changes

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 10c9556..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
 instance, in the way that the Reader monad allowed us to add a layer
 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
+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?]


@@ -30,7 +32,7 @@ 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,
 
        let t1 = Node (Node (Leaf 2, Leaf 3),
                       Node (Leaf 5, Node (Leaf 7,

-                                             Leaf 11)))
+                                           Leaf 11)))

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


@@ -50,9 +52,9 @@ Our first task will be to replace each leaf with its double:
            | Node (l, r) -> Node (tree_map leaf_modifier l,
                                   tree_map leaf_modifier r);;
 
            | Node (l, r) -> Node (tree_map leaf_modifier l,
                                   tree_map leaf_modifier r);;
 

-`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:
+`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;;
        tree_map double t1;;
 
        let double i = i + i;;
        tree_map double t1;;


@@ -71,14 +73,15 @@ structure of the tree unchanged.  For instance:
                14   22
 
 We could have built the doubling operation right into the `tree_map`
                14   22
 
 We could have built the doubling operation right into the `tree_map`

-code.  However, because we've left what to do to each leaf as 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`:
+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 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


@@ -92,40 +95,54 @@ 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 2  f 3  f 5  .
-                    __|___
-                    |    |
-                   f 7  f 11
+       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 `f` returns an `int
-tree` in which each leaf `i` has been replaced with `f i`.
-
-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 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 *)
+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_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;;
+       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`:
 
 
 It would be a simple matter to turn an *integer* into an `int reader`:
 

-       let int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;;
-       int_readerize 2 (fun i -> i + i);;
+       let asker : int -> int reader =
+         fun (a : int) ->
+           fun (modifier : int -> int) -> modifier a;;
+       asker 2 (fun i -> i + i);;

        - : int = 4
 
        - : int = 4
 

-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?
+`asker a` is a monadic box that waits for an an environment (here, the argument `modifier`) and returns what that environment maps `a` to.
+
+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.
 
 A tree is not the kind of thing that we can apply a
 function of type `int -> int` to.
 


@@ -139,7 +156,7 @@ But we can do this:
                                 reader_unit (Node (l', r'))));;
 
 This function says: give me a function `f` that knows how to turn
                                 reader_unit (Node (l', r'))));;
 
 This function says: give me a function `f` that knows how to turn

-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
+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:
 
                      ------------
 turn an `'a tree` into an `'b tree reader`.  That is, if you show me how to do this:
 
                      ------------


@@ -173,17 +190,17 @@ Then we can expect that supplying it to our `int tree reader` will double all th
 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.
 
 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 int_readerize t1 double;;
+       # 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
 we apply the very same `int tree reader` (namely, `tree_monadize
        - : 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, `tree_monadize

-int_readerize t1`) to a different `int -> int` function---say, the
+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:
 

-       # tree_monadize int_readerize t1 square;;
+       # 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)))
 


@@ -210,50 +227,95 @@ modification whatsoever, except for replacing the (parametric) type
 
 Then we can count the number of leaves in the tree:
 
 
 Then we can count the number of leaves in the tree:
 

-       # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;;
+       # let incrementer = fun a ->
+           fun s -> (a, s+1);;
+       
+       # tree_monadize incrementer t1 0;;

        - : int tree * int =
        (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5)
        
        - : int tree * int =
        (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)

 
 

-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.
+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.

 
 

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

 
 

-       # tree_monadize (fun i -> [ [i; square i] ]) t1;;
-       - : int list tree list =
-       [Node
-         (Node (Leaf [2; 4], Leaf [3; 9]),
-          Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]
+Later, we will talk more about controlling the order in which nodes are visited.

 
 

-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. 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.
+One more revealing example before getting down to business: replacing
+`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)))
+       ]

 
 

-<!--
-FIXME: We don't make it clear why the fun has to be int -> int list list, instead of int -> int list
--->

 
 

+Unlike the previous cases, instead of turning a tree into a function
+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;;
+       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 continuation_unit a = fun k -> k a;;
        let continuation_bind u f = fun k -> u (fun a -> f a k);;
        

-       let rec tree_monadize (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
            | Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b))
            | Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' ->
            match t with
            | Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b))
            | Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' ->


@@ -265,10 +327,21 @@ We use the Continuation monad described above, and insert the
 
 So for example, we compute:
 
 
 So for example, we compute:
 

-       # tree_monadize (fun a -> fun k -> a :: k 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. 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?
+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
 simulate some of the computations performed above.  To see how, first
 
 The Continuation monad is amazingly flexible; we can use it to
 simulate some of the computations performed above.  To see how, first


@@ -288,30 +361,107 @@ interesting functions for the first argument of `tree_monadize`:
        - : 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 *)
-       # 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 *)
        # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);;
        - : int = 5
 
        (* Counting leaves *)
        # 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
-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`.

 
 

-       type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;
+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).
 
