week9 reference tweak

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index d3ccc98..0d9e33d 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.
-
-From an engineering standpoint, we'll build a tree transformer that
+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, copying the tree 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 machine 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
 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
-of intensionality to an extensional grammar, but we have not yet seen
+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:
 
 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?]


@@ -28,14 +30,14 @@ 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)))

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

-       _|__  _|__
+       _|_   _|__

        |  |  |  |
        2  3  5  .
                _|__
        |  |  |  |
        2  3  5  .
                _|__


@@ -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,
-and maps that function over all the leaves in the tree, leaving the
-structure of the tree unchanged.  For instance:
+`tree_map` takes a tree and 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;;
 
        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)))
        


@@ -70,243 +72,427 @@ 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
-more flexible.  So the next step---combining the tree transducer with
-a reader monad---is to have the treemap function return a (monadized)
-tree that is ready to accept any `int->int` function and produce the
+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
+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.
 
 updated tree.
 

-\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
-
-       \f    .
-         ____|____
-         |       |
-         .       .
-       __|__   __|__
-       |   |   |   |
-       f2  f3  f5  .
-                 __|___
-                 |    |
-                 f7  f11
+       fun e ->    .
+              _____|____
+              |        |
+              .        .
+            __|___   __|___
+            |    |   |    |
+           e 2  e 3  e 5  .
+                        __|___
+                        |    |
+                       e 7  e 11

 
 That is, we want to transform the ordinary tree `t1` (of type `int
 
 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 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`.
-
-       type 'a reader = (int->int) -> 'a;;  (* mnemonic: e for environment *)
-       let reader_unit (x : 'a) : 'a reader = fun _ -> x;;
-       let reader_bind (u: 'a reader) (f : 'a -> 'c reader) : 'c reader = fun e -> f (u e) e;;
+tree`) into a reader monadic object of type `(int -> int) -> int
+tree`: something that, when you apply it to an `int -> int` function
+`e` returns an `int tree` in which each leaf `i` has been replaced
+with `e 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 `e`) 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;;
+       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 would be a simple matter to turn an *integer* into an `int reader`:
+
+       let asker : int -> int reader =
+         fun (a : int) ->
+           fun (modifier : int -> int) -> modifier a;;
+       asker 2 (fun i -> i + i);;
+       - : int = 4

 
 

-It's easy to figure out how to turn an `int` into an `int reader`:
+`asker a` is a monadic box that waits for an an environment (here, the argument `modifier`) and returns what that environment maps `a` to.

 
 

-       let int2int_reader (x : 'a): 'b reader = fun (op : 'a -> 'b) -> op x;;
-       int2int_reader 2 (fun i -> i + i);;
-       - : int = 4
+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.

 
 

-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
-function of type `int->int` to.
+But we can do this:

 
 

-       let rec treemonadizer (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+       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, such as `asker` or `reader_unit`---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:

 
 

-       # treemonadizer int2int_reader t1 (fun i -> i + i);;
+                     ____________
+         .           |    .     |
+       __|___  --->  |  __|___  |
+       |    |        |  |    |  |
+       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.
+
+       # tree_monadize asker 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
+asker 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 asker 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 transducer 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;;
+       # let incrementer = fun a ->
+           fun s -> (a, s+1);;
+       
+       # tree_monadize incrementer 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)

        
        

-           .
-        ___|___
-        |     |
-        .     .
-       _|__  _|__
-       |  |  |  |
-       2  3  5  .
-               _|__
-               |  |
-               7  11
+               .
+            ___|___
+            |     |
+            .     .
+        (  _|__  _|__     ,   5 )
+           |  |  |  |
+           2  3  5  .
+                   _|__
+                   |  |
+                   7  11
+
+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 `incrementer`
+takes an argument `a` and wraps it in an State monadic box that
+increments the store and leaves behind a wrapped `a`. 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 store-incrementing for each of its leaves.
+
+We can use the state monad to annotate 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 -> ((a,s+1), s+1)) t1 0;;
+       - : int tree * int =
+         (Node
+           (Node (Leaf (2, 1), Leaf (3, 2)),
+            Node
+             (Leaf (5, 3),
+              Node (Leaf (7, 4), Leaf (11, 5)))),
+         5)
+
+The key thing to notice is that instead of just wrapping `a` in the
+monadic box, we wrap a pair of `a` and the current store.
+
+Reversing the annotation 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' ->     (* R first *)
+                              state_bind (tree_monadize f l) (fun l'->    (* Then L  *)
+                                state_unit (Node (l', r'))));;
+       
+       # tree_monadize_rev (fun a -> fun s -> ((a,s+1), s+1)) t1 0;;
+       - : int tree * int =
+         (Node
+           (Node (Leaf (2, 5), Leaf (3, 4)),
+            Node
+             (Leaf (5, 3),
+              Node (Leaf (7, 2), Leaf (11, 1)))),
+         5)

 
 

-Notice that we've counted each internal node twice---it's a good
-exercise to adjust the code to count each node once.
+Later, we will talk more about controlling the order in which nodes are visited.

 
 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` lets us do:
+
+       # let decider i = if i = 2 then [20; 21] else [i];;
+       # tree_monadize decider t1;;
+       - : int tree List_monad.m =
+       [
+         Node (Node (Leaf 20, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)));
+         Node (Node (Leaf 21, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
+       ]

 
 

-       # treemonadizer (fun x -> [ [x; square x] ]) t1;;
-       - : int list tree list =
-       [Node
-         (Node (Leaf [2; 4], Leaf [3; 9]),
-          Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]

 
 Unlike the previous cases, instead of turning a tree into a function
 
 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.
+from some input to a result, this monadized tree gives us back a list of trees,
+one for each choice of `int`s for its leaves.

