edits

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 000772a..2ec15d6 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -22,7 +22,7 @@ the utility of replacing one monad with other.
 First, we'll be needing a lot of trees for the remainder of the
 course.  Here again is a type constructor for leaf-labeled, binary trees:
 
 First, we'll be needing a lot of trees for the remainder of the
 course.  Here again is a type constructor for leaf-labeled, binary trees:
 

-    type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
+    type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree);;

 
 [How would you adjust the type constructor to allow for labels on the
 internal nodes?]
 
 [How would you adjust the type constructor to allow for labels on the
 internal nodes?]


@@ -81,7 +81,7 @@ in place of `double`:
 
        let square i = i * i;;
        tree_map square t1;;
 
        let square i = i * i;;
        tree_map square t1;;

-       - : int tree =ppp
+       - : int tree =

        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
 Note that what `tree_map` does is take some unchanging contextual
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
 Note that what `tree_map` does is take some unchanging contextual


@@ -248,6 +248,37 @@ increments the state. When we give that same operations to our
 `tree_monadize` function, it then wraps an `int tree` in a box, one
 that does the same state-incrementing for each of its leaves.
 
 `tree_monadize` function, it then wraps an `int tree` in a box, one
 that does the same state-incrementing for each of its leaves.
 

+We can use the state monad to replace leaves with a number
+corresponding to that leave's ordinal position.  When we do so, we
+reveal the order in which the monadic tree forces evaluation:
+
+        # tree_monadize (fun a -> fun s -> (s+1, s+1)) t1 0;;
+        - : int tree * int =
+        (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5)
+
+The key thing to notice is that instead of copying `a` into the
+monadic box, we throw away the `a` and put a copy of the state in
+instead.
+
+Reversing the order requires reversing the order of the state_bind
+operations.  It's not obvious that this will type correctly, so think
+it through:
+
+       let rec tree_monadize_rev (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
+           match t with
+           | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b))
+           | Node (l, r) -> state_bind (tree_monadize f r) (fun r' ->
+                              state_bind (tree_monadize f l) (fun l' ->
+                                state_unit (Node (l', r'))));;
+
+        # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;;
+        - : int tree * int =
+        (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5)
+
+We will need below to depend on controlling the order in which nodes
+are visited when we use the continuation monad to solve the
+same-fringe problem.
+

 One more revealing example before getting down to business: replacing
 `state` everywhere in `tree_monadize` with `list` gives us
 
 One more revealing example before getting down to business: replacing
 `state` everywhere in `tree_monadize` with `list` gives us
 


@@ -327,6 +358,30 @@ generalizing the type of the Continuation monad to
 
 If you want to see how to parameterize the definition of the `tree_monadize` function, so that you don't have to keep rewriting it for each new monad, see [this code](/code/tree_monadize.ml).
 
 
 If you want to see how to parameterize the definition of the `tree_monadize` function, so that you don't have to keep rewriting it for each new monad, see [this code](/code/tree_monadize.ml).
 

+Using continuations to solve the same fringe problem
+----------------------------------------------------
+
+We've seen two solutions to the same fringe problem so far.  
+The simplest is to map each tree to a list of its leaves, then compare
+the lists.  But if the fringes differ in an early position, we've
+wasted our time visiting the rest of the tree. 
+
+The second solution was to use tree zippers and mutable state to
+simulate coroutines.  We would unzip the first tree until we found the
+next leaf, then store the zipper structure in the mutable variable
+while we turned our attention to the other tree.  Because we stop as
+soon as we find the first mismatched leaf, this solution does not have
+the flaw just mentioned of the solution that maps both trees to a list
+of leaves before beginning comparison.
+
+Since zippers are just continuations reified, we expect that the
+solution in terms of zippers can be reworked using continuations, and
+this is indeed the case.  To make this work in the most convenient
+way, we need to use the fully general type for continuations just mentioned.
+
+tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);;
+
+

 
 The Binary Tree monad
 ---------------------
 
 The Binary Tree monad
 ---------------------

diff --git a/monad_transformers.mdwn b/monad_transformers.mdwn
index efe35b2..b72c035 100644 (file)
--- a/monad_transformers.mdwn
+++ b/monad_transformers.mdwn
@@ -10,7 +10,12 @@ So far, we've defined monads as single-layered things. Though in the Groenendijk
        let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader =
                fun e -> (fun v -> f v e) (u e);;
 
        let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader =
                fun e -> (fun v -> f v e) (u e);;
 

-We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`, in order to factor out the parts where any Reader monad is being supplied as an argument to another function. Then if we want instead to add a Reader layer to some arbitrary other monad M, with its own M.unit and M.bind, here's how we do it:
+We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v
+e) (u e)`, in order to factor out the parts where any Reader monad is
+being supplied as an argument to another function, as illustrated in
+the `bind` function in the following example. Then if we want instead
+to add a Reader layer to some arbitrary other monad M, with its own
+M.unit and M.bind, here's how we do it:

 
        (* monadic operations for the ReaderT monadic transformer *)
 
 
        (* monadic operations for the ReaderT monadic transformer *)
 


@@ -29,7 +34,11 @@ We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`,
        let bind (u : ('a, M) readerT) (f : 'a -> ('b, M) readerT) : ('b, M) readerT =
                fun e -> M.bind (u e) (fun v -> f v e);;
 
        let bind (u : ('a, M) readerT) (f : 'a -> ('b, M) readerT) : ('b, M) readerT =
                fun e -> M.bind (u e) (fun v -> f v e);;
 

-Notice the key differences: where before we just returned `a`, now we instead return `M.unit a`. Where before we just supplied value `u e` of type `'a reader` as an argument to a function, now we instead `M.bind` the `'a reader` to that function. Notice also the differences in the types.
+Notice the key differences: where before we just returned `a`, now we
+instead return `M.unit a`. Where before we just supplied value `u e`
+of type `'a reader` as an argument to a function, now we instead
+`M.bind` the `'a reader` to that function. Notice also the differences
+in the types.

 
 What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad:
 
 
 What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad: