Merge branch 'master' of ssh://server.philosophy.fas.nyu.edu/Users/lambda/lambda

[lambda.git] / week11.mdwn

These notes may change in the next few days (today is 30 Nov 2010).
The material here benefited from many discussions with Ken Shan.

##[[Tree and List Zippers]]##

##[[Coroutines and Aborts]]##

##[[From Lists to Continuations]]##

##Same-fringe using a zipper-based coroutine##

Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees (here I *am* drawing the labels in the diagram):

            .                .
           / \              / \
          .   3            1   .
         / \                  / \
        1   2                2   3

have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter new ways to approach the problem of determining when two trees have the same fringe.


Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:

            .                .
           / \              / \
          .   3            1   .
         / \                  / \
        1   2                2   3

you won't move upwards at the same steps. Keep comparing "the next leafs" until they are different, or you exhaust the leafs of only one of the trees (then again the trees have different fringes), or you exhaust the leafs of both trees at the same time, without having found leafs with different labels. In this last case, the trees have the same fringe.

If your trees are very big---say, millions of leaves---you can imagine how this would be quicker and more memory-efficient than traversing each tree to construct a list of its fringe, and then comparing the two lists so built to see if they're equal. For one thing, the zipper method can abort early if the fringes diverge early, without needing to traverse or build a list containing the rest of each tree's fringe.

Let's sketch the implementation of this. We won't provide all the details for an implementation of the tree zipper, but we will sketch an interface for it.

First, we define a type for leaf-labeled, binary trees:

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

Next, the interface for our tree zippers. We'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:

        # type blah = Blah of (int * int * (char -> bool));;

and then having to remember which element in the triple was which:

        # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
        Error: This expression has type int * (char -> bool) * int
       but an expression was expected of type int * int * (char -> bool)
        # (* damnit *)
        # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
        val b1 : blah = Blah (1, 2, <fun>)

records let you attach descriptive labels to the components of the tuple:

        # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
        # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
        val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
        # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
        val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}

These were the strategies to extract the components of an unlabeled tuple:

        let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)

        let (h, w, test) = b1;; (* works for arbitrary tuples *)

        match b1 with
        | (h, w, test) -> ...;; (* same as preceding *)

Here is how you can extract the components of a labeled record:

        let h = b2.height;; (* handy! *)

        let {height = h; weight = w; char_tester = test} = b2
        in (* go on to use h, w, and test ... *)

        match test with
        | {height = h; weight = w; char_tester = test} ->
                (* go on to use h, w, and test ... *)

Anyway, using record types, we might define the tree zipper interface like so:

        type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot
        and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;

        type 'a zipper = { level : 'a starred_level; filler: 'a tree };;

        let rec move_botleft (z : 'a zipper) : 'a zipper =
            (* returns z if the targetted node in z has no children *)
            (* else returns move_botleft (zipper which results from moving down and left in z) *)

<!--
            let {level; filler} = z
            in match filler with
            | Leaf _ -> z
            | Node(left, right) ->
                let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
                in move_botleft zdown
            ;;
-->

        let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
            (* if it's possible to move right in z, returns Some (the result of doing so) *)
            (* else if it's not possible to move any further up in z, returns None *)
            (* else returns move_right_or_up (result of moving up in z) *)

<!--
            let {level; filler} = z
            in match level with
            | Starring_Left {parent; sibling = right} -> Some {level = Starring_Right {parent; sibling = filler}; filler = right}
            | Root -> None
            | Starring_Right {parent; sibling = left} ->
                let z' = {level = parent; filler = Node(left, filler)}
                in move_right_or_up z'
            ;;
-->

The following function takes an 'a tree and returns an 'a zipper focused on its root:

        let new_zipper (t : 'a tree) : 'a zipper =
            {level = Root; filler = t}
            ;;

Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:

        let make_fringe_enumerator (t: 'a tree) =
            (* create a zipper targetting the botleft of t *)
            let zbotleft = move_botleft (new_zipper t)
            (* create a refcell initially pointing to zbotleft *)
            in let zcell = ref (Some zbotleft)
            (* construct the next_leaf function *)
            in let next_leaf () : 'a option =
                match !zcell with
                | Some z -> (
                    (* extract label of currently-targetted leaf *)
                    let Leaf current = z.filler
                    (* update zcell to point to next leaf, if there is one *)
                    in let () = zcell := match move_right_or_up z with
                        | None -> None
                        | Some z' -> Some (move_botleft z')
                    (* return saved label *)
                    in Some current
                | None -> (* we've finished enumerating the fringe *)
                    None
                )
            (* return the next_leaf function *)
            in next_leaf
            ;;

