edits

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 20039c2..2ec15d6 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -3,24 +3,26 @@
 Manipulating trees with monads
 ------------------------------
 
 Manipulating trees with monads
 ------------------------------
 

-This topic develops an idea based on a detailed suggestion of Ken
-Shan's.  We'll build a series of functions that operate on trees,
-doing various things, including replacing leaves, counting nodes, and
-converting a tree to a list of leaves.  The end result will be an
-application for continuations.
+This topic develops an idea based on a suggestion of Ken Shan's.
+We'll build a series of functions that operate on trees, doing various
+things, including updating leaves with a Reader monad, counting nodes
+with a State monad, replacing leaves with a List monad, and converting
+a tree into a list of leaves with a Continuation monad.  It will turn
+out that the continuation monad can simulate the behavior of each of
+the other monads.

 
 From an engineering standpoint, we'll build a tree transformer that
 deals in monads.  We can modify the behavior of the system by swapping
 one monad for another.  We've already seen how adding a monad can add
 a layer of funtionality without disturbing the underlying system, for
 
 From an engineering standpoint, we'll build a tree transformer that
 deals in monads.  We can modify the behavior of the system by swapping
 one monad for another.  We've already seen how adding a monad can add
 a layer of funtionality without disturbing the underlying system, for

-instance, in the way that the reader monad allowed us to add a layer
+instance, in the way that the Reader monad allowed us to add a layer

 of intensionality to an extensional grammar, but we have not yet seen
 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:
 
 of intensionality to an extensional grammar, but we have not yet seen
 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:
 

-    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?]


@@ -28,9 +30,9 @@ internal nodes?]
 We'll be using trees where the nodes are integers, e.g.,
 
 
 We'll be using trees where the nodes are integers, e.g.,
 
 

-       let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
-                      (Node ((Leaf 5),(Node ((Leaf 7),
-                                             (Leaf 11))))))
+       let t1 = Node (Node (Leaf 2, Leaf 3),
+                      Node (Leaf 5, Node (Leaf 7,
+                                           Leaf 11)))

            .
         ___|___
         |     |
            .
         ___|___
         |     |


@@ -44,18 +46,18 @@ We'll be using trees where the nodes are integers, e.g.,
 
 Our first task will be to replace each leaf with its double:
 
 
 Our first task will be to replace each leaf with its double:
 

-       let rec treemap (newleaf : 'a -> 'b) (t : 'a tree) : 'b tree =
+       let rec tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree =

          match t with
          match t with

-           | Leaf x -> Leaf (newleaf x)
-           | Node (l, r) -> Node ((treemap newleaf l),
-                                  (treemap newleaf r));;
+           | Leaf i -> Leaf (leaf_modifier i)
+           | Node (l, r) -> Node (tree_map leaf_modifier l,
+                                  tree_map leaf_modifier r);;

 
 

-`treemap` takes a function that transforms old leaves into new leaves,
+`tree_map` takes a function that transforms old leaves into new leaves,

 and maps that function over all the leaves in the tree, leaving the
 structure of the tree unchanged.  For instance:
 
        let double i = i + i;;
 and maps that function over all the leaves in the tree, leaving the
 structure of the tree unchanged.  For instance:
 
        let double i = i + i;;

-       treemap double t1;;
+       tree_map double t1;;

        - : int tree =
        Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
        
        - : int tree =
        Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
        


@@ -70,28 +72,30 @@ structure of the tree unchanged.  For instance:
                |    |
                14   22
 
                |    |
                14   22
 

-We could have built the doubling operation right into the `treemap`
-code.  However, because what to do to each leaf is a parameter, we can
-decide to do something else to the leaves without needing to rewrite
-`treemap`.  For instance, we can easily square each leaf instead by
-supplying the appropriate `int -> int` operation in place of `double`:
+We could have built the doubling operation right into the `tree_map`
+code.  However, because we've made what to do to each leaf a
+parameter, we can decide to do something else to the leaves without
+needing to rewrite `tree_map`.  For instance, we can easily square
+each leaf instead by supplying the appropriate `int -> int` operation
+in place of `double`:

 
 

-       let square x = x * x;;
-       treemap square t1;;
-       - : int tree =ppp
+       let square i = i * i;;
+       tree_map square t1;;
+       - : int tree =

        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 

-Note that what `treemap` does is take some global, contextual
+Note that what `tree_map` does is take some unchanging contextual

 information---what to do to each leaf---and supplies that information
 information---what to do to each leaf---and supplies that information

-to each subpart of the computation.  In other words, `treemap` has the
-behavior of a reader monad.  Let's make that explicit.
+to each subpart of the computation.  In other words, `tree_map` has the
+behavior of a Reader monad.  Let's make that explicit.