 
 Now for the main point.  What if we wanted to convert a tree to a list
 of leaves?
 
 
 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);;
+       type ('r,'a) continuation = ('a -> 'r) -> 'r;;
+       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 -> ('r,'b) continuation) (t : 'a tree) : ('r,'b tree) 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 `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.

 
 

-We use the continuation monad described above, and insert the
-`continuation` type in the appropriate place in the `treemonadizer` code.
-We then compute:
+So for example, we compute:

 
 

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

        - : 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 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?
+
+Soon we'll return to the same-fringe problem.  Since the
+simple but inefficient way to solve it is to map each tree to a list
+of its leaves, this transformation is on the path to a more efficient
+solution.  We'll just have to figure out how to postpone computing the
+tail of the list until it's needed...

 
 

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

        (* Counting leaves *)
        (* 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
 
        - : int = 5
 

-We could simulate the tree state example too, but it would require
-generalizing the type of the continuation monad to
+It's not immediately obvious to us how to simulate the List monadization of the tree using this technique.
+
+We could simulate the tree annotating example by setting the relevant 
+type to `(store -> 'result, 'a) continuation`.
+
+Andre Filinsky has proposed that the continuation monad is
+able to simulate any other monad (Google for "mother of all monads").
+
+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).
+
+The idea of using continuations to characterize natural language meaning
+------------------------------------------------------------------------

 
 

-       type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
+We might a philosopher or a linguist be interested in continuations,
+especially if efficiency of computation is usually not an issue?
+Well, the application of continuations to the same-fringe problem
+shows that continuations can manage order of evaluation in a
+well-controlled manner.  In a series of papers, one of us (Barker) and
+Ken Shan have argued that a number of phenomena in natural langauge
+semantics are sensitive to the order of evaluation.  We can't
+reproduce all of the intricate arguments here, but we can give a sense
+of how the analyses use continuations to achieve an analysis of
+natural language meaning.

 
 

-The binary tree monad
----------------------
+**Quantification and default quantifier scope construal**.  

 
 

-Of course, by now you may have realized that we have discovered a new
-monad, the binary tree monad:
+We saw in the copy-string example ("abSd") and in the same-fringe example that
+local properties of a structure (whether a character is `'S'` or not, which
+integer occurs at some leaf position) can control global properties of
+the computation (whether the preceeding string is copied or not,
+whether the computation halts or proceeds).  Local control of
+surrounding context is a reasonable description of in-situ
+quantification.
+
+    (1) John saw everyone yesterday.
+
+This sentence means (roughly)
+
+    forall x . yesterday(saw x) john
+
+That is, the quantifier *everyone* contributes a variable in the
+direct object position, and a universal quantifier that takes scope
+over the whole sentence.  If we have a lexical meaning function like
+the following:
+
+       let lex (s:string) k = match s with 
+         | "everyone" -> Node (Leaf "forall x", k "x")
+         | "someone" -> Node (Leaf "exists y", k "y")
+         | _ -> k s;;
+
+Then we can crudely approximate quantification as follows:
+
+       # let sentence1 = Node (Leaf "John", 
+                                                 Node (Node (Leaf "saw", 
+                                                                         Leaf "everyone"), 
+                                                               Leaf "yesterday"));;
+
+       # tree_monadize lex sentence1 (fun x -> x);;
+       - : string tree =
+       Node
+        (Leaf "forall x",
+         Node (Leaf "John", Node (Node (Leaf "saw", Leaf "x"), Leaf "yesterday")))
+
+In order to see the effects of evaluation order, 
+observe what happens when we combine two quantifiers in the same
+sentence:
+
+       # let sentence2 = Node (Leaf "everyone", Node (Leaf "saw", Leaf "someone"));;
+       # tree_monadize lex sentence2 (fun x -> x);;
+       - : string tree =
+       Node
+        (Leaf "forall x",
+         Node (Leaf "exists y", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
+
+The universal takes scope over the existential.  If, however, we
+replace the usual `tree_monadizer` with `tree_monadizer_rev`, we get
+inverse scope:
+
+       # tree_monadize_rev lex sentence2 (fun x -> x);;
+       - : string tree =
+       Node
+        (Leaf "exists y",
+         Node (Leaf "forall x", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
+
+There are many crucially important details about quantification that
+are being simplified here, and the continuation treatment used here is not
+scalable for a number of reasons.  Nevertheless, it will serve to give
+an idea of how continuations can provide insight into the behavior of
+quantifiers.
+
+
+The Tree monad
+==============
+
+Of course, by now you may have realized that we are working with a new
+monad, the binary, leaf-labeled 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`:
-
-\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
+except that each leaf `a` has been replaced with the tree returned by `f a`:

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

-                     |   |                     |   |
-                     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
 


@@ -318,35 +504,32 @@ 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)))  ))
-

                                                   .
                                               ____|____
                  .               .            |       |
        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

-built the exact same final tree.
+build the exact same final tree.

 
 So binary trees are a monad.
 
 
 So binary trees are a monad.
 


@@ -355,3 +538,9 @@ 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\_monadize?
+--------------------------------------------
+
+Our different implementations of `tree_monadize` above were different *layerings* of the Tree monad with other monads (Reader, State, List, and Continuation). We'll explore that further here: [[Monad Transformers]].
+