1 These notes may change in the next few days (today is 30 Nov 2010).
2 The material here benefited from many discussions with Ken Shan.
4 ##[[Tree and List Zippers]]##
6 ##[[Coroutines and Aborts]]##
8 ##[[From Lists to Continuations]]##
10 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:
12 let find_nth' (test : 'a -> bool) (n : int) (lst : 'a list) (default : 'a) : ('a * 'a * 'a) ->
13 let rec helper (predecessor : 'a) n lst =
15 | [] -> failwith "not found"
16 | x :: xs when test x -> (if n = 1
17 then (predecessor, x, match xs with [] -> default | y::ys -> y)
18 else helper x (n - 1) xs
20 | x :: xs -> helper x n xs
21 in helper default n lst;;
23 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...?
25 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.
27 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:
29 [10; 20; 30; 40; 50; 60; 70; 80; 90]
31 we might imagine the list "broken" at position 3 like this (positions are numbered starting from 0):
40 Then if we move one step forward in the list, it would be "broken" at position 4:
48 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".
50 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:
59 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.)
61 We had some discussio in seminar of the right way to understand the "zipper" metaphor. I think it's best to think of the tab of the zipper being here:
73 And imagine that you're just seeing the left half of a real-zipper, rotated 60 degrees counter-clockwise. When the list is all "zipped up", we've "move backwards" to the state where the first element is targetted:
75 ([], [10; 20; 30; 40; 50; 60; 70; 80; 90])
77 However you understand the "zipper" metaphor, 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:
79 [10; 20; 30; *; 50; 60; 70; 80; 90], * filled by 40
81 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.
86 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.
88 It's important to set some ground rules for what will follow. If you don't understand these ground rules you will get confused. First off, for many uses of trees one wants some of the nodes or leafs in the tree to be *labeled* with additional information. It's important not to conflate the label with the node itself. Numerically one and the same piece of information---for example, the same `int`---could label two nodes of the tree without those nodes thereby being identical, as here:
97 The leftmost leaf and the rightmost leaf have the same label; but they are different leafs. The leftmost leaf has a sibling leaf with the label 2; the rightmost leaf has no siblings that are leafs. Sometimes when one is diagramming trees, one will annotate the nodes with the labels, as above. Other times, when one is diagramming trees, one will instead want to annotate the nodes with tags to make it easier to refer to particular parts of the tree. So for instance, I could diagram the same tree as above as:
106 Here I haven't drawn what the labels are. The leftmost leaf, the node tagged "3" in this diagram, doesn't have the label `3`. It has the label 1, as we said before. I just haven't put that into the diagram. The node tagged "2" doesn't have the label `2`. It doesn't have any label. The tree in this example only has information labeling its leafs, not any of its inner nodes. The identity of its inner nodes is exhausted by their position in the tree.
108 That is a second thing to note. In what follows, we'll only be working with *leaf-labeled* trees. In some uses of trees, one also wants labels on inner nodes. But we won't be discussing any such trees now. Our trees only have labels on their leafs. The diagrams below will tag all of the nodes, as in the second diagram above, and won't display what the leafs' labels are.
110 Final introductory comment: in particular applications, you may only need to work with binary trees---trees where internal nodes always have exactly two subtrees. That is what we'll work with in the homework, for example. But to get the guiding idea of how tree zippers work, it's helpful first to think about trees that permit nodes to have many subtrees. So that's how we'll start.
112 Suppose we have the following tree:
122 20 50 80 91 92 93 94 95 96
125 This is a leaf-labeled tree whose labels aren't displayed. The `9200` and so on are tags to make it easier for us to refer to particular parts of the tree.
127 Suppose we want to represent that we're *at* the node marked `50`. We might use the following metalanguage notation to specify this:
129 {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
131 This is modeled on the notation suggested above for list zippers. Here `subtree 20` refers to the whole subtree rooted at node `20`:
137 Similarly for `subtree 50` and `subtree 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`:
139 {parent = ...; siblings = [*; subtree 920; subtree 950]}, * filled by subtree 500
141 And the parent of that targetted subtree should intuitively be a tree targetted on `node 9200`:
143 {parent = None; siblings = [*]}, * filled by tree 9200
145 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:
152 }, * filled by tree 9200;
153 siblings = [*; subtree 920; subtree 950]
154 }, * filled by subtree 500;
155 siblings = [subtree 20; *; subtree 80]
156 }, * filled by subtree 50
158 In fact, there's some redundancy in this structure, at the points where we have `* filled by tree 9200` and `* filled by subtree 500`. Since node 9200 doesn't have any label attached to it, the subtree rooted in it is determined by the rest of this structure; and so too with `subtree 500`. So we could really work with:
166 siblings = [*; subtree 920; subtree 950]
168 siblings = [subtree 20; *; subtree 80]
169 }, * filled by subtree 50
172 We still do need to keep track of what fills the outermost targetted position---`* filled by subtree 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
174 For simplicity, I'll continue to use the abbreviated form:
176 {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
178 But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**. They should already seem intuitively similar to list zippers, at least in what we're using them to represent. I think it may also be helpful to call them **targetted trees**, though, and so will be switching back and forth between these different terms.
180 Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:
182 {parent = ...; siblings = [*; subtree 50; subtree 80]}, * filled by subtree 20
184 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.