Here's an example of `make_fringe_enumerator` in action:

        # let tree1 = Leaf 1;;
        val tree1 : int tree = Leaf 1
        # let next1 = make_fringe_enumerator tree1;;
        val next1 : unit -> int option = <fun>
        # next1 ();;
        - : int option = Some 1
        # next1 ();;
        - : int option = None
        # next1 ();;
        - : int option = None
        # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
        val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
        # let next2 = make_fringe_enumerator tree2;;
        val next2 : unit -> int option = <fun>
        # next2 ();;
        - : int option = Some 1
        # next2 ();;
        - : int option = Some 2
        # next2 ();;
        - : int option = Some 3
        # next2 ();;
        - : int option = None
        # next2 ();;
        - : int option = None

You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will keep returning `None`.

Using these fringe enumerators, we can write our `same_fringe` function like this:

        let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
                let next1 = make_fringe_enumerator t1
                in let next2 = make_fringe_enumerator t2
                in let rec loop () : bool =
                        match next1 (), next2 () with
                        | Some a, Some b when a = b -> loop ()
                        | None, None -> true
                        | _ -> false
                in loop ()
                ;;

The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.

The technique illustrated here with our fringe enumerators is a powerful and important one. It's an example of what's sometimes called **cooperative threading**. A "thread" is a subprogram that the main computation spawns off. Threads are called "cooperative" when the code of the main computation and the thread fixes when control passes back and forth between them. (When the code doesn't control this---for example, it's determined by the operating system or the hardware in ways that the programmer can't predict---that's called "preemptive threading.") Cooperative threads are also sometimes called *coroutines* or *generators*.

With cooperative threads, one typically yields control to the thread, and then back again to the main program, multiple times. Here's the pattern in which that happens in our `same_fringe` function:

        main program            next1 thread            next2 thread
        ------------            ------------            ------------
        start next1
        (paused)                        starting
        (paused)                        calculate first leaf
        (paused)                        <--- return it
        start next2                     (paused)                        starting
        (paused)                        (paused)                        calculate first leaf
        (paused)                        (paused)                        <-- return it
        compare leaves          (paused)                        (paused)
        call loop again         (paused)                        (paused)
        call next1 again        (paused)                        (paused)
        (paused)                        calculate next leaf     (paused)
        (paused)                        <-- return it           (paused)
        ... and so on ...

If you want to read more about these kinds of threads, here are some links:

<!-- *  [[!wikipedia Computer_multitasking]]
*       [[!wikipedia Thread_(computer_science)]] -->

*       [[!wikipedia Coroutine]]
*       [[!wikipedia Iterator]]
*       [[!wikipedia Generator_(computer_science)]]
*       [[!wikipedia Fiber_(computer_science)]]
<!-- *  [[!wikipedia Green_threads]]
*       [[!wikipedia Protothreads]] -->

The way we built cooperative threads here crucially relied on two heavyweight tools. First, it relied on our having a data structure (the tree zipper) capable of being a static snapshot of where we left off in the tree whose fringe we're enumerating. Second, it relied on our using mutable reference cells so that we could update what the current snapshot (that is, tree zipper) was, so that the next invocation of the `next_leaf` function could start up again where the previous invocation left off.

It's possible to build cooperative threads without using those tools, however. Some languages have a native syntax for them. Here's how we'd write the same-fringe solution above using native coroutines in the language Lua:

        > function fringe_enumerator (tree)
            if tree.leaf then
                coroutine.yield (tree.leaf)
            else
                fringe_enumerator (tree.left)
                fringe_enumerator (tree.right)
            end
        end
        
        > function same_fringe (tree1, tree2)
            local next1 = coroutine.wrap (fringe_enumerator)
            local next2 = coroutine.wrap (fringe_enumerator)
            local function loop (leaf1, leaf2)
                if leaf1 or leaf2 then
                    return leaf1 == leaf2 and loop( next1(), next2() )
                elseif not leaf1 and not leaf2 then
                    return true
                else
                    return false
                end
            end
            return loop (next1(tree1), next2(tree2))
        end
        
        > return same_fringe ( {leaf=1}, {leaf=2})
        false
        
        > return same_fringe ( {leaf=1}, {leaf=1})
        true
        
        > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
            {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
        true

We're going to think about the underlying principles to this execution pattern, and instead learn how to implement it from scratch---without necessarily having zippers to rely on.


##Exceptions and Aborts##

To get a better understanding of how that execution patter works, we'll add yet a second execution pattern to our plate, and then think about what they have in common.

