3 Recall [[the recent homework assignment|/exercises/assignment12]] where you solved the same-fringe problem with a `make_fringe_enumerator` function, or in the Scheme version using streams instead of zippers, with a `lazy-flatten` function.
5 The technique illustrated in those solutions 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*.
7 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:
9 main program next1 thread next2 thread
10 ------------ ------------ ------------
13 (paused) calculate first leaf
14 (paused) <--- return it
15 start next2 (paused) starting
16 (paused) (paused) calculate first leaf
17 (paused) (paused) <-- return it
18 compare leaves (paused) (paused)
19 call loop again (paused) (paused)
20 call next1 again (paused) (paused)
21 (paused) calculate next leaf (paused)
22 (paused) <-- return it (paused)
25 If you want to read more about these kinds of threads, here are some links:
27 <!-- * [[!wikipedia Computer_multitasking]]
28 * [[!wikipedia Thread_(computer_science)]] -->
30 * [[!wikipedia Coroutine]]
31 * [[!wikipedia Iterator]]
32 * [[!wikipedia Generator_(computer_science)]]
33 * [[!wikipedia Fiber_(computer_science)]]
34 <!-- * [[!wikipedia Green_threads]]
35 * [[!wikipedia Protothreads]] -->
37 The way we built cooperative threads using `make_fringe_enumerator` 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 either required us to manually save and restore the thread's snapshotted state (a tree zipper); or else we had to use a mutable reference cell to save and restore that state for us. Using the saved state, the next invocation of the `next_leaf` function could start up again where the previous invocation left off.
39 It's possible to build cooperative threads without using those tools, however. Already our [[solution using streams|/exercises/assignment12#streams2]] uses neither zippers nor any mutation. Instead it saves the thread's state in explicitly-created thunks, and resumes the thread by forcing the thunk.
41 Some languages have a native syntax for coroutines. Here's how we'd write the same-fringe solution above using native coroutines in the language Lua:
43 > function fringe_enumerator (tree)
45 coroutine.yield (tree.leaf)
47 fringe_enumerator (tree.left)
48 fringe_enumerator (tree.right)
52 > function same_fringe (tree1, tree2)
53 local next1 = coroutine.wrap (fringe_enumerator)
54 local next2 = coroutine.wrap (fringe_enumerator)
55 local function loop (leaf1, leaf2)
56 if leaf1 or leaf2 then
57 return leaf1 == leaf2 and loop( next1(), next2() )
58 elseif not leaf1 and not leaf2 then
64 return loop (next1(tree1), next2(tree2))
67 > return same_fringe ( {leaf=1}, {leaf=2} )
70 > return same_fringe ( {leaf=1}, {leaf=1} )
73 > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
74 {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
77 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 or dedicated native syntax to rely on.
80 ##Exceptions and Aborts##
82 To get a better understanding of how that execution pattern works, we'll add yet a second execution pattern to our plate, and then think about what they have in common.
84 While writing OCaml code, you've probably come across errors. In fact, you've probably come across errors of several sorts. One sort of error comes about when you've got syntax errors and the OCaml interpreter isn't even able to parse your code. A second sort of error is type errors, as in:
88 Error: This expression has type int list
89 but an expression was expected of type string list
91 Type errors are also detected and reported before OCaml attempts to execute or evaluate your code. But you may also have encountered a third kind of error, that arises while your program is running. For example:
94 Exception: Division_by_zero.
96 Exception: Failure "nth".
98 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 standard implementation of `List.nth`:
101 if n < 0 then invalid_arg "List.nth" else
102 let rec nth_aux l n =
104 | [] -> failwith "nth"
105 | a::l -> if n = 0 then a else nth_aux l (n-1)
108 (The Juli8 version of `List.nth` only differs in sometimes raising a different error.) Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:
110 raise (Invalid_argument "List.nth");;
111 raise (Failure "nth");;
113 where `Invalid_argument "List.nth"` is a value of type `exn`, and so too `Failure "nth"`. When you have some value `bad` of type `exn` and evaluate the expression:
117 the effect is for the program to immediately stop without evaluating any further code:
119 # let xcell = ref 0;;
120 val xcell : int ref = {contents = 0}
121 # let bad = Failure "test"
124 Exception: Failure "test".
128 Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
130 I said when you evaluate the expression:
134 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. (In OCaml 4.02 and higher, there is also a more inclusive construct that combines these, `match ... with PATTERN -> ... | exception PATTERN -> ...`.)