186 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:
189 parent = {parent = ...; siblings = [*; subtree 50; subtree 80]};
190 siblings = [*; leaf 2; leaf 3]
191 }, * filled by leaf 1
193 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:
200 We'll call this new untargetted tree `subtree 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 `subtree 20'` being the subtree that fills that parent's target position `*`:
204 siblings = [*; subtree 50; subtree 80]
205 }, * filled by subtree 20'
207 Or, spelling that structure out fully:
215 siblings = [*; subtree 920; subtree 950]
217 siblings = [*; subtree 50; subtree 80]
218 }, * filled by subtree 20'
220 Moving upwards yet again would get us:
227 siblings = [*; subtree 920; subtree 950]
228 }, * filled by subtree 500'
230 where `subtree 500'` refers to a tree built from a root node whose children are given by the list `[*; subtree 50; subtree 80]`, with `subtree 20'` inserted into the `*` position. Moving upwards yet again would get us:
235 }, * filled by tree 9200'
237 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.
239 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:
241 * [[!wikipedia Zipper (data structure)]]
242 * 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.
243 * As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.
246 ##Same-fringe using a zipper-based coroutine##
248 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):
256 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.
259 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:
267 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.
269 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.
271 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.
273 First, we define a type for leaf-labeled, binary trees:
275 type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
277 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:
279 # type blah = Blah of (int * int * (char -> bool));;
281 and then having to remember which element in the triple was which:
283 # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
284 Error: This expression has type int * (char -> bool) * int
285 but an expression was expected of type int * int * (char -> bool)
287 # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
288 val b1 : blah = Blah (1, 2, <fun>)
290 records let you attach descriptive labels to the components of the tuple:
292 # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
293 # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
294 val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
295 # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
296 val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}
298 These were the strategies to extract the components of an unlabeled tuple:
300 let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
302 let (h, w, test) = b1;; (* works for arbitrary tuples *)
305 | (h, w, test) -> ...;; (* same as preceding *)
307 Here is how you can extract the components of a labeled record:
309 let h = b2.height;; (* handy! *)
311 let {height = h; weight = w; char_tester = test} = b2
312 in (* go on to use h, w, and test ... *)
315 | {height = h; weight = w; char_tester = test} ->
316 (* go on to use h, w, and test ... *)
318 Anyway, using record types, we might define the tree zipper interface like so:
320 type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot
321 and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;
323 type 'a zipper = { level : 'a starred_level; filler: 'a tree };;
325 let rec move_botleft (z : 'a zipper) : 'a zipper =
326 (* returns z if the targetted node in z has no children *)
327 (* else returns move_botleft (zipper which results from moving down and left in z) *)
330 let {level; filler} = z
333 | Node(left, right) ->
334 let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
335 in move_botleft zdown
339 let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
340 (* if it's possible to move right in z, returns Some (the result of doing so) *)
341 (* else if it's not possible to move any further up in z, returns None *)
342 (* else returns move_right_or_up (result of moving up in z) *)
345 let {level; filler} = z
347 | Starring_Left {parent; sibling = right} -> Some {level = Starring_Right {parent; sibling = filler}; filler = right}
349 | Starring_Right {parent; sibling = left} ->
350 let z' = {level = parent; filler = Node(left, filler)}
351 in move_right_or_up z'
355 The following function takes an 'a tree and returns an 'a zipper focused on its root:
357 let new_zipper (t : 'a tree) : 'a zipper =
358 {level = Root; filler = t}
361 Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
363 let make_fringe_enumerator (t: 'a tree) =
364 (* create a zipper targetting the botleft of t *)
365 let zbotleft = move_botleft (new_zipper t)
366 (* create a refcell initially pointing to zbotleft *)
367 in let zcell = ref (Some zbotleft)
368 (* construct the next_leaf function *)
369 in let next_leaf () : 'a option =
372 (* extract label of currently-targetted leaf *)
373 let Leaf current = z.filler
374 (* update zcell to point to next leaf, if there is one *)
375 in let () = zcell := match move_right_or_up z with
377 | Some z' -> Some (move_botleft z')
378 (* return saved label *)
380 | None -> (* we've finished enumerating the fringe *)
383 (* return the next_leaf function *)
387 Here's an example of `make_fringe_enumerator` in action:
389 # let tree1 = Leaf 1;;
390 val tree1 : int tree = Leaf 1
391 # let next1 = make_fringe_enumerator tree1;;
392 val next1 : unit -> int option = <fun>
394 - : int option = Some 1
396 - : int option = None
398 - : int option = None
399 # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
400 val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
401 # let next2 = make_fringe_enumerator tree2;;
402 val next2 : unit -> int option = <fun>
404 - : int option = Some 1
406 - : int option = Some 2
408 - : int option = Some 3
410 - : int option = None
412 - : int option = None
414 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`.
416 Using these fringe enumerators, we can write our `same_fringe` function like this:
418 let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
419 let next1 = make_fringe_enumerator t1
420 in let next2 = make_fringe_enumerator t2
421 in let rec loop () : bool =
422 match next1 (), next2 () with
423 | Some a, Some b when a = b -> loop ()
429 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.
431 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*.
433 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:
435 main program next1 thread next2 thread
436 ------------ ------------ ------------
439 (paused) calculate first leaf
440 (paused) <--- return it
441 start next2 (paused) starting
442 (paused) (paused) calculate first leaf
443 (paused) (paused) <-- return it
444 compare leaves (paused) (paused)
445 call loop again (paused) (paused)
446 call next1 again (paused) (paused)
447 (paused) calculate next leaf (paused)
448 (paused) <-- return it (paused)
451 If you want to read more about these kinds of threads, here are some links:
453 <!-- * [[!wikipedia Computer_multitasking]]
454 * [[!wikipedia Thread_(computer_science)]] -->
456 * [[!wikipedia Coroutine]]
457 * [[!wikipedia Iterator]]
458 * [[!wikipedia Generator_(computer_science)]]
459 * [[!wikipedia Fiber_(computer_science)]]
460 <!-- * [[!wikipedia Green_threads]]
461 * [[!wikipedia Protothreads]] -->
463 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.
