11 @@ that the following two axioms
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. Thsi will become more clear once we have
+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.