With cooperative threads, one typically yields control to the thread, and then back again to the main program, multiple times. Here's the pattern in which that happens in our `same_fringe` function:
- main program next1 thread next2 thread
- ------------ ------------ ------------
- start next1
- (paused) starting
- (paused) calculate first leaf
- (paused) <--- return it
- start next2 (paused) starting
- (paused) (paused) calculate first leaf
- (paused) (paused) <-- return it
- compare leaves (paused) (paused)
- call loop again (paused) (paused)
- call next1 again (paused) (paused)
- (paused) calculate next leaf (paused)
- (paused) <-- return it (paused)
- ... and so on ...
+ main program next1 thread next2 thread
+ ------------ ------------ ------------
+ start next1
+ (paused) starting
+ (paused) calculate first leaf
+ (paused) <--- return it
+ start next2 (paused) starting
+ (paused) (paused) calculate first leaf
+ (paused) (paused) <-- return it
+ compare leaves (paused) (paused)
+ call loop again (paused) (paused)
+ call next1 again (paused) (paused)
+ (paused) calculate next leaf (paused)
+ (paused) <-- return it (paused)
+ ... and so on ...
If you want to read more about these kinds of threads, here are some links:
-<!-- * [[!wikipedia Computer_multitasking]]
-* [[!wikipedia Thread_(computer_science)]] -->
+<!-- * [[!wikipedia Computer_multitasking]]
+* [[!wikipedia Thread_(computer_science)]] -->
-* [[!wikipedia Coroutine]]
-* [[!wikipedia Iterator]]
-* [[!wikipedia Generator_(computer_science)]]
-* [[!wikipedia Fiber_(computer_science)]]
-<!-- * [[!wikipedia Green_threads]]
-* [[!wikipedia Protothreads]] -->
+* [[!wikipedia Coroutine]]
+* [[!wikipedia Iterator]]
+* [[!wikipedia Generator_(computer_science)]]
+* [[!wikipedia Fiber_(computer_science)]]
+<!-- * [[!wikipedia Green_threads]]
+* [[!wikipedia Protothreads]] -->
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.
-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.
-
-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:
-
- > function fringe_enumerator (tree)
- if tree.leaf then
- coroutine.yield (tree.leaf)
- else
- fringe_enumerator (tree.left)
- fringe_enumerator (tree.right)
- end
- end
-
- > function same_fringe (tree1, tree2)
- local next1 = coroutine.wrap (fringe_enumerator)
- local next2 = coroutine.wrap (fringe_enumerator)
- local function loop (leaf1, leaf2)
- if leaf1 or leaf2 then
- return leaf1 == leaf2 and loop( next1(), next2() )
- elseif not leaf1 and not leaf2 then
- return true
- else
- return false
- end
- end
- return loop (next1(tree1), next2(tree2))
- end
-
- > return same_fringe ( {leaf=1}, {leaf=2} )
- false
-
- > return same_fringe ( {leaf=1}, {leaf=1} )
- true
-
- > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
- {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
- true
+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 the code of explicitly-created thunks, and resumes the thread by forcing the thunk.
+
+Some languages have a native syntax for coroutines. Here's how we'd write the same-fringe solution using native coroutines in the language Lua:
+
+
+ > function fringe_enumerator (tree)
+ if tree.leaf then
+ coroutine.yield (tree.leaf)
+ else
+ fringe_enumerator (tree.left)
+ fringe_enumerator (tree.right)
+ end
+ end
+
+ > function same_fringe (tree1, tree2)
+ -- coroutine.wrap turns a function into a coroutine
+ local next1 = coroutine.wrap (fringe_enumerator)
+ local next2 = coroutine.wrap (fringe_enumerator)
+ local function loop (leaf1, leaf2)
+ if leaf1 or leaf2 then
+ return leaf1 == leaf2 and loop( next1(), next2() )
+ elseif not leaf1 and not leaf2 then
+ return true
+ else
+ return false
+ end
+ end
+ return loop (next1(tree1), next2(tree2))
+ end
+
+ > return same_fringe ( {leaf=1}, {leaf=2} )
+ false
+
+ > return same_fringe ( {leaf=1}, {leaf=1} )
+ true
+
+ > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
+ {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
+ true
We're going to think about the underlying principles to this execution pattern, and instead learn how to implement it from scratch---without necessarily having zippers or dedicated native syntax to rely on.
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:
- # let lst = [1; 2] in
- "a" :: lst;;
- Error: This expression has type int list
- but an expression was expected of type string list
+ # let lst = [1; 2] in
+ "a" :: lst;;
+ ---
+ Error: This expression has type int list
+ but an expression was expected of type string list
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:
- # 1/0;;
- Exception: Division_by_zero.