139 else if x = 2 then raise (Failure "two")
140 else raise (Failure "three")
142 with Failure "two" -> 20
144 val foo : int -> int = <fun>
150 Exception: Failure "three".
152 Notice what happens here. If we call `foo 1`, then the code between `try` and `with` evaluates to `110`, 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.
154 So what I should have said is that when you evaluate the expression:
158 *and that exception is never caught*, then the effect is for the program to immediately stop.
160 **Trivia**: what's the type of the `raise (Failure "two")` in:
163 else raise (Failure "two")
168 else raise (Failure "two")
170 So now what do you expect the type of this to be:
172 fun x -> raise (Failure "two")
176 (fun x -> raise (Failure "two") : 'a -> 'a)
178 Remind you of anything we discussed earlier? (At one point earlier in term we were asking whether you could come up with any functions of type `'a -> 'a` other than the identity function.)
182 Of course, it's possible to handle errors in other ways too. There's no reason why the implementation of `List.nth` *had* to raise an exception. 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 the exception-raising pattern now.
184 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:
188 (raise (Failure "blah")
190 with Failure "fooey" -> 10
191 with Failure "blah" -> 20;;
194 The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
200 with Failure "blah" -> 20
202 raise (Failure "blah")
206 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 that function, and we find a matching `try ... with ...` block in `foo`. This catches the error and so then the `try ... with ...` block in `foo` (the code that called `bar` in the first place) will evaluate to `20`.
208 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:
218 then if we called `foo 1`, we'd get the result `110`. If we called `foo 2`, on the other hand, we'd get `20` (note, not `120`). 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.
220 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`. In Lua, the preceding example would be written:
227 return 20 -- abort early
229 return value + 100 -- in a language like Scheme, you could omit the `return` here
230 -- but in Lua, a function's normal result must always be explicitly `return`ed
239 Okay, so that's our second execution pattern.
241 ##What do these have in common?##
243 In both of these patterns --- coroutines and exceptions/aborts --- 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 data structure 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.
245 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:
255 we can imagine a box:
258 +---try begin----------------+
259 | (if x = 1 then 10 |
262 +---end----------------------+
265 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.
268 What would a "snapshot of the code outside the box" look like? Well, let's rearrange the code somewhat. It should be equivalent to this:
272 +---try begin----------------+
273 | (if x = 1 then 10 |
276 +---end----------------------+
277 in (foo_result) + 1000;;
279 and we can think of the code starting with `let foo_result = ...` as a function, with the box being its parameter, like this:
284 or, spelling out the gap `< >` as a bound variable:
288 in (foo_result) + 1000
290 That function is our "snapshot". Normally what happens is that code *inside* the box delivers up a value, and that value gets supplied as an argument to the snapshot-function just described. That is, our code is essentially working like this:
293 in let snapshot = fun box ->
295 in (foo_result) + 1000
298 else ... (* we'll come back to this part *)
302 But now how should the `abort 20` part, that we ellided here, work? What should happen when we try to evaluate that?
304 Well, that's when we use the snapshot code in an unusual way. If we encounter an `abort 20`, we should abandon the code we're currently executing, and instead just supply `20` to the snapshot we saved when we entered the box. That is, something like this:
307 in let snapshot = fun box ->
309 in (foo_result) + 1000
316 Except that isn't quite right, yet---in this fragment, after the `snapshot 20` code is finished, we'd pick up again inside `let value = (...) + 100 in snapshot value`. That's not what we want. We don't want to pick up again there. We want instead to do this:
319 in let snapshot = fun box ->
321 in (foo_result) + 1000
324 else snapshot 20 THEN STOP
328 We can get that by some further rearranging of the code:
331 in let snapshot = fun box ->
333 in (foo_result) + 1000
334 in let continue_normally = fun from_value ->
335 let value = from_value + 100
338 if x = 1 then continue_normally 10
341 And this is indeed what is happening, at a fundamental level, when you use an expression like `abort 20`.
344 # #require "delimcc";;
346 # let reset body = let p = new_prompt () in push_prompt p (body p);;
348 let snapshot = fun box ->
350 in (foo_result) + 1000
351 in let continue_normally = fun from_value ->
352 let value = from_value + 100
354 in if x = 1 then continue_normally 10
358 +===try begin================+
359 | (if x = 1 then 10 |
362 +===end======================+
366 let foo x = reset(fun p () ->
368 if x = 1 then k 10 else 20)
382 A similar kind of "snapshotting" lets coroutines keep track of where they left off, so that they can start up again at that same place.