While writing OCaml code, you've probably come across errors. In fact, you've probably come across errors of two sorts. One sort of error comes about when you've got syntax errors or type errors and the OCaml interpreter isn't even able to understand your code:

        # let lst = [1; 2] in
          "a" :: lst;;
        Error: This expression has type int list
                   but an expression was expected of type string list

But you may also have encountered other kinds of error, that arise while your program is running. For example:

        # 1/0;;
        Exception: Division_by_zero.
        # List.nth [1;2] 10;;
        Exception: Failure "nth".

These "Exceptions" are **run-time errors**. OCaml will automatically detect some of them, like when you attempt to divide by zero. Other exceptions are *raised* by code. For instance, here is the implementation of `List.nth`:

        let nth l n =
          if n < 0 then invalid_arg "List.nth" else
          let rec nth_aux l n =
                match l with
                | [] -> failwith "nth"
                | a::l -> if n = 0 then a else nth_aux l (n-1)
          in nth_aux l n

Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:

        raise (Invalid_argument "List.nth");;
        raise (Failure "nth");;

where `Invalid_argument "List.nth"` is a value of type `exn`, and so too `Failure "nth"`. When you have some value `ex` of type `exn` and evaluate the expression:

        raise ex

the effect is for the program to immediately stop without evaluating any further code:

        # let xcell = ref 0;;
        val xcell : int ref = {contents = 0}
        # let ex = Failure "test"
          in let _ = raise ex
          in xcell := 1;;
        Exception: Failure "test".
        # !xcell;;
        - : int = 0

Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.

I said when you evaluate the expression:

        raise ex

the effect is for the program to immediately stop. That's not exactly true. You can also programmatically arrange to *catch* errors, without the program necessarily stopping. In OCaml we do that with a `try ... with PATTERN -> ...` construct, analogous to the `match ... with PATTERN -> ...` construct:

        # let foo x =
            try
                if x = 1 then 10
                else if x = 2 then raise (Failure "two")
                else raise (Failure "three")
            with Failure "two" -> 20
            ;;
        val foo : int -> int = <fun>
        # foo 1;;
        - : int = 10
        # foo 2;;
        - : int = 20
        # foo 3;;
        Exception: Failure "three".

Notice what happens here. If we call `foo 1`, then the code between `try` and `with` evaluates to `10`, with no exceptions being raised. That then is what the entire `try ... with ...` block evaluates to; and so too what `foo 1` evaluates to. If we call `foo 2`, then the code between `try` and `with` raises an exception `Failure "two"`. The pattern in the `with` clause matches that exception, so we get instead `20`. If we call `foo 3`, we again raise an exception. This exception isn't matched by the `with` block, so it percolates up to the top of the program, and then the program immediately stops.

So what I should have said is that when you evaluate the expression:

        raise ex

*and that exception is never caught*, then the effect is for the program to immediately stop.

Of course, it's possible to handle errors in other ways too. There's no reason why the implementation of `List.nth` *had* to do things this way. They might instead have returned `Some a` when the list had an nth member `a`, and `None` when it does not. But it's pedagogically useful for us to think about this pattern now.

When an exception is raised, it percolates up through the code that called it, until it finds a surrounding `try ... with ...` that matches it. That might not be the first `try ... with ...` that it encounters. For example:

        # try
            try
                raise (Failure "blah")
            with Failure "fooey" -> 10
          with Failure "blah" -> 20;;
        - : int = 20

The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:

        # let foo b x =
            try
                b x
            with Failure "blah" -> 20
        in let bar x =
            raise (Failure "blah")
        in foo bar 0;;
        - : int = 20

Here we call `foo bar 0`, and `foo` in turn calls `bar 0`, and `bar` raises the exception. Since there's no matching `try ... with ...` block in `bar`, we percolate back up the history of *who called this function?* and find a matching `try ... with ...` block in `foo`. This catches the error and so then the `try ... with ...` block in `foo` that called `bar` in the first place will evaluate to `20`.

OK, now this exception-handling apparatus does exemplify the second execution pattern we want to focus on. But it may bring it into clearer focus if we simplify the pattern even more. Imagine we could write code like this instead:

        # let foo x =
            try
                (if x = 1 then 10
                else abort 20) + 1
            end
            ;;

then if we called `foo 1`, we'd get the result `11`. If we called `foo 2`, on the other hand, we'd get `20` (note, not `21`). This exemplifies the same interesting "jump out of this part of the code" behavior that the `try ... raise ... with ...` code does, but without the details of matching which exception was raised, and handling the exception to produce a new result.