- # List.nth [1;2] 10;;
- Exception: Failure "nth".
+ # 1/0;;
+ Exception: Division_by_zero.
+ # List.nth [1;2] 10;;
+ Exception: Failure "nth".
-These "Exceptions" are **run-time errors**. OCaml will automatically detect some of them, like when you attempt to divide by zero. Other exceptions are *raised* by code. For instance, here is the standard implementation of `List.nth`:
+These "Exceptions" are **run-time errors**. OCaml will automatically detect some of them, like when you attempt to divide by zero. Other exceptions are manually *raised* by code. For instance, here is the standard implementation of `List.nth`:
- let nth l n =
- if n < 0 then invalid_arg "List.nth" else
- let rec nth_aux l n =
- match l with
- | [] -> failwith "nth"
- | a::l -> if n = 0 then a else nth_aux l (n-1)
- in nth_aux l n
+ let nth l n =
+ if n < 0 then invalid_arg "List.nth" else
+ let rec nth_aux l n =
+ match l with
+ | [] -> failwith "nth"
+ | a::l -> if n = 0 then a else nth_aux l (n-1)
+ in nth_aux l n
(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:
- raise (Invalid_argument "List.nth");;
- raise (Failure "nth");;
+ raise (Invalid_argument "List.nth");;
+ raise (Failure "nth");;
-where `Invalid_argument "List.nth"` is a value of type `exn`, and so too `Failure "nth"`. When you have some value `bad` of type `exn` and evaluate the expression:
+where `Invalid_argument "List.nth"` constructs a value of type `exn`, and so too `Failure "nth"`. When you have some value `bad` of type `exn` and evaluate the expression:
- raise bad
+ raise bad
the effect is for the program to immediately stop without evaluating any further code:
- # let xcell = ref 0;;
- val xcell : int ref = {contents = 0}
- # let bad = Failure "test"
- in let _ = raise bad
- in xcell := 1;;
- Exception: Failure "test".
- # !xcell;;
- - : int = 0
+ # let xcell = ref 0;;
+ val xcell : int ref = {contents = 0}
+ # let bad = Failure "test"
+ in let _ = raise bad
+ in xcell := 1;;
+ Exception: Failure "test".
+ # !xcell;;
+ - : int = 0
Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
I said when you evaluate the expression:
- raise bad
+ raise bad
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 -> ...`.)
- # let foo x =
- try
- (if x = 1 then 10
- else if x = 2 then raise (Failure "two")
- else raise (Failure "three")
- ) + 100
- with Failure "two" -> 20
- ;;
- val foo : int -> int = <fun>
- # foo 1;;
- - : int = 110
- # foo 2;;
- - : int = 20
- # foo 3;;
- Exception: Failure "three".
+ # let foo x =
+ try
+ (if x = 1 then 10
+ else if x = 2 then raise (Failure "two")
+ else raise (Failure "three")
+ ) + 100
+ with Failure "two" -> 20
+ ;;
+ val foo : int -> int = <fun>
+ # foo 1;;
+ - : int = 110
+ # foo 2;;
+ - : int = 20
+ # foo 3;;
+ Exception: Failure "three".
Notice what happens here. If we call `foo 1`, then the code between `try` and `with` evaluates to `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.
So what I should have said is that when you evaluate the expression:
- raise bad
+ raise bad
*and that exception is never caught*, then the effect is for the program to immediately stop.
**Trivia**: what's the type of the `raise (Failure "two")` in:
- if x = 1 then 10
- else raise (Failure "two")
+ if x = 1 then 10
+ else raise (Failure "two")
What's its type in:
- if x = 1 then "ten"
- else raise (Failure "two")
+ if x = 1 then "ten"
+ else raise (Failure "two")
So now what do you expect the type of this to be:
- fun x -> raise (Failure "two")
+ fun x -> raise (Failure "two")
How about this:
- (fun x -> raise (Failure "two") : 'a -> 'a)
+ (fun x -> raise (Failure "two") : 'a -> 'a)
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.)