 
 

-In general, we're on a journey of making our treemap function more and
+In general, we're on a journey of making our `tree_map` function more and

 more flexible.  So the next step---combining the tree transformer with
 more flexible.  So the next step---combining the tree transformer with

-a reader monad---is to have the treemap function return a (monadized)
+a Reader monad---is to have the `tree_map` function return a (monadized)

 tree that is ready to accept any `int -> int` function and produce the
 updated tree.
 
 tree that is ready to accept any `int -> int` function and produce the
 updated tree.
 

+\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11))))

 
        \f      .
           _____|____
 
        \f      .
           _____|____


@@ -105,103 +109,180 @@ updated tree.
                    f 7  f 11
 
 That is, we want to transform the ordinary tree `t1` (of type `int
                    f 7  f 11
 
 That is, we want to transform the ordinary tree `t1` (of type `int

-tree`) into a reader object of type `(int -> int) -> int tree`: something
-that, when you apply it to an `int -> int` function `f` returns an `int
-tree` in which each leaf `x` has been replaced with `f x`.
-
-With previous readers, we always knew which kind of environment to
-expect: either an assignment function (the original calculator
-simulation), a world (the intensionality monad), an integer (the
-Jacobson-inspired link monad), etc.  In this situation, it will be
-enough for now to expect that our reader will expect a function of
-type `int -> int`.
+tree`) into a reader monadic object of type `(int -> int) -> int
+tree`: something that, when you apply it to an `int -> int` function
+`f` returns an `int tree` in which each leaf `i` has been replaced
+with `f i`.
+
+[Application note: this kind of reader object could provide a model
+for Kaplan's characters.  It turns an ordinary tree into one that
+expects contextual information (here, the `λ f`) that can be
+used to compute the content of indexicals embedded arbitrarily deeply
+in the tree.]
+
+With our previous applications of the Reader monad, we always knew
+which kind of environment to expect: either an assignment function, as
+in the original calculator simulation; a world, as in the
+intensionality monad; an individual, as in the Jacobson-inspired link
+monad; etc.  In the present case, we expect that our "environment"
+will be some function of type `int -> int`. "Looking up" some `int` in
+the environment will return us the `int` that comes out the other side
+of that function.

 
        type 'a reader = (int -> int) -> 'a;;  (* mnemonic: e for environment *)
        let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
        let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> f (u e) e;;
 
 
        type 'a reader = (int -> int) -> 'a;;  (* mnemonic: e for environment *)
        let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
        let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> f (u e) e;;
 

-It's easy to figure out how to turn an `int` into an `int reader`:
+It would be a simple matter to turn an *integer* into an `int reader`:

 
 

-       let int2int_reader (x : 'a) : 'b reader = fun (op : 'a -> 'b) -> op x;;
-       int2int_reader 2 (fun i -> i + i);;
+       let int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;;
+       int_readerize 2 (fun i -> i + i);;

        - : int = 4
 
        - : int = 4
 

-But what do we do when the integers are scattered over the leaves of a
-tree?  A binary tree is not the kind of thing that we can apply a
+But how do we do the analagous transformation when our `int`s are scattered over the leaves of a tree? How do we turn an `int tree` into a reader?
+A tree is not the kind of thing that we can apply a

 function of type `int -> int` to.
 
 function of type `int -> int` to.
 

-       let rec treemonadizer (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+But we can do this:
+
+       let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =

            match t with
            match t with

-           | Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x'))
-           | Node (l, r) -> reader_bind (treemonadizer f l) (fun x ->
-                              reader_bind (treemonadizer f r) (fun y ->
-                                reader_unit (Node (x, y))));;
+           | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b))
+           | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' ->
+                              reader_bind (tree_monadize f r) (fun r' ->
+                                reader_unit (Node (l', r'))));;