Many programming languages have this simplified exceution pattern, either instead of or alongside a `try ... with ...`-like pattern. In Lua and many other languages, `abort` is instead called `return`. The preceding example would be written:

        > function foo(x)
            local value
            if (x == 1) then
                value = 10
            else
                return 20
            end
            return value + 1
        end
        
        > return foo(1)
        11
        
        > return foo(2)
        20

Okay, so that's our second execution pattern.

##What do these have in common?##

In both of these patterns, we need to have some way to take a snapshot of where we are in the evaluation of a complex piece of code, so that we might later resume execution at that point. In the coroutine example, the two threads need to have a snapshot of where they were in the enumeration of their tree's leaves. In the abort example, we need to have a snapshot of where to pick up again if some embedded piece of code aborts. Sometimes we might distill that snapshot into a datastructure like a zipper. But we might not always know how to do so; and learning how to think about these snapshots without the help of zippers will help us see patterns and similarities we might otherwise miss.

A more general way to think about these snapshots is to think of the code we're taking a snapshot of as a *function.* For example, in this code:

        let foo x =
            try
                (if x = 1 then 10
                else abort 20) + 1
            end
        in (foo 2) + 1;;

we can imagine a box:

        let foo x =
        +---------------------------+
        |   try                     |
        |       (if x = 1 then 10   |
        |       else abort 20) + 1  |
        |   end                     |
        +---------------------------+
        in (foo 2) + 1;;

and as we're about to enter the box, we want to take a snapshot of the code *outside* the box. If we decide to abort, we'd be aborting to that snapshotted code.

<!--
# #require "delimcc";;
# open Delimcc;;
# let reset body = let p = new_prompt () in push_prompt p (body p);;
val reset : ('a Delimcc.prompt -> unit -> 'a) -> 'a = <fun>
# let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 1) + 100;;
- : int = 111
# let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 2) + 100;;
- : int = 120
-->




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

In coming weeks, we'll learn about a different way to create threads, that relies on **continuations** rather than on those two tools. All of these tools are inter-related. As Oleg says, "Zipper can be viewed as a delimited continuation reified as a data structure." These different tools are also inter-related with monads. Many of these tools can be used to define the others. We'll explore some of the connections between them in the remaining weeks, but we encourage you to explore more.


##Introducing Continuations##

A continuation is "the rest of the program." Or better: an **delimited continuation** is "the rest of the program, up to a certain boundary." An **undelimited continuation** is "the rest of the program, period."

Even if you haven't read specifically about this notion (for example, even if you haven't read Chris and Ken's work on using continuations in natural language semantics), you'll have brushed shoulders with it already several times in this course.

A naive semantics for atomic sentences will say the subject term is of type `e`, and the predicate of type `e -> t`, and that the subject provides an argument to the function expressed by the predicate.

Monatague proposed we instead take subject terms to be of type `(e -> t) -> t`, and that now it'd be the predicate (still of type `e -> t`) that provides an argument to the function expressed by the subject.

If all the subject did then was supply an `e` to the `e -> t` it receives as an argument, we wouldn't have gained anything we weren't already able to do. But of course, there are other things the subject can do with the `e -> t` it receives as an argument. For instance, it can check whether anything in the domain satisfies that `e -> t`; or whether most things do; and so on.

This inversion of who is the argument and who is the function receiving the argument is paradigmatic of working with continuations. We did the same thing ourselves back in the early days of the seminar, for example in our implementation of pairs. In the untyped lambda calculus, we identified the pair `(x, y)` with a function:

        \handler. handler x y

A pair-handling function would accept the two elements of a pair as arguments, and then do something with one or both of them. The important point here is that the handler was supplied as an argument to the pair. Eventually, the handler would itself be supplied with arguments. But only after it was supplied as an argument to the pair. This inverts the order you'd expect about what is the data or argument, and what is the function that operates on it.

Consider a complex computation, such as:

        1 + 2 * (1 - g (3 + 4))

Part of this computation---`3 + 4`---leads up to supplying `g` with an argument. The rest of the computation---`1 + 2 * (1 - ___)`---waits for the result of applying `g` to that argument and will go on to do something with it (inserting the result into the `___` slot). That "rest of the computation" can be regarded as a function:

        \result. 1 + 2 * (1 - result)

This function will be applied to whatever is the result of `g (3 + 4)`. So this function can be called the *continuation* of that application of `g`. For some purposes, it's useful to be able to invert the function/argument order here, and rather than supplying the result of applying `g` to the continuation, we instead supply the continuation to `g`. Well, not to `g` itself, since `g` only wants a single `int` argument. But we might build some `g`-like function which accepts not just an `int` argument like `g` does, but also a continuation argument.