465 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:
467 > function fringe_enumerator (tree)
469 coroutine.yield (tree.leaf)
471 fringe_enumerator (tree.left)
472 fringe_enumerator (tree.right)
476 > function same_fringe (tree1, tree2)
477 local next1 = coroutine.wrap (fringe_enumerator)
478 local next2 = coroutine.wrap (fringe_enumerator)
479 local function loop (leaf1, leaf2)
480 if leaf1 or leaf2 then
481 return leaf1 == leaf2 and loop( next1(), next2() )
482 elseif not leaf1 and not leaf2 then
488 return loop (next1(tree1), next2(tree2))
491 > return same_fringe ( {leaf=1}, {leaf=2})
494 > return same_fringe ( {leaf=1}, {leaf=1})
497 > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
498 {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
501 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.
504 ##Exceptions and Aborts##
506 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.
508 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:
510 # let lst = [1; 2] in
512 Error: This expression has type int list
513 but an expression was expected of type string list
515 But you may also have encountered other kinds of error, that arise while your program is running. For example:
518 Exception: Division_by_zero.
519 # List.nth [1;2] 10;;
520 Exception: Failure "nth".
522 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`:
525 if n < 0 then invalid_arg "List.nth" else
526 let rec nth_aux l n =
528 | [] -> failwith "nth"
529 | a::l -> if n = 0 then a else nth_aux l (n-1)
532 Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:
534 raise (Invalid_argument "List.nth");;
535 raise (Failure "nth");;
537 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:
541 the effect is for the program to immediately stop without evaluating any further code:
543 # let xcell = ref 0;;
544 val xcell : int ref = {contents = 0}
545 # let ex = Failure "test"
548 Exception: Failure "test".
552 Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
554 I said when you evaluate the expression:
558 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:
563 else if x = 2 then raise (Failure "two")
564 else raise (Failure "three")
565 with Failure "two" -> 20
567 val foo : int -> int = <fun>
573 Exception: Failure "three".
575 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.
577 So what I should have said is that when you evaluate the expression:
581 *and that exception is never caught*, then the effect is for the program to immediately stop.
583 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.
585 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:
589 raise (Failure "blah")
590 with Failure "fooey" -> 10
591 with Failure "blah" -> 20;;
594 The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
599 with Failure "blah" -> 20
601 raise (Failure "blah")
605 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`.
607 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:
616 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.
618 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:
636 Okay, so that's our second execution pattern.
638 ##What do these have in common?##
640 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.
642 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:
651 we can imagine a box:
654 +---------------------------+
656 | (if x = 1 then 10 |
657 | else abort 20) + 1 |
659 +---------------------------+
662 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.
665 # #require "delimcc";;
667 # let reset body = let p = new_prompt () in push_prompt p (body p);;
668 val reset : ('a Delimcc.prompt -> unit -> 'a) -> 'a = <fun>
669 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 1) + 100;;
671 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 2) + 100;;
678 --------------------------------------
680 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.
683 ##Introducing Continuations##
685 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."
687 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.
689 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.
691 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.
693 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.
695 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:
697 \handler. handler x y
699 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.
701 Consider a complex computation, such as:
703 1 + 2 * (1 - g (3 + 4))
705 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:
707 \result. 1 + 2 * (1 - result)
709 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.
711 Go back and read the material on "Aborting a Search Through a List" in [[Week4]] for an example of doing this.
713 In very general terms, the strategy is to work with functions like this:
717 ... if you want to abort early, supply an argument to k ...
718 ... do more stuff ...
720 in let gcon = fun result -> 1 + 2 * (1 - result)
721 in gcon (g' gcon (3 + 4))
723 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.
725 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:
729 ... if you want to abort early, supply an argument to k ...
730 ... do more stuff ...
732 in let gcon = fun result ->
733 let final_value = 1 + 2 * (1 - result)
734 in end_program_with final_value
735 in gcon (g' gcon (3 + 4))
737 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.)
739 So now, guess what would be the result of doing the following:
743 in let gcon = fun result ->
744 let final_value = (1, result)
745 in end_program_with final_value
746 in gcon (g' gcon (3 + 4))
748 <!-- (1, 7) ... explain why not (1, 8) -->
751 Refunctionalizing zippers: from lists to continuations
752 ------------------------------------------------------
754 If zippers are continuations reified (defuntionalized), then one route
755 to continuations is to re-functionalize a zipper. Then the
756 concreteness and understandability of the zipper provides a way of
757 understanding and equivalent treatment using continuations.
759 Let's work with lists of chars for a change. To maximize readability, we'll
760 indulge in an abbreviatory convention that "abSd" abbreviates the
761 list `['a'; 'b'; 'S'; 'd']`.
763 We will set out to compute a deceptively simple-seeming **task: given a
764 string, replace each occurrence of 'S' in that string with a copy of
765 the string up to that point.**
767 We'll define a function `t` (for "task") that maps strings to their
777 In linguistic terms, this is a kind of anaphora
778 resolution, where `'S'` is functioning like an anaphoric element, and
779 the preceding string portion is the antecedent.
781 This deceptively simple task gives rise to some mind-bending complexity.
782 Note that it matters which 'S' you target first (the position of the *
783 indicates the targeted 'S'):
814 ~~> t "aSbaaaSbaabab"
819 Aparently, this task, as simple as it is, is a form of computation,
820 and the order in which the `'S'`s get evaluated can lead to divergent
823 For now, we'll agree to always evaluate the leftmost `'S'`, which
824 guarantees termination, and a final string without any `'S'` in it.
