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 ##Same-fringe using a zipper-based coroutine##
12 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):
20 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.
23 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:
31 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.
33 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.
35 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.
37 First, we define a type for leaf-labeled, binary trees:
39 type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
41 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:
43 # type blah = Blah of (int * int * (char -> bool));;
45 and then having to remember which element in the triple was which:
47 # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
48 Error: This expression has type int * (char -> bool) * int
49 but an expression was expected of type int * int * (char -> bool)
51 # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
52 val b1 : blah = Blah (1, 2, <fun>)
54 records let you attach descriptive labels to the components of the tuple:
56 # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
57 # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
58 val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
59 # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
60 val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}
62 These were the strategies to extract the components of an unlabeled tuple:
64 let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
66 let (h, w, test) = b1;; (* works for arbitrary tuples *)
69 | (h, w, test) -> ...;; (* same as preceding *)
71 Here is how you can extract the components of a labeled record:
73 let h = b2.height;; (* handy! *)
75 let {height = h; weight = w; char_tester = test} = b2
76 in (* go on to use h, w, and test ... *)
79 | {height = h; weight = w; char_tester = test} ->
80 (* go on to use h, w, and test ... *)
82 Anyway, using record types, we might define the tree zipper interface like so:
84 type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot
85 and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;
87 type 'a zipper = { level : 'a starred_level; filler: 'a tree };;
89 let rec move_botleft (z : 'a zipper) : 'a zipper =
90 (* returns z if the targetted node in z has no children *)
91 (* else returns move_botleft (zipper which results from moving down and left in z) *)
94 let {level; filler} = z
97 | Node(left, right) ->
98 let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
103 let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
104 (* if it's possible to move right in z, returns Some (the result of doing so) *)
105 (* else if it's not possible to move any further up in z, returns None *)
106 (* else returns move_right_or_up (result of moving up in z) *)
109 let {level; filler} = z
111 | Starring_Left {parent; sibling = right} -> Some {level = Starring_Right {parent; sibling = filler}; filler = right}
113 | Starring_Right {parent; sibling = left} ->
114 let z' = {level = parent; filler = Node(left, filler)}
115 in move_right_or_up z'
119 The following function takes an 'a tree and returns an 'a zipper focused on its root:
121 let new_zipper (t : 'a tree) : 'a zipper =
122 {level = Root; filler = t}
125 Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
127 let make_fringe_enumerator (t: 'a tree) =
128 (* create a zipper targetting the botleft of t *)
129 let zbotleft = move_botleft (new_zipper t)
130 (* create a refcell initially pointing to zbotleft *)
131 in let zcell = ref (Some zbotleft)
132 (* construct the next_leaf function *)
133 in let next_leaf () : 'a option =
136 (* extract label of currently-targetted leaf *)
137 let Leaf current = z.filler
138 (* update zcell to point to next leaf, if there is one *)
139 in let () = zcell := match move_right_or_up z with
141 | Some z' -> Some (move_botleft z')
142 (* return saved label *)
144 | None -> (* we've finished enumerating the fringe *)
147 (* return the next_leaf function *)
151 Here's an example of `make_fringe_enumerator` in action:
153 # let tree1 = Leaf 1;;
154 val tree1 : int tree = Leaf 1
155 # let next1 = make_fringe_enumerator tree1;;
156 val next1 : unit -> int option = <fun>
158 - : int option = Some 1
160 - : int option = None
162 - : int option = None
163 # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
164 val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
165 # let next2 = make_fringe_enumerator tree2;;
166 val next2 : unit -> int option = <fun>
168 - : int option = Some 1
170 - : int option = Some 2
172 - : int option = Some 3
174 - : int option = None
176 - : int option = None
178 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`.
180 Using these fringe enumerators, we can write our `same_fringe` function like this:
182 let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
183 let next1 = make_fringe_enumerator t1
184 in let next2 = make_fringe_enumerator t2
185 in let rec loop () : bool =
186 match next1 (), next2 () with
187 | Some a, Some b when a = b -> loop ()
193 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.
195 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*.
197 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:
199 main program next1 thread next2 thread
200 ------------ ------------ ------------
203 (paused) calculate first leaf
204 (paused) <--- return it
205 start next2 (paused) starting
206 (paused) (paused) calculate first leaf
207 (paused) (paused) <-- return it
208 compare leaves (paused) (paused)
209 call loop again (paused) (paused)
210 call next1 again (paused) (paused)
211 (paused) calculate next leaf (paused)
212 (paused) <-- return it (paused)
215 If you want to read more about these kinds of threads, here are some links:
217 <!-- * [[!wikipedia Computer_multitasking]]
218 * [[!wikipedia Thread_(computer_science)]] -->
220 * [[!wikipedia Coroutine]]
221 * [[!wikipedia Iterator]]
222 * [[!wikipedia Generator_(computer_science)]]
223 * [[!wikipedia Fiber_(computer_science)]]
224 <!-- * [[!wikipedia Green_threads]]
225 * [[!wikipedia Protothreads]] -->
227 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.