Go back and read the material on "Aborting a Search Through a List" in [[Week4]] for an example of doing this.

In very general terms, the strategy is to work with functions like this:

        let g' k (i : int) =
                ... do stuff ...
                ... if you want to abort early, supply an argument to k ...
                ... do more stuff ...
                ... normal result
        in let gcon = fun result -> 1 + 2 * (1 - result)
        in gcon (g' gcon (3 + 4))

It's a convention to use variables like `k` for continuation arguments. If the function `g'` never supplies an argument to its contination argument `k`, but instead just finishes evaluating to a normal result, that normal result will be delivered to `g'`'s continuation `gcon`, just as happens when we don't pass around any explicit continuation variables.

The above snippet of OCaml code doesn't really capture what happens when we pass explicit continuation variables. For suppose that inside `g'`, we do supply an argument to `k`. That would go into the `result` parameter in `gcon`. But then what happens once we've finished evaluating the application of `gcon` to that `result`? In the OCaml snippet above, the final value would then bubble up through the context in the body of `g'` where `k` was applied, and eventually out to the final line of the snippet, where it once again supplied an argument to `gcon`. That's not what happens with a real continuation. A real continuation works more like this:

        let g' k (i : int) =
                ... do stuff ...
                ... if you want to abort early, supply an argument to k ...
                ... do more stuff ...
                ... normal result
        in let gcon = fun result ->
                let final_value = 1 + 2 * (1 - result)
                in end_program_with final_value
        in gcon (g' gcon (3 + 4))

So once we've finished evaluating the application of `gcon` to a `result`, the program is finished. (This is how undelimited continuations behave. We'll discuss delimited continuations later.)

So now, guess what would be the result of doing the following:

        let g' k (i : int) =
                1 + k i
        in let gcon = fun result ->
                let final_value = (1, result)
                in end_program_with final_value
        in gcon (g' gcon (3 + 4))

<!-- (1, 7) ... explain why not (1, 8) -->

Rethinking the list monad
-------------------------

To construct a monad, the key element is to settle on a type
constructor, and the monad more or less naturally follows from that.
We'll remind you of some examples of how monads follow from the type
constructor in a moment.  This will involve some review of familair
material, but it's worth doing for two reasons: it will set up a
pattern for the new discussion further below, and it will tie together
some previously unconnected elements of the course (more specifically,
version 3 lists and monads).

For instance, take the **Reader Monad**.  Once we decide that the type
constructor is

    type 'a reader = env -> 'a

then the choice of unit and bind is natural:

    let r_unit (a : 'a) : 'a reader = fun (e : env) -> a

The reason this is a fairly natural choice is that because the type of
an `'a reader` is `env -> 'a` (by definition), the type of the
`r_unit` function is `'a -> env -> 'a`, which is an instance of the
type of the *K* combinator.  So it makes sense that *K* is the unit
for the reader monad.

Since the type of the `bind` operator is required to be

    r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)

We can reason our way to the traditional reader `bind` function as
follows. We start by declaring the types determined by the definition
of a bind operation:

    let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...

Now we have to open up the `u` box and get out the `'a` object in order to
feed it to `f`.  Since `u` is a function from environments to
objects of type `'a`, the way we open a box in this monad is
by applying it to an environment:

<pre>
        ... f (u e) ...
</pre>

This subexpression types to `'b reader`, which is good.  The only
problem is that we made use of an environment `e` that we didn't already have,
so we must abstract over that variable to balance the books:

        fun e -> f (u e) ...

[To preview the discussion of the Curry-Howard correspondence, what
we're doing here is constructing an intuitionistic proof of the type,
and using the Curry-Howard labeling of the proof as our bind term.]

This types to `env -> 'b reader`, but we want to end up with `env ->
'b`.  Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to an environment.  So we end up as follows:

<pre>
r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e         
</pre>

And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does.

[The bind we cite here is a condensed version of the careful `let a = u e in ...`
constructions we provided in earlier lectures.  We use the condensed
version here in order to emphasize similarities of structure across
monads.]

The **State Monad** is similar.  Once we've decided to use the following type constructor:

    type 'a state = store -> ('a, store)

Then our unit is naturally:

    let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)

And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box:

    let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = 
                ... f (...) ...