 
 This function says: give me a function `f` that knows how to turn
 
 This function says: give me a function `f` that knows how to turn

-something of type `'a` into an `'b reader`, and I'll show you how to
-turn an `'a tree` into an `'a tree reader`.  In more fanciful terms,
-the `treemonadizer` function builds plumbing that connects all of the
-leaves of a tree into one connected monadic network; it threads the
-monad through the leaves.
+something of type `'a` into an `'b reader`---this is a function of the same type that you could bind an `'a reader` to---and I'll show you how to
+turn an `'a tree` into an `'b tree reader`.  That is, if you show me how to do this:
+
+                     ------------
+         1     --->  |    1     |
+                     ------------
+
+then I'll give you back the ability to do this:
+
+                     ____________
+         .           |    .     |
+       __|___  --->  |  __|___  |
+       |    |        |  |    |  |
+       1    2        |  1    2  |
+                     ------------
+
+And how will that boxed tree behave? Whatever actions you perform on it will be transmitted down to corresponding operations on its leaves. For instance, our `int reader` expects an `int -> int` environment. If supplying environment `e` to our `int reader` doubles the contained `int`:
+
+                     ------------
+         1     --->  |    1     |  applied to e  ~~>  2
+                     ------------
+
+Then we can expect that supplying it to our `int tree reader` will double all the leaves:
+
+                     ____________
+         .           |    .     |                      .
+       __|___  --->  |  __|___  | applied to e  ~~>  __|___
+       |    |        |  |    |  |                    |    |
+       1    2        |  1    2  |                    2    4
+                     ------------
+
+In more fanciful terms, the `tree_monadize` function builds plumbing that connects all of the leaves of a tree into one connected monadic network; it threads the
+`'b reader` monad through the original tree's leaves.

 
 

-       # treemonadizer int2int_reader t1 (fun i -> i + i);;
+       # tree_monadize int_readerize t1 double;;

        - : int tree =
        Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
 
 Here, our environment is the doubling function (`fun i -> i + i`).  If
        - : int tree =
        Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
 
 Here, our environment is the doubling function (`fun i -> i + i`).  If

-we apply the very same `int tree reader` (namely, `treemonadizer
-int2int_reader t1`) to a different `int -> int` function---say, the
+we apply the very same `int tree reader` (namely, `tree_monadize
+int_readerize t1`) to a different `int -> int` function---say, the

 squaring function, `fun i -> i * i`---we get an entirely different
 result:
 
 squaring function, `fun i -> i * i`---we get an entirely different
 result:
 

-       # treemonadizer int2int_reader t1 (fun i -> i * i);;
+       # tree_monadize int_readerize t1 square;;

        - : int tree =
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
        - : int tree =
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 

-Now that we have a tree transformer that accepts a monad as a
+Now that we have a tree transformer that accepts a *reader* monad as a

 parameter, we can see what it would take to swap in a different monad.
 parameter, we can see what it would take to swap in a different monad.

-For instance, we can use a state monad to count the number of nodes in
+
+For instance, we can use a State monad to count the number of leaves in

 the tree.
 
        type 'a state = int -> 'a * int;;
 the tree.
 
        type 'a state = int -> 'a * int;;

-       let state_unit x i = (x, i+.5);;
-       let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
+       let state_unit a = fun s -> (a, s);;
+       let state_bind u f = fun s -> let (a, s') = u s in f a s';;

 
 

-Gratifyingly, we can use the `treemonadizer` function without any
+Gratifyingly, we can use the `tree_monadize` function without any

 modification whatsoever, except for replacing the (parametric) type
 modification whatsoever, except for replacing the (parametric) type

-`reader` with `state`:
+`'b reader` with `'b state`, and substituting in the appropriate unit and bind:

 
 

-       let rec treemonadizer (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
+       let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =

            match t with
            match t with

-           | Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x'))
-           | Node (l, r) -> state_bind (treemonadizer f l) (fun x ->
-                              state_bind (treemonadizer f r) (fun y ->
-                                state_unit (Node (x, y))));;
+           | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b))
+           | Node (l, r) -> state_bind (tree_monadize f l) (fun l' ->
+                              state_bind (tree_monadize f r) (fun r' ->
+                                state_unit (Node (l', r'))));;

 
 

-Then we can count the number of nodes in the tree:
+Then we can count the number of leaves in the tree:

 
 

-       # treemonadizer state_unit t1 0;;
+       # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;;

        - : int tree * int =
        - : int tree * int =

-       (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
+       (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5)

        
            .
         ___|___
         |     |
         .     .
        
            .
         ___|___
         |     |
         .     .

-       _|__  _|__
+       _|__  _|__         , 5

        |  |  |  |
        2  3  5  .
                _|__
                |  |
                7  11
 
        |  |  |  |
        2  3  5  .
                _|__
                |  |
                7  11
 

-Notice that we've counted each internal node twice---it's a good
-exercise to adjust the code to count each node once.
+Note that the value returned is a pair consisting of a tree and an
+integer, 5, which represents the count of the leaves in the tree.
+
+Why does this work? Because the operation `fun a -> fun s -> (a, s+1)`
+takes an `int` and wraps it in an `int state` monadic box that
+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.
+
+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
 
 One more revealing example before getting down to business: replacing

-`state` everywhere in `treemonadizer` with `list` gives us
+`state` everywhere in `tree_monadize` with `list` gives us

 
 

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


@@ -209,103 +290,138 @@ One more revealing example before getting down to business: replacing
 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with
 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with

-a list of `int`'s.
+a list of `int`'s. We might also have done this with a Reader monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List monad an operation like `fun -> [ i; [2*i; 3*i] ]`. Use small trees for your experiment.
+
+[Why is the argument to `tree_monadize` `int -> int list list` instead
+of `int -> int list`?  Well, as usual, the List monad bind operation
+will erase the outer list box, so if we want to replace the leaves
+with lists, we have to nest the replacement lists inside a disposable
+box.]