When an exception is raised, it percolates up through the code that called it, until it finds a surrounding `try ... with ...` that matches it. That might not be the first `try ... with ...` that it encounters. For example:
- # try
- try
- (raise (Failure "blah")
- ) + 100
- with Failure "fooey" -> 10
- with Failure "blah" -> 20;;
- - : int = 20
+ # try
+ try
+ (raise (Failure "blah")
+ ) + 100
+ with Failure "fooey" -> 10
+ with Failure "blah" -> 20;;
+ - : int = 20
The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
- # let foo b x =
- try
- (b x
- ) + 100
- with Failure "blah" -> 20
- in let bar x =
- raise (Failure "blah")
- in foo bar 0;;
- - : int = 20
+ # let foo b x =
+ try
+ (b x
+ ) + 100
+ with Failure "blah" -> 20
+ in let bar x =
+ raise (Failure "blah")
+ in foo bar 0;;
+ - : int = 20
Here we call `foo bar 0`, and `foo` in turn calls `bar 0`, and `bar` raises the exception. Since there's no matching `try ... with ...` block in `bar`, we percolate back up the history of who called 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`.
OK, now this exception-handling apparatus does exemplify the second execution pattern we want to focus on. But it may bring it into clearer focus if we **simplify the pattern** even more. Imagine we could write code like this instead:
- # let foo x =
- try begin
- (if x = 1 then 10
- else abort 20
- ) + 100
- end
- ;;
+ # let foo x =
+ try begin
+ (if x = 1 then 10
+ else abort 20
+ ) + 100
+ end
+ ;;
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.
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:
- > function foo(x)
- local value
- if (x == 1) then
- value = 10
- else
- return 20 -- abort early
- end
- return value + 100 -- in a language like Scheme, you could omit the `return` here
- -- but in Lua, a function's normal result must always be explicitly `return`ed
- end
-
- > return foo(1)
- 110
-
- > return foo(2)
- 20
+ > function foo(x)
+ local value
+ if (x == 1) then
+ value = 10
+ else
+ return 20 -- abort early
+ end
+ return value + 100 -- in a language like Scheme, you could omit the `return` here
+ -- but in Lua, a function's normal result must always be explicitly `return`ed
+ end
+
+ > return foo(1)
+ 110
+
+ > return foo(2)
+ 20
Okay, so that's our second execution pattern.
A more general way to think about these snapshots is to think of the code we're taking a snapshot of as a *function.* For example, in this code:
- let foo x =
- try begin
- (if x = 1 then 10
- else abort 20
- ) + 100
- end
- in (foo 2) + 1;;
+ let foo x = (* same definition as before *)
+ try begin
+ (if x = 1 then 10
+ else abort 20
+ ) + 100
+ end
+ in (foo 2) + 1000;; (* this line is new *)
we can imagine a box:
- let foo x =
- +---try begin----------------+
- | (if x = 1 then 10 |
- | else abort 20 |
- | ) + 100 |
- +---end----------------------+
- in (foo 2) + 1000;;
+ let foo x =
+ +---try begin----------------+
+ | (if x = 1 then 10 |
+ | else abort 20 |
+ | ) + 100 |
+ +---end----------------------+
+ in (foo 2) + 1000;;
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.
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:
- let x = 2
- in let foo_result =
- +---try begin----------------+
- | (if x = 1 then 10 |
- | else abort 20 |
- | ) + 100 |
- +---end----------------------+
- in (foo_result) + 1000;;
+ let x = 2
+ in let foo_result =
+ +---try begin----------------+
+ | (if x = 1 then 10 |
+ | else abort 20 |
+ | ) + 100 |
+ +---end----------------------+
+ in (foo_result) + 1000;;
and we can think of the code starting with `let foo_result = ...` as a function, with the box being its parameter, like this:
or, spelling out the gap `< >` as a bound variable:
- fun box ->
- let foo_result = box
- in (foo_result) + 1000
+ fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
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:
- let x = 2
- in let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let value =
- (if x = 1 then 10
- else ... (* we'll come back to this part *)
- ) + 100
- in shapshot value;;
+ let x = 2
+ in let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let foo_applied_to_x =
+ (if x = 1 then 10
+ else ... (* we'll come back to this part *)
+ ) + 100
+ in shapshot foo_applied_to_x;;
But now how should the `abort 20` part, that we ellided here, work? What should happen when we try to evaluate that?