But unlocking the `u` box is a little more complicated.  As before, we
need to posit a state `s` that we can apply `u` to.  Once we do so,
however, we won't have an `'a`, we'll have a pair whose first element
is an `'a`.  So we have to unpack the pair:

        ... let (a, s') = u s in ... (f a) ...

Abstracting over the `s` and adjusting the types gives the result:

        let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = 
                fun (s : store) -> let (a, s') = u s in f a s'

The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we
won't pause to explore it here, though conceptually its unit and bind
follow just as naturally from its type constructor.

Our other familiar monad is the **List Monad**, which we were told
looks like this:

    type 'a list = ['a];;
    l_unit (a : 'a) = [a];;
    l_bind u f = List.concat (List.map f u);;

Thinking through the list monad will take a little time, but doing so
will provide a connection with continuations.

Recall that `List.map` takes a function and a list and returns the
result to applying the function to the elements of the list:

    List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]

and List.concat takes a list of lists and erases the embdded list
boundaries:

    List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]

And sure enough, 

    l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]

Now, why this unit, and why this bind?  Well, ideally a unit should
not throw away information, so we can rule out `fun x -> []` as an
ideal unit.  And units should not add more information than required,
so there's no obvious reason to prefer `fun x -> [x,x]`.  In other
words, `fun x -> [x]` is a reasonable choice for a unit.

As for bind, an `'a list` monadic object contains a lot of objects of
type `'a`, and we want to make use of each of them (rather than
arbitrarily throwing some of them away).  The only
thing we know for sure we can do with an object of type `'a` is apply
the function of type `'a -> 'a list` to them.  Once we've done so, we
have a collection of lists, one for each of the `'a`'s.  One
possibility is that we could gather them all up in a list, so that
`bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`.  But that restricts
the object returned by the second argument of `bind` to always be of
type `'b list list`.  We can elimiate that restriction by flattening
the list of lists into a single list: this is
just List.concat applied to the output of List.map.  So there is some logic to the
choice of unit and bind for the list monad.  

Yet we can still desire to go deeper, and see if the appropriate bind
behavior emerges from the types, as it did for the previously
considered monads.  But we can't do that if we leave the list type as
a primitive Ocaml type.  However, we know several ways of implementing
lists using just functions.  In what follows, we're going to use type
3 lists, the right fold implementation (though it's important and
intriguing to wonder how things would change if we used some other
strategy for implementating lists).  These were the lists that made
lists look like Church numerals with extra bits embdded in them:

    empty list:                fun f z -> z
    list with one element:     fun f z -> f 1 z
    list with two elements:    fun f z -> f 2 (f 1 z)
    list with three elements:  fun f z -> f 3 (f 2 (f 1 z))

and so on.  To save time, we'll let the OCaml interpreter infer the
principle types of these functions (rather than inferring what the
types should be ourselves):

        # fun f z -> z;;
        - : 'a -> 'b -> 'b = <fun>
        # fun f z -> f 1 z;;
        - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
        # fun f z -> f 2 (f 1 z);;
        - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
        # fun f z -> f 3 (f 2 (f 1 z))
        - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>

We can see what the consistent, general principle types are at the end, so we
can stop. These types should remind you of the simply-typed lambda calculus
types for Church numerals (`(o -> o) -> o -> o`) with one extra type
thrown in, the type of the element a the head of the list
(in this case, an int).

So here's our type constructor for our hand-rolled lists:

    type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b

Generalizing to lists that contain any kind of element (not just
ints), we have

    type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b

So an `('a, 'b) list'` is a list containing elements of type `'a`,
where `'b` is the type of some part of the plumbing.  This is more
general than an ordinary OCaml list, but we'll see how to map them
into OCaml lists soon.  We don't need to fully grasp the role of the `'b`'s
in order to proceed to build a monad:

    l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z

No problem.  Arriving at bind is a little more complicated, but
exactly the same principles apply, you just have to be careful and
systematic about it.

    l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list'  = ...

Unpacking the types gives:

    l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
            (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
            : ('c -> 'd -> 'd) -> 'd -> 'd = ...

Perhaps a bit intimiating.
But it's a rookie mistake to quail before complicated types. You should
be no more intimiated by complex types than by a linguistic tree with
deeply embedded branches: complex structure created by repeated
application of simple rules.

[This would be a good time to try to build your own term for the types
just given.  Doing so (or attempting to do so) will make the next
paragraph much easier to follow.]

As usual, we need to unpack the `u` box.  Examine the type of `u`.
This time, `u` will only deliver up its contents if we give `u` an
argument that is a function expecting an `'a` and a `'b`. `u` will 
fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:

        ... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...