 
 Now for the main point.  What if we wanted to convert a tree to a list
 of leaves?
 
        type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
 
 Now for the main point.  What if we wanted to convert a tree to a list
 of leaves?
 
        type ('a, 'r) continuation = ('a -> 'r) -> 'r;;

-       let continuation_unit x c = c x;;
-       let continuation_bind u f c = u (fun a -> f a c);;
+       let continuation_unit a = fun k -> k a;;
+       let continuation_bind u f = fun k -> u (fun a -> f a k);;

        
        

-       let rec treemonadizer (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =
+       let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =

            match t with
            match t with

-           | Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x'))
-           | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x ->
-                              continuation_bind (treemonadizer f r) (fun y ->
-                                continuation_unit (Node (x, y))));;
+           | Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b))
+           | Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' ->
+                              continuation_bind (tree_monadize f r) (fun r' ->
+                                continuation_unit (Node (l', r'))));;

 
 

-We use the continuation monad described above, and insert the
-`continuation` type in the appropriate place in the `treemonadizer` code.
-We then compute:
+We use the Continuation monad described above, and insert the
+`continuation` type in the appropriate place in the `tree_monadize` code. Then if we give the `tree_monadize` function an operation that converts `int`s into `'b`-wrapping Continuation monads, it will give us back a way to turn `int tree`s into corresponding `'b tree`-wrapping Continuation monads.

 
 

-       # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
+So for example, we compute:
+
+       # tree_monadize (fun a -> fun k -> a :: k a) t1 (fun t -> []);;

        - : int list = [2; 3; 5; 7; 11]
 
        - : int list = [2; 3; 5; 7; 11]
 

-We have found a way of collapsing a tree into a list of its leaves.
+We have found a way of collapsing a tree into a list of its leaves. Can you trace how this is working? Think first about what the operation `fun a -> fun k -> a :: k a` does when you apply it to a plain `int`, and the continuation `fun _ -> []`. Then given what we've said about `tree_monadize`, what should we expect `tree_monadize (fun a -> fun k -> a :: k a` to do?

 
 

-The continuation monad is amazingly flexible; we can use it to
+The Continuation monad is amazingly flexible; we can use it to

 simulate some of the computations performed above.  To see how, first
 simulate some of the computations performed above.  To see how, first

-note that an interestingly uninteresting thing happens if we use the
-continuation unit as our first argument to `treemonadizer`, and then
+note that an interestingly uninteresting thing happens if we use
+`continuation_unit` as our first argument to `tree_monadize`, and then

 apply the result to the identity function:
 
 apply the result to the identity function:
 

-       # treemonadizer continuation_unit t1 (fun x -> x);;
+       # tree_monadize continuation_unit t1 (fun t -> t);;

        - : int tree =
        Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
 
 That is, nothing happens.  But we can begin to substitute more
        - : int tree =
        Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
 
 That is, nothing happens.  But we can begin to substitute more

-interesting functions for the first argument of `treemonadizer`:
+interesting functions for the first argument of `tree_monadize`:

 
        (* Simulating the tree reader: distributing a operation over the leaves *)
 
        (* Simulating the tree reader: distributing a operation over the leaves *)

-       # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
+       # tree_monadize (fun a -> fun k -> k (square a)) t1 (fun t -> t);;

        - : int tree =
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
        (* Simulating the int list tree list *)
        - : int tree =
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
        (* Simulating the int list tree list *)

-       # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);;
+       # tree_monadize (fun a -> fun k -> k [a; square a]) t1 (fun t -> t);;

        - : int list tree =
        Node
         (Node (Leaf [2; 4], Leaf [3; 9]),
          Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
 
        (* Counting leaves *)
        - : int list tree =
        Node
         (Node (Leaf [2; 4], Leaf [3; 9]),
          Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
 
        (* Counting leaves *)

-       # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
+       # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);;

        - : int = 5
 
 We could simulate the tree state example too, but it would require
        - : int = 5
 
 We could simulate the tree state example too, but it would require

-generalizing the type of the continuation monad to
+generalizing the type of the Continuation monad to
+
+       type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;
+
+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.

