manip trees: split off last section into monad_transformers

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index c92065c..000772a 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -3,17 +3,19 @@
 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.
 
 of intensionality to an extensional grammar, but we have not yet seen
 the utility of replacing one monad with other.
 


@@ -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)))

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


@@ -71,24 +73,25 @@ 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;;
        - : int tree =ppp
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
 
        let square i = i * i;;
        tree_map square t1;;
        - : int tree =ppp
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 

-Note that what `tree_map` 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
 to each subpart of the computation.  In other words, `tree_map` has the
 information---what to do to each leaf---and supplies that information
 to each subpart of the computation.  In other words, `tree_map` has the

-behavior of a reader monad.  Let's make that explicit.
+behavior of a Reader monad.  Let's make that explicit.

 
 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
 
 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)
+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.
 


@@ -106,16 +109,25 @@ 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 `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, it will be
-enough to expect that our "environment" will be some 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;;
 
        type 'a reader = (int -> int) -> 'a;;  (* mnemonic: e for environment *)
        let reader_unit (a : 'a) : 'a reader = fun _ -> a;;


@@ -135,16 +147,44 @@ But we can do this:
 
        let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
            match t with
 
        let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
            match t with

-           | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i'))
-           | Node (l, r) -> reader_bind (tree_monadize f l) (fun x ->
-                              reader_bind (tree_monadize 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 `'b tree reader`.  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
+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.
 
        # tree_monadize int_readerize t1 double;;
 `'b reader` monad through the original tree's leaves.
 
        # tree_monadize int_readerize t1 double;;


@@ -161,10 +201,10 @@ result:
        - : 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 reader 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 leaves 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;;


@@ -177,14 +217,14 @@ modification whatsoever, except for replacing the (parametric) type
 
        let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
            match t with
 
        let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
            match t with

-           | Leaf i -> state_bind (f i) (fun i' -> state_unit (Leaf i'))
-           | Node (l, r) -> state_bind (tree_monadize f l) (fun x ->
-                              state_bind (tree_monadize 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 leaves in the tree:
 
 
 Then we can count the number of leaves in the tree:
 

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


@@ -192,13 +232,21 @@ Then we can count the number of leaves in the tree:
         ___|___
         |     |
         .     .
         ___|___
         |     |
         .     .

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

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

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


@@ -211,12 +259,13 @@ 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.
-
-<!--
-FIXME: We don't make it clear why the fun has to be int -> int list list, instead of int -> int list
--->
+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?
 
 Now for the main point.  What if we wanted to convert a tree to a list
 of leaves?


@@ -227,27 +276,28 @@ of leaves?
        
        let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =
            match t with
        
        let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =
            match t with

-           | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i'))
-           | Node (l, r) -> continuation_bind (tree_monadize f l) (fun x ->
-                              continuation_bind (tree_monadize 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 `tree_monadize` code.
-We then compute:
+So for example, we compute:

 
 

-       # tree_monadize (fun a k -> a :: (k a)) t1 (fun t -> []);;
+       # 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
 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:
 
 simulate some of the computations performed above.  To see how, first
 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:
 

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

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


@@ -255,41 +305,42 @@ That is, nothing happens.  But we can begin to substitute more
 interesting functions for the first argument of `tree_monadize`:
 
        (* Simulating the tree reader: distributing a operation over the leaves *)
 interesting functions for the first argument of `tree_monadize`:
 
        (* Simulating the tree reader: distributing a operation over the leaves *)

-       # tree_monadize (fun a k -> k (square a)) t1 (fun i -> i);;
+       # 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 *)

-       # tree_monadize (fun a k -> k [a; square a]) t1 (fun i -> i);;
+       # 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 *)

-       # tree_monadize (fun a k -> 1 + k a) t1 (fun i -> 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).
 
 
 
        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).
 
 

-The binary tree monad
+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 (a: 'a) = Leaf a;;
+       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
        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));;
+           | 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:
 


@@ -364,3 +415,29 @@ 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]].
+