826 This is a task well-suited to using a zipper. We'll define a function
827 `tz` (for task with zippers), which accomplishes the task by mapping a
828 char list zipper to a char list. We'll call the two parts of the
829 zipper `unzipped` and `zipped`; we start with a fully zipped list, and
830 move elements to the zipped part by pulling the zipped down until the
831 entire list has been unzipped (and so the zipped half of the zipper is empty).
834 type 'a list_zipper = ('a list) * ('a list);;
836 let rec tz (z:char list_zipper) =
837 match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
838 | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
839 | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
841 # tz ([], ['a'; 'b'; 'S'; 'd']);;
842 - : char list = ['a'; 'b'; 'a'; 'b'; 'd']
844 # tz ([], ['a'; 'S'; 'b'; 'S']);;
845 - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
848 Note that this implementation enforces the evaluate-leftmost rule.
851 One way to see exactly what is going on is to watch the zipper in
852 action by tracing the execution of `tz`. By using the `#trace`
853 directive in the Ocaml interpreter, the system will print out the
854 arguments to `tz` each time it is (recurcively) called. Note that the
855 lines with left-facing arrows (`<--`) show (recursive) calls to `tz`,
856 giving the value of its argument (a zipper), and the lines with
857 right-facing arrows (`-->`) show the output of each recursive call, a
863 # tz ([], ['a'; 'b'; 'S'; 'd']);;
864 tz <-- ([], ['a'; 'b'; 'S'; 'd'])
865 tz <-- (['a'], ['b'; 'S'; 'd']) (* Pull zipper *)
866 tz <-- (['b'; 'a'], ['S'; 'd']) (* Pull zipper *)
867 tz <-- (['b'; 'a'; 'b'; 'a'], ['d']) (* Special step *)
868 tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], []) (* Pull zipper *)
869 tz --> ['a'; 'b'; 'a'; 'b'; 'd'] (* Output reversed *)
870 tz --> ['a'; 'b'; 'a'; 'b'; 'd']
871 tz --> ['a'; 'b'; 'a'; 'b'; 'd']
872 tz --> ['a'; 'b'; 'a'; 'b'; 'd']
873 tz --> ['a'; 'b'; 'a'; 'b'; 'd']
874 - : char list = ['a'; 'b'; 'a'; 'b'; 'd']
877 The nice thing about computations involving lists is that it's so easy
878 to visualize them as a data structure. Eventually, we want to get to
879 a place where we can talk about more abstract computations. In order
880 to get there, we'll first do the exact same thing we just did with
881 concrete zipper using procedures.
883 Think of a list as a procedural recipe: `['a'; 'b'; 'S'; 'd']`
884 is the result of the computation `a::(b::(S::(d::[])))` (or, in our old
885 style, `makelist a (makelist b (makelist S (makelist c empty)))`).
886 The recipe for constructing the list goes like this:
889 (0) Start with the empty list []
890 (1) make a new list whose first element is 'd' and whose tail is the list constructed in step (0)
891 (2) make a new list whose first element is 'S' and whose tail is the list constructed in step (1)
892 -----------------------------------------
893 (3) make a new list whose first element is 'b' and whose tail is the list constructed in step (2)
894 (4) make a new list whose first element is 'a' and whose tail is the list constructed in step (3)
897 What is the type of each of these steps? Well, it will be a function
898 from the result of the previous step (a list) to a new list: it will
899 be a function of type `char list -> char list`. We'll call each step
900 (or group of steps) a **continuation** of the recipe. So in this
901 context, a continuation is a function of type `char list -> char
902 list`. For instance, the continuation corresponding to the portion of
903 the recipe below the horizontal line is the function `fun (tail:char
904 list) -> a::(b::tail)`.
906 This means that we can now represent the unzipped part of our
907 zipper--the part we've already unzipped--as a continuation: a function
908 describing how to finish building the list. We'll write a new
909 function, `tc` (for task with continuations), that will take an input
910 list (not a zipper!) and a continuation and return a processed list.
911 The structure and the behavior will follow that of `tz` above, with
912 some small but interesting differences. We've included the orginal
913 `tz` to facilitate detailed comparison:
916 let rec tz (z:char list_zipper) =
917 match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
918 | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
919 | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
921 let rec tc (l: char list) (c: (char list) -> (char list)) =
922 match l with [] -> List.rev (c [])
923 | 'S'::zipped -> tc zipped (fun x -> c (c x))
924 | target::zipped -> tc zipped (fun x -> target::(c x));;
926 # tc ['a'; 'b'; 'S'; 'd'] (fun x -> x);;
927 - : char list = ['a'; 'b'; 'a'; 'b']
929 # tc ['a'; 'S'; 'b'; 'S'] (fun x -> x);;
930 - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
933 To emphasize the parallel, I've re-used the names `zipped` and
934 `target`. The trace of the procedure will show that these variables
935 take on the same values in the same series of steps as they did during
936 the execution of `tz` above. There will once again be one initial and
937 four recursive calls to `tc`, and `zipped` will take on the values
938 `"bSd"`, `"Sd"`, `"d"`, and `""` (and, once again, on the final call,
939 the first `match` clause will fire, so the the variable `zipper` will
940 not be instantiated).
942 I have not called the functional argument `unzipped`, although that is
943 what the parallel would suggest. The reason is that `unzipped` is a
944 list, but `c` is a function. That's the most crucial difference, the
945 point of the excercise, and it should be emphasized. For instance,
946 you can see this difference in the fact that in `tz`, we have to glue
947 together the two instances of `unzipped` with an explicit (and
948 relatively inefficient) `List.append`.
