X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek3_combinatory_logic.mdwn;h=1c6537e4b2547aa880f95f6a8aa2691c69c071f0;hp=a176603cfc57571957c0a5b16f407a83b16ed2b5;hb=662133800820b7973cc895c474ec52e98b74a9b1;hpb=d765cd3f1ae243cedf9ae80fcbf9f1051ff72c52

diff --git a/topics/week3_combinatory_logic.mdwn b/topics/week3_combinatory_logic.mdwn
index a176603c..1c6537e4 100644
--- a/topics/week3_combinatory_logic.mdwn
+++ b/topics/week3_combinatory_logic.mdwn
@@ -12,34 +12,33 @@ 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`
+>   **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`.
+>   **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.
+>   **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-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`.
+>   **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`.)
+>   **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 flipped 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 (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`?)
-
->	**&omega;** (that is, lower-case omega) is defined to be: `\x. x x` <!-- aka M -->
-
-<!-- missing: T, L -->
+>   **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)
@@ -48,19 +47,23 @@ reorder arguments
 C is \uvx.uxv (curried_flip)
 T is \xy.yx
 
-S, K, I, B known
+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 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.
-(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
+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.)
+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.
@@ -70,15 +73,16 @@ natural language denotation is a combinator.
 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]
+<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>
 
-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.
 
@@ -96,38 +100,38 @@ 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,
+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) ~~> IIX ~~> IX ~~> X
+    II(IX) ~~> I(IX) ~~> IX ~~> X
 
-The reduction rule for K is also straightforward:
+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
+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:
+`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.
+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 is equivalent to a
-certain crafty combination of Ss and Ks:
+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.
+So the combinator `SKK` is equivalent to the combinator `I`. (Really, it could be `SKy` for any `y`.)
 
 These reduction rule have the same status with respect to Combinatory
 Logic as beta reduction and eta reduction, etc., have with respect to
@@ -138,30 +142,30 @@ 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.
 
-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.
+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
+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
+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
+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.
+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
@@ -177,19 +181,19 @@ Assume that for any lambda term T, [T] is the equivalent combinatory logic 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]]
 
-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
 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.)
+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
@@ -203,7 +207,7 @@ 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:
+Let's check that the translation of the `false` boolean behaves as expected by feeding it two arbitrary arguments:
 
     KIXY ~~> IY ~~> Y
 
@@ -263,14 +267,14 @@ 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
+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:
+The combinators `K` and `S` correspond to two well-known axioms of sentential logic:
 
 ###A connection between Combinatory Logic and Sentential Logic###
 
@@ -285,7 +289,7 @@ 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
+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.