229 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:
231 > function fringe_enumerator (tree)
233 coroutine.yield (tree.leaf)
235 fringe_enumerator (tree.left)
236 fringe_enumerator (tree.right)
240 > function same_fringe (tree1, tree2)
241 local next1 = coroutine.wrap (fringe_enumerator)
242 local next2 = coroutine.wrap (fringe_enumerator)
243 local function loop (leaf1, leaf2)
244 if leaf1 or leaf2 then
245 return leaf1 == leaf2 and loop( next1(), next2() )
246 elseif not leaf1 and not leaf2 then
252 return loop (next1(tree1), next2(tree2))
255 > return same_fringe ( {leaf=1}, {leaf=2})
258 > return same_fringe ( {leaf=1}, {leaf=1})
261 > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
262 {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
265 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.
268 ##Exceptions and Aborts##
270 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.
272 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:
274 # let lst = [1; 2] in
276 Error: This expression has type int list
277 but an expression was expected of type string list
279 But you may also have encountered other kinds of error, that arise while your program is running. For example:
282 Exception: Division_by_zero.
283 # List.nth [1;2] 10;;
284 Exception: Failure "nth".
286 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`:
289 if n < 0 then invalid_arg "List.nth" else
290 let rec nth_aux l n =
292 | [] -> failwith "nth"
293 | a::l -> if n = 0 then a else nth_aux l (n-1)
296 Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:
298 raise (Invalid_argument "List.nth");;
299 raise (Failure "nth");;
301 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:
305 the effect is for the program to immediately stop without evaluating any further code:
307 # let xcell = ref 0;;
308 val xcell : int ref = {contents = 0}
309 # let ex = Failure "test"
312 Exception: Failure "test".
316 Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
318 I said when you evaluate the expression:
322 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:
327 else if x = 2 then raise (Failure "two")
328 else raise (Failure "three")
329 with Failure "two" -> 20
331 val foo : int -> int = <fun>
337 Exception: Failure "three".
339 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.
341 So what I should have said is that when you evaluate the expression:
345 *and that exception is never caught*, then the effect is for the program to immediately stop.
347 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.
349 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:
353 raise (Failure "blah")
354 with Failure "fooey" -> 10
355 with Failure "blah" -> 20;;
358 The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
363 with Failure "blah" -> 20
365 raise (Failure "blah")
369 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`.
371 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:
380 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.
382 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:
400 Okay, so that's our second execution pattern.
402 ##What do these have in common?##
404 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.
406 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:
415 we can imagine a box:
418 +---------------------------+
420 | (if x = 1 then 10 |
421 | else abort 20) + 1 |
423 +---------------------------+
426 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.
429 # #require "delimcc";;
431 # let reset body = let p = new_prompt () in push_prompt p (body p);;
432 val reset : ('a Delimcc.prompt -> unit -> 'a) -> 'a = <fun>
433 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 1) + 100;;
435 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 2) + 100;;
442 --------------------------------------
444 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.
447 ##Introducing Continuations##
449 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."
451 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.
453 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.
455 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.
457 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.
459 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:
461 \handler. handler x y
463 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.
465 Consider a complex computation, such as:
467 1 + 2 * (1 - g (3 + 4))
469 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:
471 \result. 1 + 2 * (1 - result)
473 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.
475 Go back and read the material on "Aborting a Search Through a List" in [[Week4]] for an example of doing this.
477 In very general terms, the strategy is to work with functions like this:
481 ... if you want to abort early, supply an argument to k ...
482 ... do more stuff ...
484 in let gcon = fun result -> 1 + 2 * (1 - result)
485 in gcon (g' gcon (3 + 4))
487 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.
489 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:
493 ... if you want to abort early, supply an argument to k ...
494 ... do more stuff ...
496 in let gcon = fun result ->
497 let final_value = 1 + 2 * (1 - result)
498 in end_program_with final_value
499 in gcon (g' gcon (3 + 4))
501 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.)
503 So now, guess what would be the result of doing the following:
507 in let gcon = fun result ->
508 let final_value = (1, result)
509 in end_program_with final_value
510 in gcon (g' gcon (3 + 4))
512 <!-- (1, 7) ... explain why not (1, 8) -->
514 ##[[List Monad as Continuation Monad]]##
516 ##[[Manipulating Trees with Monads]]##