949 In the `tc` version of the task, we simply compose `c` with itself:
950 `c o c = fun x -> c (c x)`.
952 Why use the identity function as the initial continuation? Well, if
953 you have already constructed the initial list `"abSd"`, what's the next
954 step in the recipe to produce the desired result, i.e, the very same
955 list, `"abSd"`? Clearly, the identity continuation.
957 A good way to test your understanding is to figure out what the
958 continuation function `c` must be at the point in the computation when
959 `tc` is called with the first argument `"Sd"`. Two choices: is it
960 `fun x -> a::b::x`, or it is `fun x -> b::a::x`? The way to see if
961 you're right is to execute the following command and see what happens:
963 tc ['S'; 'd'] (fun x -> 'a'::'b'::x);;
965 There are a number of interesting directions we can go with this task.
966 The reason this task was chosen is because it can be viewed as a
967 simplified picture of a computation using continuations, where `'S'`
968 plays the role of a control operator with some similarities to what is
969 often called `shift`. In the analogy, the input list portrays a
970 sequence of functional applications, where `[f1; f2; f3; x]` represents
971 `f1(f2(f3 x))`. The limitation of the analogy is that it is only
972 possible to represent computations in which the applications are
973 always right-branching, i.e., the computation `((f1 f2) f3) x` cannot
974 be directly represented.
976 One possibile development is that we could add a special symbol `'#'`,
977 and then the task would be to copy from the target `'S'` only back to
978 the closest `'#'`. This would allow the task to simulate delimited
979 continuations with embedded prompts.
981 The reason the task is well-suited to the list zipper is in part
982 because the list monad has an intimate connection with continuations.
983 The following section explores this connection. We'll return to the
984 list task after talking about generalized quantifiers below.
987 Rethinking the list monad
988 -------------------------
990 To construct a monad, the key element is to settle on a type
991 constructor, and the monad more or less naturally follows from that.
992 We'll remind you of some examples of how monads follow from the type
993 constructor in a moment. This will involve some review of familair
994 material, but it's worth doing for two reasons: it will set up a
995 pattern for the new discussion further below, and it will tie together
996 some previously unconnected elements of the course (more specifically,
997 version 3 lists and monads).
999 For instance, take the **Reader Monad**. Once we decide that the type
1002 type 'a reader = env -> 'a
1004 then the choice of unit and bind is natural:
1006 let r_unit (a : 'a) : 'a reader = fun (e : env) -> a
1008 The reason this is a fairly natural choice is that because the type of
1009 an `'a reader` is `env -> 'a` (by definition), the type of the
1010 `r_unit` function is `'a -> env -> 'a`, which is an instance of the
1011 type of the *K* combinator. So it makes sense that *K* is the unit
1012 for the reader monad.
1014 Since the type of the `bind` operator is required to be
1016 r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)
1018 We can reason our way to the traditional reader `bind` function as
1019 follows. We start by declaring the types determined by the definition
1020 of a bind operation:
1022 let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...
1024 Now we have to open up the `u` box and get out the `'a` object in order to
1025 feed it to `f`. Since `u` is a function from environments to
1026 objects of type `'a`, the way we open a box in this monad is
1027 by applying it to an environment:
1033 This subexpression types to `'b reader`, which is good. The only
1034 problem is that we made use of an environment `e` that we didn't already have,
1035 so we must abstract over that variable to balance the books:
1037 fun e -> f (u e) ...
1039 [To preview the discussion of the Curry-Howard correspondence, what
1040 we're doing here is constructing an intuitionistic proof of the type,
1041 and using the Curry-Howard labeling of the proof as our bind term.]
1043 This types to `env -> 'b reader`, but we want to end up with `env ->
1044 '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:
1047 r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e
1050 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.
1052 [The bind we cite here is a condensed version of the careful `let a = u e in ...`
1053 constructions we provided in earlier lectures. We use the condensed
1054 version here in order to emphasize similarities of structure across
1057 The **State Monad** is similar. Once we've decided to use the following type constructor:
1059 type 'a state = store -> ('a, store)
1061 Then our unit is naturally:
1063 let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
1065 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:
1067 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
1070 But unlocking the `u` box is a little more complicated. As before, we
1071 need to posit a state `s` that we can apply `u` to. Once we do so,
1072 however, we won't have an `'a`, we'll have a pair whose first element
1073 is an `'a`. So we have to unpack the pair:
1075 ... let (a, s') = u s in ... (f a) ...
1077 Abstracting over the `s` and adjusting the types gives the result:
1079 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
1080 fun (s : store) -> let (a, s') = u s in f a s'
1082 The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we
1083 won't pause to explore it here, though conceptually its unit and bind
1084 follow just as naturally from its type constructor.
1086 Our other familiar monad is the **List Monad**, which we were told
1089 type 'a list = ['a];;
1090 l_unit (a : 'a) = [a];;
1091 l_bind u f = List.concat (List.map f u);;
1093 Thinking through the list monad will take a little time, but doing so
1094 will provide a connection with continuations.
1096 Recall that `List.map` takes a function and a list and returns the
1097 result to applying the function to the elements of the list:
1099 List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
1101 and List.concat takes a list of lists and erases the embdded list
1104 List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
1108 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
1110 Now, why this unit, and why this bind? Well, ideally a unit should
1111 not throw away information, so we can rule out `fun x -> []` as an
1112 ideal unit. And units should not add more information than required,
1113 so there's no obvious reason to prefer `fun x -> [x,x]`. In other
1114 words, `fun x -> [x]` is a reasonable choice for a unit.