 
 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
+------------------------------------------------------------------------
+
+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.
+
+**Quantification and default quantifier scope construal**.  
+
+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 Binary Tree monad
----------------------

 
 

-Of course, by now you may have realized that we have discovered a new
-monad, the Binary Tree monad:
+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);;
        let tree_unit (a: 'a) : 'a tree = Leaf a;;
 
        type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
        let tree_unit (a: 'a) : 'a tree = Leaf a;;


@@ -328,22 +478,20 @@ 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`,
 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 `f a`:
-
-\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5)))
+except that each leaf `a` has been replaced with the tree returned by `f a`:

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

-                     |   |                     |   |
+                     |   |                    /\   |

                      a1  .                   f a1  .
                         _|__                     __|__
                      a1  .                   f a1  .
                         _|__                     __|__

-                        |  |                     |   |
+                        |  |                     |   /\

                         .  a5                    .  f a5
           bind         _|__       f   =        __|__
                         .  a5                    .  f a5
           bind         _|__       f   =        __|__

-                       |  |                    |   |
+                       |  |                    |   /\

                        .  a4                   .  f a4
                      __|__                   __|___
                        .  a4                   .  f a4
                      __|__                   __|___

-                     |   |                   |    |
+                     |   |                  /\    /\

                      a2  a3                f a2  f a3
 
 Given this equivalence, the right identity law
                      a2  a3                f a2  f a3
 
 Given this equivalence, the right identity law


@@ -361,9 +509,6 @@ As for the associative law,
 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)))))
-

                                                   .
                                               ____|____
                  .               .            |       |
                                                   .
                                               ____|____
                  .               .            |       |


@@ -384,7 +529,7 @@ Now when we bind this tree to `g`, we get
 
 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.
 


@@ -394,135 +539,8 @@ called a
 that is intended to represent non-deterministic computations as a tree.
 
 
 that is intended to represent non-deterministic computations as a tree.
 
 

-What's this have to do with tree\_mondadize?
+What's this have to do with tree\_monadize?

 --------------------------------------------
 
 --------------------------------------------
 

-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. So far, we've defined monads as single-layered things. (Though in the Groenendijk, Stokhoff, and Veltmann application, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way.) But in practice, one often wants to combine the abilities of several monads. Corresponding to each monad like Reader, there's a corresponding ReaderT **monad transformer**. That takes an existing monad M and adds a Reader monad layer to it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad:
-
-       (* monadic operations for the Reader monad *)
-
-       type 'a reader =
-               env -> 'a;;
-       let unit (a : 'a) : 'a reader =
-               fun e -> a;;
-       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)` 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:
-
-       (* monadic operations for the ReaderT monadic transformer *)
-
-       (* We're not giving valid OCaml code, but rather something
-        * that's conceptually easier to digest.
-        * How you really need to write this in OCaml is more circuitious...
-        * see http://lambda.jimpryor.net/code/tree_monadize.ml for some details. *)
-
-       type ('a, M) readerT =
-               env -> 'a M;;
-       (* this is just an 'a M reader; but that doesn't generalize *)
-
-       let unit (a : 'a) : ('a, M) readerT =
-               fun e -> M.unit a;;
-
-       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 return `M.unit a`. Where before we just supplied a 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:
-
-       type 'a identity = 'a;;
-       let unit (a : 'a) : 'a = a;;
-       let bind (u : 'a) (f : 'a -> 'b) : 'b = f u;;
-
-and you used the ReaderT transformer to add a Reader monad layer to the Identity monad. What do you suppose you would get?
-
-The relations between the State monad and the StateT monadic transformer are parallel:
-
-       (* monadic operations for the State monad *)
-
-       type 'a state =
-               store -> ('a * store);;
-
-       let unit (a : 'a) : 'a state =
-               fun s -> (a, s);;
-
-       let bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
-               fun s -> (fun (a, s') -> f a s') (u s);;
-
-We've used `(fun (a, s') -> f a s') (u s)` instead of the more familiar `let (a, s') = u s in f a s'` in order to factor out the part where a value of type `'a state` is supplied as an argument to a function. Now StateT will be:
-
-       (* monadic operations for the StateT monadic transformer *)
-
-       type ('a, M) stateT =
-               store -> ('a * store) M;;
-       (* notice this is not an 'a M state *)
-
-       let unit (a : 'a) : ('a, M) stateT =
-               fun s -> M.unit (a, s);;
-
-       let bind (u : ('a, M) stateT) (f : 'a -> ('b, M) stateT) : ('b, M) stateT =
-               fun s -> M.bind (u s) (fun (a, s') -> f a s');;
-
-Do you see the pattern? Where ordinarily we'd return an `'a` value, now we instead return an `'a M` value. Where ordinarily we'd supply a `'a state` value as an argument to a function, now we instead `M.bind` it to that function.
-
-Okay, now let's do the same thing for our Tree monad.
-
-       (* monadic operations for the Tree monad *)
-
-       type 'a tree =
-               Leaf of 'a | Node of ('a tree) * ('a tree);;
-
-       let unit (a: 'a) : 'a tree =
-               Leaf a;;
-
-       let rec bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
-           match u with
-           | Leaf a -> (fun b -> Leaf b) (f a)
-           | Node (l, r) -> (fun l' r' -> Node (l', r')) (bind l f) (bind r f);;
-
-       (* monadic operations for the TreeT monadic transformer *)
-
-       type ('a, M) treeT =
-               'a tree M;;
-
-       let unit (a: 'a) : ('a, M) tree =
-               M.unit (Leaf a);;
-
-       let rec bind (u : ('a, M) tree) (f : 'a -> ('b, M) tree) : ('b, M) tree =
-           match u with
-           | Leaf a -> M.bind (f a) (fun b -> M.unit (Leaf b))
-           | Node (l, r) -> M.bind (bind l f) (fun l' ->
-                                                       M.bind (bind r f) (fun r' ->
-                                                               M.unit (Node (l', r'));;
-
-Compare this definition of `bind` for the TreeT monadic transformer to our earlier definition of `tree_monadize`, specialized for the Reader monad:
-
-       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'))));;
-
+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]].