 
 

-       type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
+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.

 
 

-The binary tree monad
+tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);;
+
+
+
+The Binary Tree monad

 ---------------------
 
 Of course, by now you may have realized that we have discovered a new
 ---------------------
 
 Of course, by now you may have realized that we have discovered a new

-monad, the binary tree monad:
+monad, the Binary Tree monad.  Just as mere lists are in fact a monad,
+so are trees.  Here is the type constructor, unit, and bind:

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

-       let tree_unit (x: 'a) = Leaf x;;
+       let tree_unit (a: 'a) : 'a tree = Leaf a;;

        let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
            match u with
        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));;
+           | Leaf a -> f a
+           | Node (l, r) -> Node (tree_bind l f, tree_bind r f);;

 
 For once, let's check the Monad laws.  The left identity law is easy:
 
 
 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
+    Left identity: bind (unit a) f = bind (Leaf a) f = f a

 
 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`,
 
 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`:
+except that each leaf `a` has been replaced with `f a`:

 
 

-\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
+\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5)))

 
                        .                         .
                      __|__                     __|__
                      |   |                     |   |
 
                        .                         .
                      __|__                     __|__
                      |   |                     |   |

-                     a1  .                    fa1  .
+                     a1  .                   f a1  .

                         _|__                     __|__
                         |  |                     |   |
                         _|__                     __|__
                         |  |                     |   |

-                        .  a5                    .  fa5
+                        .  a5                    .  f a5

           bind         _|__       f   =        __|__
                        |  |                    |   |
           bind         _|__       f   =        __|__
                        |  |                    |   |

-                       .  a4                   .  fa4
+                       .  a4                   .  f a4

                      __|__                   __|___
                      |   |                   |    |
                      __|__                   __|___
                      |   |                   |    |

-                     a2  a3                 fa2  fa3
+                     a2  a3                f a2  f a3

 
 Given this equivalence, the right identity law
 
 
 Given this equivalence, the right identity law
 


@@ -317,31 +433,31 @@ falls out once we realize that
 
 As for the associative law,
 
 
 As for the associative law,
 

-       Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
+       Associativity: bind (bind u f) g = bind u (\a. bind (f a) g)

 
 we'll give an example that will show how an inductive proof would
 proceed.  Let `f a = Node (Leaf a, Leaf a)`.  Then
 
 
 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)))  ))
+\tree (. (. (. (. (a1) (a2)))))
+\tree (. (. (. (. (a1) (a1)) (. (a1) (a1)))))

 
                                                   .
                                               ____|____
                  .               .            |       |
        bind    __|__   f  =    __|_    =      .       .
                |   |           |   |        __|__   __|__
 
                                                   .
                                               ____|____
                  .               .            |       |
        bind    __|__   f  =    __|_    =      .       .
                |   |           |   |        __|__   __|__

-               a1  a2         fa1 fa2       |   |   |   |
+               a1  a2        f a1 f a2      |   |   |   |

                                             a1  a1  a1  a1
 
 Now when we bind this tree to `g`, we get
 
                                             a1  a1  a1  a1
 
 Now when we bind this tree to `g`, we get
 

-                  .
-              ____|____
-              |       |
-              .       .
-            __|__   __|__
-            |   |   |   |
-           ga1 ga1 ga1 ga1
+                   .
+              _____|______
+              |          |
+              .          .
+            __|__      __|__
+            |   |      |   |
+          g a1 g a1  g a1 g a1

 
 At this point, it should be easy to convince yourself that
 using the recipe on the right hand side of the associative law will
 
 At this point, it should be easy to convince yourself that
 using the recipe on the right hand side of the associative law will


@@ -354,3 +470,29 @@ 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.
 
 [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.
 

+
+What's this have to do with tree\_mondadize?
+--------------------------------------------
+
+So we've defined a Tree monad:
+
+       type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
+       let tree_unit (a: 'a) : 'a tree = Leaf a;;
+       let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
+           match u with
+           | Leaf a -> f a
+           | Node (l, r) -> Node (tree_bind l f, tree_bind r f);;
+
+What's this have to do with the `tree_monadize` functions we defined earlier?
+
+       let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+           match t with
+           | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b))
+           | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' ->
+                              reader_bind (tree_monadize f r) (fun r' ->
+                                reader_unit (Node (l', r'))));;
+
+... and so on for different monads?
+
+The answer is that each of those `tree_monadize` functions is adding a Tree monad *layer* to a pre-existing Reader (and so on) monad. We discuss that further here: [[Monad Transformers]].
+