edits

[lambda.git] / zipper-lists-continuations.mdwn

diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn
index 4aef298..08170a3 100644 (file)
--- a/zipper-lists-continuations.mdwn
+++ b/zipper-lists-continuations.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.
 
 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
 -------------------------
 
 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);;
 
     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]]
 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]
 
 
     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
 
     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!
 
 
 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
 ----------------------------------------------------------
 
 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
 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
 
 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 non-determinism on the list monad side.
+continuations?  Well, strictly speaking, what we have done is shown
+that one particular implementation of lists---the left fold
+implementation---gives 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
 -------------------------
 
 Refunctionalizing zippers
 -------------------------


@@ -561,13 +577,13 @@ _|__  _|__
 </pre>
 
 Notice that we've counted each internal node twice---it's a good
 </pre>
 
 Notice that we've counted each internal node twice---it's a good

-excerice to adjust the code to count each node once.
+exercise to adjust the code to count each node once.

 
 One more revealing example before getting down to business: replacing
 `state` everywhere in `treemonadizer` with `list` gives us
 
 <pre>
 
 One more revealing example before getting down to business: replacing
 `state` everywhere in `treemonadizer` with `list` gives us
 
 <pre>

-# treemonadizer (fun x -> [[x; square x]]) t1;;
+# treemonadizer (fun x -> [ [x; square x] ]) t1;;

 - : int list tree list =
 [Node
   (Node (Leaf [2; 4], Leaf [3; 9]),
 - : int list tree list =
 [Node
   (Node (Leaf [2; 4], Leaf [3; 9]),


@@ -642,3 +658,94 @@ generalizing the type of the continuation monad to
 
     type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
 
 
     type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
 

+The binary tree monad
+---------------------
+
+Of course, by now you may have realized that we have discovered a new
+monad, the binary tree monad:
+
+<pre>
+type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
+let tree_unit (x:'a) = Leaf x;;
+let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree = 
+  match u with Leaf x -> f x 
+             | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;
+</pre>
+
+For once, let's check the Monad laws.  The left identity law is easy:
+
+    Left identity: bind (unit a) f = bind (Leaf a) f = fa
+
+To check the other two laws, we need to make the following
+observation: it is easy to prove based on `tree_bind` by a simple
+induction on the structure of the first argument that the tree
+resulting from `bind u f` is a tree with the same strucure as `u`,
+except that each leaf `a` has been replaced with `fa`:
+
+\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
+<pre>
+                .                         .       
+              __|__                     __|__   
+              |   |                     |   |   
+              a1  .                    fa1  .   
+                 _|__                     __|__ 
+                 |  |                     |   | 
+                 .  a5                    .  fa5
+   bind         _|__       f   =        __|__   
+                |  |                    |   |   
+                .  a4                   .  fa4  
+              __|__                   __|___   
+              |   |                   |    |   
+              a2  a3                 fa2  fa3         
+</pre>   
+
+Given this equivalence, the right identity law
+
+    Right identity: bind u unit = u
+
+falls out once we realize that
+
+    bind (Leaf a) unit = unit a = Leaf a
+
+As for the associative law,
+
+    Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
+
+we'll give an example that will show how an inductive proof would
+proceed.  Let `f a = Node (Leaf a, Leaf a)`.  Then
+
+\tree (. (. (. (. (a1)(a2)))))
+\tree (. (. (. (. (a1) (a1)) (. (a1) (a1)))  ))
+<pre>
+                                           .
+                                       ____|____
+          .               .            |       |
+bind    __|__   f  =    __|_    =      .       .
+        |   |           |   |        __|__   __|__
+        a1  a2         fa1 fa2       |   |   |   |
+                                     a1  a1  a1  a1  
+</pre>
+
+Now when we bind this tree to `g`, we get
+
+<pre>
+           .
+       ____|____
+       |       |
+       .       .
+     __|__   __|__
+     |   |   |   |
+    ga1 ga1 ga1 ga1  
+</pre>
+
+At this point, it should be easy to convince yourself that
+using the recipe on the right hand side of the associative law will
+built the exact same final tree.
+
+So binary trees are a monad.
+
+Haskell combines this monad with the Option monad to provide a monad
+called a
+[SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)
+that is intended to 
+represent non-deterministic computations as a tree.