-[[!toc]]
+These notes may change in the next few days (today is 30 Nov 2010).
+The material here benefited from many discussions with Ken Shan.
-Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees:
+##[[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):
. .
/ \ / \
/ \ / \
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 two new ways to approach the problem of determining when two trees have the same fringe.
-
-##List Zippers##
-
-Say you've got some moderately-complex function for searching through a list, for example:
-
- let find_nth (test : 'a -> bool) (n : int) (lst : 'a list) : (int * 'a) ->
- let rec helper (position : int) n lst =
- match lst with
- | [] -> failwith "not found"
- | x :: xs when test x -> (if n = 1
- then (position, x)
- else helper (position + 1) (n - 1) xs
- )
- | x :: xs -> helper (position + 1) n xs
- in helper 0 n lst;;
-
-This searches for the `n`th element of a list that satisfies the predicate `test`, and returns a pair containing the position of that element, and the element itself. Good. But now what if you wanted to retrieve a different kind of information, such as the `n`th element matching `test`, together with its preceding and succeeding elements? In a real situation, you'd want to develop some good strategy for reporting when the target element doesn't have a predecessor and successor; but we'll just simplify here and report them as having some default value:
-
- let find_nth' (test : 'a -> bool) (n : int) (lst : 'a list) (default : 'a) : ('a * 'a * 'a) ->
- let rec helper (predecessor : 'a) n lst =
- match lst with
- | [] -> failwith "not found"
- | x :: xs when test x -> (if n = 1
- then (predecessor, x, match xs with [] -> default | y::ys -> y)
- else helper x (n - 1) xs
- )
- | x :: xs -> helper x n xs
- in helper default n lst;;
-
-This duplicates a lot of the structure of `find_nth`; it just has enough different code to retrieve different information when the matching element is found. But now what if you wanted to retrieve yet a different kind of information...? Ideally, there should be some way to factor out the code to find the target element---the `n`th element of the list satisfying the predicate `test`---from the code that retrieves the information you want once the target is found. We might build upon the initial `find_nth` function, since that returns the *position* of the matching element. We could hand that result off to some other function that's designed to retrieve information of a specific sort surrounding that position. But suppose our list has millions of elements, and the target element is at position 600512. The search function will already have traversed 600512 elements of the list looking for the target, then the retrieval function would have to *start again from the beginning* and traverse those same 600512 elements again. It could go a bit faster, since it doesn't have to check each element against `test` as it traverses. It already knows how far it has to travel. But still, this should seem a bit wasteful.
-
-Here's an idea. What if we had some way of representing a list as "broken" at a specific point. For example, if our base list is:
-
- [10; 20; 30; 40; 50; 60; 70; 80; 90]
-
-we might imagine the list "broken" at position 3 like this (positions are numbered starting from 0):
-
- 40;
- 30; 50;
- 20; 60;
- [10; 70;
- 80;
- 90]
-
-Then if we move one step forward in the list, it would be "broken" at position 4:
-
- 50;
- 40; 60;
- 30; 70;
- 20; 80;
- [10; 90]
-
-If we had some convenient representation of these "broken" lists, then our search function could hand *that* off to the retrieval function, and the retrieval function could start right at the position where the list was broken, without having to start at the beginning and traverse many elements to get there. The retrieval function would also be able to inspect elements both forwards and backwards from the position where the list was "broken".
-
-The kind of data structure we're looking for here is called a **list zipper**. To represent our first broken list, we'd use two lists: (1) containing the elements in the left branch, preceding the target element, *in the order reverse to their appearance in the base list*. (2) containing the target element and the rest of the list, in normal order. So:
-
- 40;
- 30; 50;
- 20; 60;
- [10; 70;
- 80;
- 90]
-
-would be represented as `([30; 20; 10], [40; 50; 60; 70; 80; 90])`. To move forward in the base list, we pop the head element `40` off of the head element of the second list in the zipper, and push it onto the first list, getting `([40; 30; 20; 10], [50; 60; 70; 80; 90])`. To move backwards again, we pop off of the first list, and push it onto the second. To reconstruct the base list, we just "move backwards" until the first list is empty. (This is supposed to evoke the image of zipping up a zipper; hence the data structure's name.)
-
-This is a very handy datastructure, and it will become even more handy when we translate it over to more complicated base structures, like trees. To help get a good conceptual grip on how to do that, it's useful to introduce a kind of symbolism for talking about zippers. This is just a metalanguage notation, for us theorists; we don't need our programs to interpret the notation. We'll use a specification like this:
-
- [10; 20; 30; *; 50; 60; 70; 80; 90], * filled by 40
-
-to represent a list zipper where the break is at position 3, and the element occupying that position is 40. For a list zipper, this is implemented using the pairs-of-lists structure described above.