1116 As for bind, an `'a list` monadic object contains a lot of objects of
1117 type `'a`, and we want to make use of each of them (rather than
1118 arbitrarily throwing some of them away). The only
1119 thing we know for sure we can do with an object of type `'a` is apply
1120 the function of type `'a -> 'a list` to them. Once we've done so, we
1121 have a collection of lists, one for each of the `'a`'s. One
1122 possibility is that we could gather them all up in a list, so that
1123 `bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`. But that restricts
1124 the object returned by the second argument of `bind` to always be of
1125 type `'b list list`. We can elimiate that restriction by flattening
1126 the list of lists into a single list: this is
1127 just List.concat applied to the output of List.map. So there is some logic to the
1128 choice of unit and bind for the list monad.
1130 Yet we can still desire to go deeper, and see if the appropriate bind
1131 behavior emerges from the types, as it did for the previously
1132 considered monads. But we can't do that if we leave the list type as
1133 a primitive Ocaml type. However, we know several ways of implementing
1134 lists using just functions. In what follows, we're going to use type
1135 3 lists, the right fold implementation (though it's important and
1136 intriguing to wonder how things would change if we used some other
1137 strategy for implementating lists). These were the lists that made
1138 lists look like Church numerals with extra bits embdded in them:
1140 empty list: fun f z -> z
1141 list with one element: fun f z -> f 1 z
1142 list with two elements: fun f z -> f 2 (f 1 z)
1143 list with three elements: fun f z -> f 3 (f 2 (f 1 z))
1145 and so on. To save time, we'll let the OCaml interpreter infer the
1146 principle types of these functions (rather than inferring what the
1147 types should be ourselves):
1150 - : 'a -> 'b -> 'b = <fun>
1151 # fun f z -> f 1 z;;
1152 - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
1153 # fun f z -> f 2 (f 1 z);;
1154 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
1155 # fun f z -> f 3 (f 2 (f 1 z))
1156 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
1158 We can see what the consistent, general principle types are at the end, so we
1159 can stop. These types should remind you of the simply-typed lambda calculus
1160 types for Church numerals (`(o -> o) -> o -> o`) with one extra type
1161 thrown in, the type of the element a the head of the list
1162 (in this case, an int).
1164 So here's our type constructor for our hand-rolled lists:
1166 type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
1168 Generalizing to lists that contain any kind of element (not just
1171 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
1173 So an `('a, 'b) list'` is a list containing elements of type `'a`,
1174 where `'b` is the type of some part of the plumbing. This is more
1175 general than an ordinary OCaml list, but we'll see how to map them
1176 into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
1177 in order to proceed to build a monad:
1179 l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z
1181 No problem. Arriving at bind is a little more complicated, but
1182 exactly the same principles apply, you just have to be careful and
1183 systematic about it.
1185 l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
1187 Unpacking the types gives:
1189 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
1190 (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
1191 : ('c -> 'd -> 'd) -> 'd -> 'd = ...
1193 Perhaps a bit intimiating.
1194 But it's a rookie mistake to quail before complicated types. You should
1195 be no more intimiated by complex types than by a linguistic tree with
1196 deeply embedded branches: complex structure created by repeated
1197 application of simple rules.
1199 [This would be a good time to try to build your own term for the types
1200 just given. Doing so (or attempting to do so) will make the next
1201 paragraph much easier to follow.]
1203 As usual, we need to unpack the `u` box. Examine the type of `u`.
1204 This time, `u` will only deliver up its contents if we give `u` an
1205 argument that is a function expecting an `'a` and a `'b`. `u` will
1206 fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
1208 ... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...
1210 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)`:
1212 ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
1214 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:
1216 ... u (fun (a : 'a) (b : 'b) -> f a k b) ...
1218 Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:
1220 fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
1222 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:
1224 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
1225 (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
1226 : ('c -> 'b -> 'b) -> 'b -> 'b =
1227 fun k -> u (fun a b -> f a k b)
1229 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.
1231 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:
1233 fun k z -> u (fun a b -> f a k b) z
1235 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?
1237 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:
1240 concat [[]; [2]; [2; 4]; [2; 4; 8]] =
1243 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
1245 fun k z -> u (fun a b -> f a k b) z
1247 do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
1254 (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.
1256 So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
1259 right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
1260 right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
1261 right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
1262 right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0
1264 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:
1266 fun k z -> u (fun a b -> f a k b) z
1268 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
1270 fun k z -> List.fold_right k (concat (map f u)) z
1274 For future reference, we might make two eta-reductions to our formula, so that we have instead:
1276 let l'_bind = fun k -> u (fun a -> f a k);;
1278 Let's make some more tests:
1281 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
1283 l'_bind (fun f z -> f 1 (f 2 z))
1284 (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
1286 Sigh. OCaml won't show us our own list. So we have to choose an `f`
1287 and a `z` that will turn our hand-crafted lists into standard OCaml
1288 lists, so that they will print out.
1290 # let cons h t = h :: t;; (* OCaml is stupid about :: *)
1291 # l'_bind (fun f z -> f 1 (f 2 z))
1292 (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
1293 - : int list = [1; 2; 2; 3]
1298 Montague's PTQ treatment of DPs as generalized quantifiers
1299 ----------------------------------------------------------
1301 We've hinted that Montague's treatment of DPs as generalized
1302 quantifiers embodies the spirit of continuations (see de Groote 2001,
1303 Barker 2002 for lengthy discussion). Let's see why.
1305 First, we'll need a type constructor. As you probably know,
1306 Montague replaced individual-denoting determiner phrases (with type `e`)
1307 with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
1308 In particular, the denotation of a proper name like *John*, which
1309 might originally denote a object `j` of type `e`, came to denote a
1310 generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
1311 Let's write a general function that will map individuals into their
1312 corresponding generalized quantifier:
1314 gqize (a : e) = fun (p : e -> t) -> p a
1316 This function is what Partee 1987 calls LIFT, and it would be
1317 reasonable to use it here, but we will avoid that name, given that we
1318 use that word to refer to other functions.
