XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipperlistscontinuations.mdwn;h=08170a31eb8e9bac740ca5f35fba36a352811831;hp=052bdf657a8440d46389f062e2bfb8c7653cf2d4;hb=cb47fc333272c7b1f1f77af11ae72643335c05dc;hpb=638a4cb56977ff574d4f8de1ff4ca6faabe7c9d0
diff git a/zipperlistscontinuations.mdwn b/zipperlistscontinuations.mdwn
index 052bdf65..08170a31 100644
 a/zipperlistscontinuations.mdwn
+++ b/zipperlistscontinuations.mdwn
@@ 1,11 +1,14 @@
+
+[[!toc]]
+
Today we're going to encounter continuations. We're going to come at
them from three different directions, and each time we're going to end
up at the same place: a particular monad, which we'll call the
continuation monad.
The three approches are:
+Much of this discussion benefited from detailed comments and
+suggestions from Ken Shan.
[[!toc]]
Rethinking the list monad

@@ 103,7 +106,10 @@ looks like this:
l_unit (a : 'a) = [a];;
l_bind u f = List.concat (List.map f u);;
Recall that `List.map` take a function and a list and returns the
+Thinking through the list monad will take a little time, but doing so
+will provide a connection with continuations.
+
+Recall that `List.map` takes a function and a list and returns the
result to applying the function to the elements of the list:
List.map (fun i > [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
@@ 117,17 +123,33 @@ And sure enough,
l_bind [1;2] (fun i > [i, i+1]) ~~> [1; 2; 2; 3]
But where is the reasoning that led us to this unit and bind?
And what is the type `['a]`? Magic.

So let's indulge ourselves in a completely useless digression and see
if we can gain some insight into the details of the List monad. Let's
choose type constructor that we can peer into, using some of the
technology we built up so laboriously during the first half of the
course. We're going to use type 3 lists, partly because we know
they'll give the result we want, but also because they're the coolest.
These were the lists that made lists look like Church numerals with
extra bits embdded in them:
+Now, why this unit, and why this bind? Well, ideally a unit should
+not throw away information, so we can rule out `fun x > []` as an
+ideal unit. And units should not add more information than required,
+so there's no obvious reason to prefer `fun x > [x,x]`. In other
+words, `fun x > [x]` is a reasonable guess for a unit.
+
+As for bind, an `'a list` monadic object contains a lot of objects of
+type `'a`, and we want to make some use of each of them. The only
+thing we know for sure we can do with an object of type `'a` is apply
+the function of type `'a > 'a list` to them. Once we've done so, we
+have a collection of lists, one for each of the `'a`'s. One
+possibility is that we could gather them all up in a list, so that
+`bind' [1;2] (fun i > [i;i]) ~~> [[1;1];[2;2]]`. But that restricts
+the object returned by the second argument of `bind` to always be of
+type `'b list list`. We can elimiate that restriction by flattening
+the list of lists into a single list. So there is some logic to the
+choice of unit and bind for the list monad.
+
+Yet we can still desire to go deeper, and see if the appropriate bind
+behavior emerges from the types, as it did for the previously
+considered monads. But we can't do that if we leave the list type is
+a primitive Ocaml type. However, we know several ways of implementing
+lists using just functions. In what follows, we're going to use type
+3 lists (the right fold implementation), though it's important to
+wonder how things would change if we used some other strategy for
+implementating lists. These were the lists that made lists look like
+Church numerals with extra bits embdded in them:
empty list: fun f z > z
list with one element: fun f z > f 1 z
@@ 279,9 +301,6 @@ lists, so that they will print out.
Ta da!
To bad this digression, though it ties together various
elements of the course, has *no relevance whatsoever* to the topic of
continuations...
Montague's PTQ treatment of DPs as generalized quantifiers

@@ 324,19 +343,16 @@ constructor and the terms from the list monad derived above:
The unit and the bind for the Montague continuation monad and the
homemade List monad are the same terms! In other words, the behavior
of the List monad and the behavior of the continuations monad are
parallel in a deep sense. To emphasize the parallel, we can
instantiate the type of the list' monad using the OCaml list type:

 type 'a c_list = ('a > 'a list) > 'a list
+parallel in a deep sense.
Have we really discovered that lists are secretly continuations? Or
have we merely found a way of simulating lists using list
continuations? Both perspectives are valid, and we can use our
intuitions about the list monad to understand continuations, and vice
versa (not to mention our intuitions about primitive recursion in
Church numerals too). The connections will be expecially relevant
when we consider indefinites and Hamblin semantics on the linguistic
side, and nondeterminism on the list monad side.
+continuations? Well, strictly speaking, what we have done is shown
+that one particular implementation of liststhe left fold
+implementationgives rise to a continuation monad fairly naturally,
+and that this monad can reproduce the behavior of the standard list
+monad. But what about other list implementations? Do they give rise
+to monads that can be understood in terms of continuations?
Refunctionalizing zippers