-
-
-##Tree Zippers##
-
-Now how could we translate a zipper-like structure over to trees? What we're aiming for is a way to keep track of where we are in a tree, in the same way that the "broken" lists let us keep track of where we are in the base list. In a particular application, you may only need to implement this for binary trees---trees where internal nodes always have exactly two subtrees. But to get the guiding idea, it's helpful first to think about trees that permit nodes to have many subtrees. Suppose we have the following tree:
-
- 1000
- / | \
- / | \
- / | \
- / | \
- / | \
- 500 920 950
- / | \ / | \ / | \
- 20 50 80 91 92 93 94 95 96
- 1 2 3 4 5 6 7 8 9
-
-and we want to represent that we're at the node marked `50`. We might use the following metalanguage notation to specify this:
-
- {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
-
-This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label associated with that node, but rather to the whole subtree rooted at that node:
-
- 20
- / | \
- 1 2 3
-
-Similarly for `node 50` and `node 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
-
- {parent = ...; siblings = [*; node 920; node 950]}, * filled by node 500
-
-And the parent of that targetted subtree should intuitively be a tree targetted on `node 1000`:
-
- {parent = None; siblings = [*]}, * filled by node 1000
-
-This tree has no parents because it's the root of the base tree. Fully spelled out, then, our tree targetted on `node 50` would be:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- }, * filled by node 1000;
- siblings = [*; node 920; node 950]
- }, * filled by node 500;
- siblings = [node 20; *; node 80]
- }, * filled by node 50
-
-In fact, there's some redundancy in this structure, at the points where we have `* filled by node 1000` and `* filled by node 500`. Most of `node 1000`---with the exception of any label attached to node `1000` itself---is determined by the rest of this structure; and so too with `node 500`. So we could really work with:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- }, label for * position (at node 1000);
- siblings = [*; node 920; node 950]
- }, label for * position (at node 500);
- siblings = [node 20; *; node 80]
- }, * filled by node 50
-
-Or, if we only had labels on the leafs of our tree:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; node 920; node 950]
- },
- siblings = [node 20; *; node 80]
- }, * filled by node 50
-
-We're understanding the `20` here in `node 20` to just be a metalanguage marker to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagrams.)
-
-We still do need to keep track of what fills the outermost targetted position---`* filled by node 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
-
-For simplicity, I'll continue to use the abbreviated form:
-
- {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
-
-But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**, for a reason that will emerge. They should already seem intuitively similar to list zippers, though, at least in what we're using them to represent. I think it may initially be more helpful to call these **targetted trees**, though, and so will be switching back and forth between this different terms.
-
-Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:
-
- {parent = ...; siblings = [*; node 50; node 80]}, * filled by node 20
-
-and similarly for moving right. If the sibling list is implemented as a list zipper, you should already know how to do that. If one were designing a tree zipper for a more restricted kind of tree, however, such as a binary tree, one would probably not represent siblings with a list zipper, but with something more special-purpose and economical.
-
-Moving downward in the tree would be a matter of constructing a tree targetted on some child of `node 20`, with the first part of the targetted tree above as its parent:
-
- {
- parent = {parent = ...; siblings = [*; node 50; node 80]};
- siblings = [*; leaf 2; leaf 3]
- }, * filled by leaf 1
-
-How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
-
- node 20
- / | \
- / | \
- leaf 1 leaf 2 leaf 3
-
-We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that fills that parent's target position `*`:
-
- {
- parent = ...;
- siblings = [*; node 50; node 80]
- }, * filled by node 20
-
-Or, spelling that structure out fully:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; node 920; node 950]
- },
- siblings = [*; node 50; node 80]
- }, * filled by node 20
-
-Moving upwards yet again would get us:
-
- {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; node 920; node 950]
- }, * filled by node 500
-
-where `node 500` refers to a tree built from a root node whose children are given by the list `[*; node 50; node 80]`, with `node 20` inserted into the `*` position. Moving upwards yet again would get us:
+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.
- {
- parent = None;
- siblings = [*]
- }, * filled by node 1000
-
-where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.
-
-We haven't given you a real implementation of the tree zipper, but only a suggestive notation. We have however told you enough that you should be able to implement it yourself. Or if you're lazy, you can read:
-
-* [[!wikipedia Zipper (data structure)]]
-* Huet, Gerard. ["Functional Pearl: The Zipper"](http://www.st.cs.uni-sb.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf) Journal of Functional Programming 7 (5): 549-554, September 1997.
-* As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.
-
-##Same-fringe using a tree zipper##
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:
Anyway, using record types, we might define the tree zipper interface like so:
- type 'a starred_tree = Root | Starring_Left of 'a starred_pair | Starring_Right of 'a starred_pair
- and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
- and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;
+ 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 {tree; filler} = z
+ let {level; filler} = z
in match filler with
| Leaf _ -> z
| Node(left, right) ->
- let zdown = {tree = Starring_Left {parent = tree; sibling = right}; filler = left}
+ let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
in move_botleft zdown
;;
-->
(* else returns move_right_or_up (result of moving up in z) *)
<!--
- let {tree; filler} = z
- in match tree with
- | Starring_Left {parent; sibling = right} -> Some {tree = Starring_Right {parent; sibling = filler}; filler = right}
+ 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' = {tree = parent; filler = Node(left, filler)}
+ 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 =
- {tree = Root; filler = t}
+ {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 root of t *)
- let zstart = new_zipper t
- in let zbotleft = move_botleft zstart
+ (* 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
- | None -> (* we've finished enumerating the fringe *)
- None
| Some z -> (
(* extract label of currently-targetted leaf *)
let Leaf current = z.filler
| 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
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.") 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:
+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
------------ ------------ ------------
(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]]##
+