In order for `u` to have the kind of argument it needs, the `... (f a) ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `(f a)`:

        ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...

Now we have an argument `b` of type `'b`, so we can supply that to `(f a) k`, getting a result of type `'b`, as we need:

        ... u (fun (a : 'a) (b : 'b) -> f a k b) ...

Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:

        fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)

This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that our bind is:

    l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
            (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
            : ('c -> 'b -> 'b) -> 'b -> 'b = 
      fun k -> u (fun a b -> f a k b)

That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.

Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:

      fun k z -> u (fun a b -> f a k b) z

Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?

Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:

        concat (map f u) =
        concat [[]; [2]; [2; 4]; [2; 4; 8]] =
        [2; 2; 4; 2; 4; 8]

Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula

      fun k z -> u (fun a b -> f a k b) z
        
do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:

        []
        [2]
        [2; 4]
        [2; 4; 8]

(or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.

So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:

        0 ==>
        right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
        right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
        right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
        right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0

which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:

      fun k z -> u (fun a b -> f a k b) z

will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as

        fun k z -> List.fold_right k (concat (map f u)) z

would.

For future reference, we might make two eta-reductions to our formula, so that we have instead:

      let l'_bind = fun k -> u (fun a -> f a k);;

Let's make some more tests:


    l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]

    l'_bind (fun f z -> f 1 (f 2 z)) 
            (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>

Sigh.  OCaml won't show us our own list.  So we have to choose an `f`
and a `z` that will turn our hand-crafted lists into standard OCaml
lists, so that they will print out.

        # let cons h t = h :: t;;  (* OCaml is stupid about :: *)
        # l'_bind (fun f z -> f 1 (f 2 z)) 
                          (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
        - : int list = [1; 2; 2; 3]

Ta da!


Montague's PTQ treatment of DPs as generalized quantifiers
----------------------------------------------------------

We've hinted that Montague's treatment of DPs as generalized
quantifiers embodies the spirit of continuations (see de Groote 2001,
Barker 2002 for lengthy discussion).  Let's see why.  

First, we'll need a type constructor.  As you probably know, 
Montague replaced individual-denoting determiner phrases (with type `e`)
with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
In particular, the denotation of a proper name like *John*, which
might originally denote a object `j` of type `e`, came to denote a
generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
Let's write a general function that will map individuals into their
corresponding generalized quantifier:

   gqize (a : e) = fun (p : e -> t) -> p a

This function is what Partee 1987 calls LIFT, and it would be
reasonable to use it here, but we will avoid that name, given that we
use that word to refer to other functions.

This function wraps up an individual in a box.  That is to say,
we are in the presence of a monad.  The type constructor, the unit and
the bind follow naturally.  We've done this enough times that we won't
belabor the construction of the bind function, the derivation is
highly similar to the List monad just given:

        type 'a continuation = ('a -> 'b) -> 'b
        c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
        c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
          fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)

Note that `c_unit` is exactly the `gqize` function that Montague used
to lift individuals into the continuation monad.

That last bit in `c_bind` looks familiar---we just saw something like
it in the List monad.  How similar is it to the List monad?  Let's
examine the type constructor and the terms from the list monad derived
above:

    type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
    l'_unit a = fun f -> f a                 
    l'_bind u f = fun k -> u (fun a -> f a k)

(We performed a sneaky but valid eta reduction in the unit term.)

The unit and the bind for the Montague continuation monad and the
homemade List monad are the same terms!  In other words, the behavior
of the List monad and the behavior of the continuations monad are
parallel in a deep sense.

Have we really discovered that lists are secretly continuations?  Or
have we merely found a way of simulating lists using list
continuations?  Well, strictly speaking, what we have done is shown
that one particular implementation of lists---the right fold
implementation---gives rise to a continuation monad fairly naturally,
and that this monad can reproduce the behavior of the standard list
monad.  But what about other list implementations?  Do they give rise
to monads that can be understood in terms of continuations?

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
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
the utility of replacing one monad with other.

First, we'll be needing a lot of trees during the remainder of the
course.  Here's a type constructor for binary trees:

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

These are trees in which the internal nodes do not have labels.  [How
would you adjust the type constructor to allow for labels on the
internal nodes?]