384 ##Continuations, finally##
386 These snapshots are called **continuations** because they represent how the computation will "continue" once some target code (in our example, the code in the box) delivers up a value.
388 You can think of them as functions that represent "how the rest of the computation proposes to continue." Except that, once we're able to get our hands on those functions, we can do exotic and unwholesome things with them. Like use them to suspend and resume a thread. Or to abort from deep inside a sub-computation: one function might pass the command to abort *it* to a subfunction, so that the subfunction has the power to jump directly to the outside caller. Or a function might *return* its continuation function to the outside caller, giving *the outside caller* the ability to "abort" the function (the function that has already returned its value---so what should happen then?) Or we may call the same continuation function *multiple times* (what should happen then?). All of these weird and wonderful possibilities await us.
390 The key idea behind working with continuations is that we're *inverting control*. In the fragment above, the code `(if x = 1 then ... else snapshot 20) + 100`---which is written as if it were to supply a value to the outside context that we snapshotted---itself *makes non-trivial use of* that snapshot. So it has to be able to refer to that snapshot; the snapshot has to somehow be available to our inside-the-box code as an *argument* or bound variable. That is: the code that is *written* like it's supplying an argument to the outside context is instead *getting that context as its own argument*. He who is written as value-supplying slave is instead become the outer context's master.
392 In fact you've already seen this several times this semester---recall how in our implementation of pairs in the untyped lambda-calculus, the handler who wanted to use the pair's components had *in the first place to be supplied to the pair as an argument*. So the exotica from the end of the seminar was already on the scene in some of our earliest steps. Recall also what we did with our [[abortable list traversals|/topics/week12_abortable_traversals]].
394 This inversion of control should also remind you of Montague's treatment of determiner phrases in ["The Proper Treatment of Quantification in Ordinary English"](http://www.blackwellpublishing.com/content/BPL_Images/Content_store/Sample_chapter/0631215417%5CPortner.pdf) (PTQ).
396 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.
398 Monatague proposed we instead take the subject term 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.
400 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.
402 This inversion of who is the argument and who is the function receiving the argument is paradigmatic of working with continuations.
404 Continuations come in many varieties. There are **undelimited continuations**, expressed in Scheme via `(call/cc (lambda (k) ...))` or the shorthand `(let/cc k ...)`. (`call/cc` is itself shorthand for `call-with-current-continuation`.) These capture "the entire rest of the computation." There are also **delimited continuations**, expressed in Scheme via `(reset ... (shift k ...) ...)` or `(prompt ... (control k ...) ...)` or any of several other operations. There are subtle differences between those that we won't be exploring in the seminar. Ken Shan has done terrific work exploring the relations of these operations to each other.
406 When working with continuations, it's easiest in the first place to write them out explicitly, the way that we explicitly wrote out the `snapshot` continuation when we transformed this:
419 in let snapshot = fun box ->
421 in (foo_result) + 1000
422 in let continue_normally = fun from_value ->
423 let value = from_value + 100
426 if x = 1 then continue_normally 10
429 Code written in the latter form is said to be written in **explicit continuation-passing style** or CPS. Later we'll talk about algorithms that mechanically convert an entire program into CPS.
431 There are also different kinds of "syntactic sugar" we can use to hide the continuation plumbing. Of course we'll be talking about how to manipulate continuations **with a Continuation monad.** We'll also talk about a style of working with continuations where they're **mostly implicit**, but special syntax allows us to distill the implicit continuaton into a first-class value (the `k` in `(let/cc k ...)` and `(shift k ...)`.
433 Various of the tools we've been introducing over the past weeks are inter-related. We saw coroutines implemented first with zippers; here we've talked in the abstract about their being implemented with continuations. Oleg says that "Zipper can be viewed as a delimited continuation reified as a data structure." Ken expresses the same idea in terms of a zipper being a "defunctionalized" continuation---that is, take something implemented as a function (a continuation) and implement the same thing as an inert data structure (a zipper).
435 Mutation, delimited continuations, and monads can also be defined in terms of each other in various ways. We find these connections fascinating but the seminar won't be able to explore them very far.
437 We recommend reading [the Yet Another Haskell Tutorial on Continuation Passing Style](http://en.wikibooks.org/wiki/Haskell/YAHT/Type_basics#Continuation_Passing_Style)---though the target language is Haskell, this discussion is especially close to material we're discussing in the seminar.