X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek4_fixed_point_combinators.mdwn;h=9c32b04ed8864c1b76cb5762fc2a086debee04a7;hp=ed22cfff6225d1841de515a7e2821532aa29be39;hb=120431d951d97e47c6b80f91f41e2c352c073703;hpb=6694165ac4d6edab602b3ad3651d0a5931b36a0e

diff --git a/topics/week4_fixed_point_combinators.mdwn b/topics/week4_fixed_point_combinators.mdwn
index ed22cfff..9c32b04e 100644
--- a/topics/week4_fixed_point_combinators.mdwn
+++ b/topics/week4_fixed_point_combinators.mdwn
@@ -178,6 +178,17 @@ Many simpler functions always *could* be defined using the resources we've so fa
 
 But functions like the Ackermann function require us to develop a more general technique for doing recursion --- and having developed it, it will often be easier to use it even in the cases where, in principle, we didn't have to.
 
+The example used to illustrate this in Chapter 9 of *The Little Schemer* is a function `looking` where:
+
+    (looking '(6 2 4 caviar 5 7 3))
+
+returns `#t`, because if we follow the path from the head of the list argument, `6`, to the sixth element of the list, `7` (the authors of that book count positions starting from 1, though generally Scheme follows the convention of counting positions starting from 0), and then proceed to the seventh element of the list, `3`, and then proceed to the third element of the list, `4`, and the proceed to the fourth element of the list, we find the `'caviar` we are looking for. On other hand, if we say:
+
+    (looking '(6 2 grits caviar 5 7 3))
+
+our path will take us from `6` to `7` to `3` to `grits`, which is not a number but not the `'caviar` we were looking for either. So this returns `#f`. It would be very difficult to define these functions without recourse to something like `letrec` or `define`, or the techniques developed below (and also in that chapter of *The Little Schemer*).
+
+
 ## Using fixed-point combinators to define recursive functions ##
 
 ### Fixed points ###
@@ -258,6 +269,7 @@ it's not complete, since we don't know what value to use for the
 symbol `LENGTH`.  Technically, it has the status of an unbound
 variable.
 
+<a id=little-h></a>
 Imagine now binding the mysterious variable, and calling the resulting
 term `h`:
 
@@ -281,7 +293,7 @@ saying that we are looking for a fixed point for `h`:
 
     h LENGTH <~~> LENGTH
 
-Replacing `h` with its definition, we have
+Replacing `h` with its definition, we have:
 
     (\xs. (empty? xs) 0 (succ (LENGTH (tail xs)))) <~~> LENGTH
 
@@ -289,7 +301,17 @@ If we can find a value for `LENGTH` that satisfies this constraint, we'll
 have a function we can use to compute the length of an arbitrary list.
 All we have to do is find a fixed point for `h`.
 
-The strategy we will present will turn out to be a general way of
+Let's reinforce this. The left-hand side has the form:
+
+    (\body. Φ[...body...]) LENGTH
+
+which beta-reduces to:
+
+    Φ[...LENGTH...]
+
+where that whole formula is convertible with the term `LENGTH` itself. In other words, the term `Φ[...LENGTH...]` contains (a term that convertible with) itself --- despite being only finitely long. (If it had to contain a term *syntactically identical to* itself, this could not be achieved.)
+
+The key to achieving all this is finding a fixed point for `h`. The strategy we will present will turn out to be a general way of
 finding a fixed point for any lambda term.
 
 
@@ -304,7 +326,7 @@ work, but examining the way in which it fails will lead to a solution.
 
     h h <~~> \xs. (empty? xs) 0 (succ (h (tail xs)))
 
-The problem is that in the subexpression `h (tail list)`, we've
+The problem is that in the subexpression `h (tail xs)`, we've
 applied `h` to a list, but `h` expects as its first argument the
 length function.
 
@@ -319,7 +341,7 @@ to discuss generalizations of this strategy.)
 
 Shifting to `H` is the key creative step.  Instead of applying `u` to a list, as happened
 when we self-applied `h`, `H` applies its argument `u` first to *itself*: `u u`.
-After `u` gets an argument, the *result* is ready to apply to a list, so we've solved the problem noted above with `h (tail list)`.
+After `u` gets an argument, the *result* is ready to apply to a list, so we've solved the problem noted above with `h (tail xs)`.
 We're not done yet, of course; we don't yet know what argument `u` to give
 to `H` that will behave in the desired way.
 
@@ -519,7 +541,7 @@ where `BODY` is equivalent to the very formula `\n. BODY n` that contains it. So
     BODY M <~~>
     ...
 
-You've written an infinite loop!
+You've written an infinite loop! (This is like the function `eternity` in Chapter 9 of *The Little Schemer*.)
 
 However, when we evaluate the application of our:
 
@@ -563,7 +585,7 @@ returns itself (a copy of `sink`); if the argument is boolean false
     sink true true false <~~> I
     sink true true true false <~~> I
 
-Evidently, then, `sink true <~~> sink`. So we want `sink` to be the fixed point
+To get this behavior, we want `sink` to be the fixed point
 of `\sink. \b. b sink I`. That is, `sink ≡ Y (\sb.bsI)`:
 
     1. sink false