We'll be using trees where the nodes are integers, e.g.,


<pre>
let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
               (Node ((Leaf 5),(Node ((Leaf 7),
                                      (Leaf 11))))))

    .
 ___|___
 |     |
 .     .
_|__  _|__
|  |  |  |
2  3  5  .
        _|__
        |  |
        7  11
</pre>

Our first task will be to replace each leaf with its double:

<pre>
let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) =
  match t with Leaf x -> Leaf (newleaf x)
             | Node (l, r) -> Node ((treemap newleaf l),
                                    (treemap newleaf r));;
</pre>
`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:

<pre>
let double i = i + i;;
treemap double t1;;
- : int tree =
Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))

    .
 ___|____
 |      |
 .      .
_|__  __|__
|  |  |   |
4  6  10  .
        __|___
        |    |
        14   22
</pre>

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`:

<pre>
let square x = x * x;;
treemap square t1;;
- : int tree =ppp
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
</pre>

Note that what `treemap` does is take some global, contextual
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.  

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
updated tree.

\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
<pre>
\f    .
  ____|____
  |       |
  .       .
__|__   __|__
|   |   |   |
f2  f3  f5  .
          __|___
          |    |
          f7  f11
</pre>

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`.

<pre>
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;;
</pre>

It's easy to figure out how to turn an `int` into an `int reader`:

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

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.

<pre>
let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
  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))));;
</pre>

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.

<pre>
# treemonadizer int2int_reader t1 (fun i -> i + i);;
- : int tree =
Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
</pre>

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

<pre>
# treemonadizer int2int_reader t1 (fun i -> i * i);;
- : int tree =
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
</pre>

Now that we have a tree transducer that accepts a monad as a
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
the tree.

<pre>
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);;
</pre>

Gratifyingly, we can use the `treemonadizer` function without any
modification whatsoever, except for replacing the (parametric) type
`reader` with `state`:

<pre>
let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
  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))));;
</pre>

Then we can count the number of nodes in the tree:

<pre>
# treemonadizer state_unit t1 0;;
- : int tree * int =
(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)

    .
 ___|___
 |     |
 .     .
_|__  _|__
|  |  |  |
2  3  5  .
        _|__
        |  |
        7  11
</pre>

Notice that we've counted each internal node twice---it's a good
exercise to adjust the code to count each node once.

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

<pre>
# 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])))]
</pre>

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.

Now for the main point.  What if we wanted to convert a tree to a list
of leaves?  

<pre>
type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
let continuation_unit x c = c x;;
let continuation_bind u f c = u (fun a -> f a c);;

let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
  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))));;
</pre>

We use the continuation monad described above, and insert the
`continuation` type in the appropriate place in the `treemonadizer` code.
We then compute:

<pre>
# treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
- : int list = [2; 3; 5; 7; 11]
</pre>

We have found a way of collapsing a tree into a list of its leaves.

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 the
continuation unit as our first argument to `treemonadizer`, and then
apply the result to the identity function:

<pre>
# treemonadizer continuation_unit t1 (fun x -> x);;
- : int tree =
Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
</pre>

That is, nothing happens.  But we can begin to substitute more
interesting functions for the first argument of `treemonadizer`:

<pre>
(* Simulating the tree reader: distributing a operation over the leaves *)
# treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
- : 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 *)
# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
- : int = 5
</pre>

We could simulate the tree state example too, but it would require
generalizing the type of the continuation monad to 

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

The binary tree monad
---------------------

Of course, by now you may have realized that we have discovered a new
monad, the binary tree monad:

<pre>
type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
let tree_unit (x:'a) = Leaf x;;
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));;
</pre>

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

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)))
<pre>
                .                         .       
              __|__                     __|__   
              |   |                     |   |   
              a1  .                    fa1  .   
                 _|__                     __|__ 
                 |  |                     |   | 
                 .  a5                    .  fa5
   bind         _|__       f   =        __|__   
                |  |                    |   |   
                .  a4                   .  fa4  
              __|__                   __|___   
              |   |                   |    |   
              a2  a3                 fa2  fa3         
</pre>   

Given this equivalence, the right identity law

    Right identity: bind u unit = u

falls out once we realize that

    bind (Leaf a) unit = unit a = Leaf a

As for the associative law,

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

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)))  ))
<pre>
                                           .
                                       ____|____
          .               .            |       |
bind    __|__   f  =    __|_    =      .       .
        |   |           |   |        __|__   __|__
        a1  a2         fa1 fa2       |   |   |   |
                                     a1  a1  a1  a1  
</pre>

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

<pre>
           .
       ____|____
       |       |
       .       .
     __|__   __|__
     |   |   |   |
    ga1 ga1 ga1 ga1  
</pre>

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.

So binary trees are a monad.

Haskell combines this monad with the Option monad to provide a monad
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.

##[[List Monad as Continuation Monad]]##

##[[Manipulating Trees with Monads]]##