1320 This function wraps up an individual in a box. That is to say,
1321 we are in the presence of a monad. The type constructor, the unit and
1322 the bind follow naturally. We've done this enough times that we won't
1323 belabor the construction of the bind function, the derivation is
1324 highly similar to the List monad just given:
1326 type 'a continuation = ('a -> 'b) -> 'b
1327 c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
1328 c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
1329 fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
1331 Note that `c_unit` is exactly the `gqize` function that Montague used
1332 to lift individuals into the continuation monad.
1334 That last bit in `c_bind` looks familiar---we just saw something like
1335 it in the List monad. How similar is it to the List monad? Let's
1336 examine the type constructor and the terms from the list monad derived
1339 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
1340 l'_unit a = fun f -> f a
1341 l'_bind u f = fun k -> u (fun a -> f a k)
1343 (We performed a sneaky but valid eta reduction in the unit term.)
1345 The unit and the bind for the Montague continuation monad and the
1346 homemade List monad are the same terms! In other words, the behavior
1347 of the List monad and the behavior of the continuations monad are
1348 parallel in a deep sense.
1350 Have we really discovered that lists are secretly continuations? Or
1351 have we merely found a way of simulating lists using list
1352 continuations? Well, strictly speaking, what we have done is shown
1353 that one particular implementation of lists---the right fold
1354 implementation---gives rise to a continuation monad fairly naturally,
1355 and that this monad can reproduce the behavior of the standard list
1356 monad. But what about other list implementations? Do they give rise
1357 to monads that can be understood in terms of continuations?
1359 Manipulating trees with monads
1360 ------------------------------
1362 This topic develops an idea based on a detailed suggestion of Ken
1363 Shan's. We'll build a series of functions that operate on trees,
1364 doing various things, including replacing leaves, counting nodes, and
1365 converting a tree to a list of leaves. The end result will be an
1366 application for continuations.
1368 From an engineering standpoint, we'll build a tree transformer that
1369 deals in monads. We can modify the behavior of the system by swapping
1370 one monad for another. We've already seen how adding a monad can add
1371 a layer of funtionality without disturbing the underlying system, for
1372 instance, in the way that the reader monad allowed us to add a layer
1373 of intensionality to an extensional grammar, but we have not yet seen
1374 the utility of replacing one monad with other.
1376 First, we'll be needing a lot of trees during the remainder of the
1377 course. Here's a type constructor for binary trees:
1379 type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
1381 These are trees in which the internal nodes do not have labels. [How
1382 would you adjust the type constructor to allow for labels on the
1385 We'll be using trees where the nodes are integers, e.g.,
1389 let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
1390 (Node ((Leaf 5),(Node ((Leaf 7),
1405 Our first task will be to replace each leaf with its double:
1408 let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) =
1409 match t with Leaf x -> Leaf (newleaf x)
1410 | Node (l, r) -> Node ((treemap newleaf l),
1411 (treemap newleaf r));;
1413 `treemap` takes a function that transforms old leaves into new leaves,
1414 and maps that function over all the leaves in the tree, leaving the
1415 structure of the tree unchanged. For instance:
1418 let double i = i + i;;
1421 Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
1435 We could have built the doubling operation right into the `treemap`
1436 code. However, because what to do to each leaf is a parameter, we can
1437 decide to do something else to the leaves without needing to rewrite
1438 `treemap`. For instance, we can easily square each leaf instead by
1439 supplying the appropriate `int -> int` operation in place of `double`:
1442 let square x = x * x;;
1445 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
1448 Note that what `treemap` does is take some global, contextual
1449 information---what to do to each leaf---and supplies that information
1450 to each subpart of the computation. In other words, `treemap` has the
1451 behavior of a reader monad. Let's make that explicit.
1453 In general, we're on a journey of making our treemap function more and
1454 more flexible. So the next step---combining the tree transducer with
1455 a reader monad---is to have the treemap function return a (monadized)
1456 tree that is ready to accept any `int->int` function and produce the
1459 \tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
1473 That is, we want to transform the ordinary tree `t1` (of type `int
1474 tree`) into a reader object of type `(int->int)-> int tree`: something
1475 that, when you apply it to an `int->int` function returns an `int
1476 tree` in which each leaf `x` has been replaced with `(f x)`.
1478 With previous readers, we always knew which kind of environment to
1479 expect: either an assignment function (the original calculator
1480 simulation), a world (the intensionality monad), an integer (the
1481 Jacobson-inspired link monad), etc. In this situation, it will be
1482 enough for now to expect that our reader will expect a function of
1486 type 'a reader = (int->int) -> 'a;; (* mnemonic: e for environment *)
1487 let reader_unit (x:'a): 'a reader = fun _ -> x;;
1488 let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;;
1491 It's easy to figure out how to turn an `int` into an `int reader`:
1494 let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
1495 int2int_reader 2 (fun i -> i + i);;
1499 But what do we do when the integers are scattered over the leaves of a
1500 tree? A binary tree is not the kind of thing that we can apply a
1501 function of type `int->int` to.
