tweak whole site: caps for Reader monad, etc

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 81dc451..e3ed6f3 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -13,7 +13,7 @@ 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
 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.
 


@@ -81,14 +81,14 @@ supplying the appropriate `int -> int` operation in place of `double`:
        - : int tree =ppp
        Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
        - : 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.
 


@@ -113,9 +113,7 @@ 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
 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`.
+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 +133,44 @@ 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;;


@@ -161,17 +187,15 @@ 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.
 

-<!-- FIXME -->
-
-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;;
        let state_unit a = fun s -> (a, s);;
 the tree.
 
        type 'a state = int -> 'a * int;;
        let state_unit a = fun s -> (a, s);;

-       let state_bind_and_count u f = fun s -> let (a, s') = u s in f a (s' + 1);;
+       let state_bind u f = fun s -> let (a, s') = u s in f a s';;

 
 Gratifyingly, we can use the `tree_monadize` function without any
 modification whatsoever, except for replacing the (parametric) type
 
 Gratifyingly, we can use the `tree_monadize` function without any
 modification whatsoever, except for replacing the (parametric) type


@@ -179,16 +203,16 @@ 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_and_count (f i) (fun i' -> state_unit (Leaf i'))
-           | Node (l, r) -> state_bind_and_count (tree_monadize f l) (fun x ->
-                              state_bind_and_count (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 nodes in the tree:
+Then we can count the number of leaves in the tree:

 
 

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

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

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

        
            .
         ___|___
        
            .
         ___|___


@@ -201,17 +225,7 @@ Then we can count the number of nodes in the tree:
                |  |
                7  11
 
                |  |
                7  11
 

-Notice that we've counted each internal node twice---it's a good
-exercise to adjust the code to count each node once.
-
-<!--
-A tree with n leaves has 2n - 1 nodes.
-This function will currently return n*1 + (n-1)*2 = 3n - 2.
-To convert b = 3n - 2 into 2n - 1, we can use: let n = (b + 2)/3 in 2*n -1
-
-But I assume Chris means here, adjust the code so that no corrections of this sort have to be applied.
--->
-
+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


@@ -224,10 +238,11 @@ One more revealing example before getting down to business: replacing
 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with
 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with

-a list of `int`'s.
+a list of `int`'s. We might also have done this with a Reader monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List monad an operation like `fun -> [ i; [2*i; 3*i] ]`. Use small trees for your experiment.
+

 
 <!--
 
 <!--

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

 -->
 
 
 -->
 
 


@@ -240,29 +255,28 @@ 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.
-We then compute:
+We use the Continuation monad described above, and insert the
+`continuation` type in the appropriate place in the `tree_monadize` code. Then if we give the `tree_monadize` function an operation that converts `int`s into `'b`-wrapping Continuation monads, it will give us back a way to turn `int tree`s into corresponding `'b tree`-wrapping Continuation monads.

 
 

-       # tree_monadize (fun a k -> a :: (k a)) t1 (fun t -> []);;
-       - : int list = [2; 3; 5; 7; 11]
+So for example, we compute:

 
 

-<!-- FIXME: what if we had fun t -> [-t]? why `t`? -->
+       # tree_monadize (fun a -> fun k -> a :: k a) t1 (fun t -> []);;
+       - : 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)))
 


@@ -270,31 +284,34 @@ 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;;
 
 
        type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;
 

-The binary tree monad
+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

 ---------------------
 
 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:

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