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:
- let x = 2
- in let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let value =
- (if x = 1 then 10
- else snapshot 20
- ) + 100
- in shapshot value;;
-
-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:
-
- let x = 2
- in let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let value =
- (if x = 1 then 10
- else snapshot 20 THEN STOP
- ) + 100
- in shapshot value;;
+ let x = 2
+ in let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let foo_applied_to_x =
+ (if x = 1 then 10
+ else snapshot 20
+ ) + 100
+ in shapshot foo_applied_to_x;;
+
+Except that isn't quite right, yet---in this fragment, after the `snapshot 20` code is finished, we'd pick up again inside `let foo_applied_to_x = (...) + 100 in snapshot foo_applied_to_x`. That's not what we want. We don't want to pick up again there. We want instead to do this:
+
+ let x = 2
+ in let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let foo_applied_to_x =
+ (if x = 1 then 10
+ else snapshot 20 THEN STOP
+ ) + 100
+ in shapshot foo_applied_to_x;;
We can get that by some further rearranging of the code:
- let x = 2
- in let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let continue_normally = fun from_value ->
- let value = from_value + 100
- in snapshot value
- in
- if x = 1 then continue_normally 10
- else snapshot 20;;
+ let x = 2
+ in let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let continue_foo_normally = fun from_value ->
+ let value = from_value + 100
+ in snapshot value
+ in (* start of foo_applied_to_x *)
+ if x = 1 then continue_foo_normally 10
+ else snapshot 20;;
-And this is indeed what is happening, at a fundamental level, when you use an expression like `abort 20`.
-
-<!--
-# #require "delimcc";;
-# open Delimcc;;
-# let reset body = let p = new_prompt () in push_prompt p (body p);;
-# let test_cps x =
- let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let continue_normally = fun from_value ->
- let value = from_value + 100
- in snapshot value
- in if x = 1 then continue_normally 10
- else snapshot 20;;
+And this is indeed what is happening, at a fundamental level, when you use an expression like `abort 20`. Here is the original code for comparison:
- let foo x =
- +===try begin================+
- | (if x = 1 then 10 |
- | else abort 20 |
- | ) + 100 |
- +===end======================+
- in (foo 2) + 1000;;
+ let foo x =
+ +---try begin----------------+
+ | (if x = 1 then 10 |
+ | else abort 20 |
+ | ) + 100 |
+ +---end----------------------+
+ in (foo 2) + 1000;;
-# let test_shift x =
- let foo x = reset(fun p () ->
- (shift p (fun k ->
- if x = 1 then k 10 else 20)
- ) + 100)
- in foo z + 1000;;
-# test_cps 1;;
-- : int = 1110
-# test_shift 1;;
-- : int = 1110
-# test_cps 2;;
-- : int = 1020
-# test_shift 2;;
-- : int = 1020
--->
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.
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:
- let foo x =
- try begin
- (if x = 1 then 10
- else abort 20
- ) + 100
- end
- in (foo 2) + 1000;;
+ let foo x =
+ +---try begin----------------+
+ | (if x = 1 then 10 |
+ | else abort 20 |
+ | ) + 100 |
+ +---end----------------------+
+ in (foo 2) + 1000;;
into this:
- let x = 2
- in let snapshot = fun box ->
- let foo_result = box
- in (foo_result) + 1000
- in let continue_normally = fun from_value ->
- let value = from_value + 100
- in snapshot value
- in
- if x = 1 then continue_normally 10
- else snapshot 20;;
+ let x = 2
+ in let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let continue_foo_normally = fun from_value ->
+ let value = from_value + 100
+ in snapshot value
+ in (* start of foo_applied_to_x *)
+ if x = 1 then continue_foo_normally 10
+ else snapshot 20;;
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.
-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 ...)`.
+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 continuation into a first-class value (the `k` in `(let/cc k ...)` and `(shift k ...)`. For reference, here's how the preceding code looks, using Scheme's `abort` or `shift` operators:
+
+ #lang racket
+ (require racket/control)
+
+ (let ([foo (lambda (x)
+ (reset
+ (+
+ (if (eqv? x 1) 10 (abort 20))
+ 100)))])
+ (+ (foo 2) 1000))
+
+ (let ([foo (lambda (x)
+ (reset
+ (+
+ (shift k
+ (if (eqv? x 1) (k 10) 20))
+ 100)))])
+ (+ (foo 2) 1000))
+
+
+<!--
+# #require "delimcc";;
+# open Delimcc;;
+# let reset body = let p = new_prompt () in push_prompt p (body p);;
+# let test_cps x =
+ let snapshot = fun box ->
+ let foo_result = box
+ in (foo_result) + 1000
+ in let continue_foo_normally = fun from_value ->
+ let value = from_value + 100
+ in snapshot value
+ in if x = 1 then continue_foo_normally 10
+ else snapshot 20;;
+
+# let test_shift x =
+ let foo x = reset(fun p () ->
+ (shift p (fun k ->
+ if x = 1 then k 10 else 20)
+ ) + 100)
+ in foo z + 1000;;
+
+# test_cps 1;;
+- : int = 1110
+# test_shift 1;;
+- : int = 1110
+# test_cps 2;;
+- : int = 1020
+# test_shift 2;;
+- : int = 1020
+-->
+
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).