1504 let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
1505 match t with Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x'))
1506 | Node (l, r) -> reader_bind (treemonadizer f l) (fun x ->
1507 reader_bind (treemonadizer f r) (fun y ->
1508 reader_unit (Node (x, y))));;
1511 This function says: give me a function `f` that knows how to turn
1512 something of type `'a` into an `'b reader`, and I'll show you how to
1513 turn an `'a tree` into an `'a tree reader`. In more fanciful terms,
1514 the `treemonadizer` function builds plumbing that connects all of the
1515 leaves of a tree into one connected monadic network; it threads the
1516 monad through the leaves.
1519 # treemonadizer int2int_reader t1 (fun i -> i + i);;
1521 Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
1524 Here, our environment is the doubling function (`fun i -> i + i`). If
1525 we apply the very same `int tree reader` (namely, `treemonadizer
1526 int2int_reader t1`) to a different `int->int` function---say, the
1527 squaring function, `fun i -> i * i`---we get an entirely different
1531 # treemonadizer int2int_reader t1 (fun i -> i * i);;
1533 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
1536 Now that we have a tree transducer that accepts a monad as a
1537 parameter, we can see what it would take to swap in a different monad.
1538 For instance, we can use a state monad to count the number of nodes in
1542 type 'a state = int -> 'a * int;;
1543 let state_unit x i = (x, i+.5);;
1544 let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
1547 Gratifyingly, we can use the `treemonadizer` function without any
1548 modification whatsoever, except for replacing the (parametric) type
1549 `reader` with `state`:
1552 let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
1553 match t with Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x'))
1554 | Node (l, r) -> state_bind (treemonadizer f l) (fun x ->
1555 state_bind (treemonadizer f r) (fun y ->
1556 state_unit (Node (x, y))));;
1559 Then we can count the number of nodes in the tree:
1562 # treemonadizer state_unit t1 0;;
1563 - : int tree * int =
1564 (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
1578 Notice that we've counted each internal node twice---it's a good
1579 exercise to adjust the code to count each node once.
1581 One more revealing example before getting down to business: replacing
1582 `state` everywhere in `treemonadizer` with `list` gives us
1585 # treemonadizer (fun x -> [ [x; square x] ]) t1;;
1586 - : int list tree list =
1588 (Node (Leaf [2; 4], Leaf [3; 9]),
1589 Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]
1592 Unlike the previous cases, instead of turning a tree into a function
1593 from some input to a result, this transformer replaces each `int` with
1596 Now for the main point. What if we wanted to convert a tree to a list
1600 type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
1601 let continuation_unit x c = c x;;
1602 let continuation_bind u f c = u (fun a -> f a c);;
1604 let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
1605 match t with Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x'))
1606 | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x ->
1607 continuation_bind (treemonadizer f r) (fun y ->
1608 continuation_unit (Node (x, y))));;
1611 We use the continuation monad described above, and insert the
1612 `continuation` type in the appropriate place in the `treemonadizer` code.
1616 # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
1617 - : int list = [2; 3; 5; 7; 11]
1620 We have found a way of collapsing a tree into a list of its leaves.
1622 The continuation monad is amazingly flexible; we can use it to
1623 simulate some of the computations performed above. To see how, first
1624 note that an interestingly uninteresting thing happens if we use the
1625 continuation unit as our first argument to `treemonadizer`, and then
1626 apply the result to the identity function:
1629 # treemonadizer continuation_unit t1 (fun x -> x);;
1631 Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
1634 That is, nothing happens. But we can begin to substitute more
1635 interesting functions for the first argument of `treemonadizer`:
1638 (* Simulating the tree reader: distributing a operation over the leaves *)
1639 # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
1641 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
1643 (* Simulating the int list tree list *)
1644 # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);;
1647 (Node (Leaf [2; 4], Leaf [3; 9]),
1648 Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
1650 (* Counting leaves *)
1651 # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
1655 We could simulate the tree state example too, but it would require
1656 generalizing the type of the continuation monad to
1658 type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
1660 The binary tree monad
1661 ---------------------
1663 Of course, by now you may have realized that we have discovered a new
1664 monad, the binary tree monad:
1667 type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
1668 let tree_unit (x:'a) = Leaf x;;
1669 let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree =
1670 match u with Leaf x -> f x
1671 | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;
1674 For once, let's check the Monad laws. The left identity law is easy:
1676 Left identity: bind (unit a) f = bind (Leaf a) f = fa
1678 To check the other two laws, we need to make the following
1679 observation: it is easy to prove based on `tree_bind` by a simple
1680 induction on the structure of the first argument that the tree
1681 resulting from `bind u f` is a tree with the same strucure as `u`,
1682 except that each leaf `a` has been replaced with `fa`:
1684 \tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
1701 Given this equivalence, the right identity law
1703 Right identity: bind u unit = u
1705 falls out once we realize that
1707 bind (Leaf a) unit = unit a = Leaf a
1709 As for the associative law,
1711 Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
1713 we'll give an example that will show how an inductive proof would
1714 proceed. Let `f a = Node (Leaf a, Leaf a)`. Then
1716 \tree (. (. (. (. (a1)(a2)))))
1717 \tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) ))
1722 bind __|__ f = __|_ = . .
1724 a1 a2 fa1 fa2 | | | |
1728 Now when we bind this tree to `g`, we get
1740 At this point, it should be easy to convince yourself that
1741 using the recipe on the right hand side of the associative law will
1742 built the exact same final tree.
1744 So binary trees are a monad.
1746 Haskell combines this monad with the Option monad to provide a monad
1748 [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)
1750 represent non-deterministic computations as a tree.
1752 ##[[List Monad as Continuation Monad]]##
1754 ##[[Manipulating Trees with Monads]]##