[[!toc]]
CPS Transforms
==============
We've already approached some tasks now by programming in **continuation-passing style.** We first did that with tuples at the start of term, and then with the v5 lists in [[week4]], and now more recently and self-consciously when discussing [aborts](/couroutines_and_aborts),
and [the "abSd" task](/from_list_zippers_to_continuations). and the use of `tree_monadize` specialized to the Continuation monad, which required us to supply an initial continuation.
In our discussion of aborts, we showed how to rearrange code like this:
let foo x =
+---try begin----------------+
| (if x = 1 then 10 |
| else abort 20 |
| ) + 100 |
+---end----------------------+
in (foo 2) + 1000;;
into a form like 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;;
How did we figure out how to rearrange that code? There are algorithms that can do this for us mechanically. These algorithms are known as **CPS transforms**, because they transform code that might not yet be in CPS form into that form.
We won't attempt to give a full CPS transform for OCaml; instead we'll just focus on the lambda calculus and a few extras, to be introduced as we proceed.
In fact there are multiple ways to do a CPS transform. Here is one:
[x] --> x
[\x. M] --> \k. k (\x. [M])
[M N] --> \k. [M] (\m. m [N] k)
And here is another:
[x] --> \k. k x
[\x. M] --> \k. k (\x. [M])
[M N] --> \k. [M] (\m. [N] (\n. m n k))
These transforms have some interesting properties. One is that---assuming we never reduce inside a lambda term, but only when redexes are present in the outermost level---the formulas generated by these transforms will always only have a single candidate redex to be reduced at any stage. In other words, the generated expressions dictate in what order the components from the original expressions will be evaluated. As it happens, the first transform above forces a *call-by-name* reduction order: assuming `M N` to be a redex, redexes inside `N` will be evaluated only after `N` has been substituted into `M`. And the second transform forces a *call-by-value* reduction order. These reduction orders will be forced no matter what the native reduction order of the interpreter is, just so long as we're only allowed to reduce redexes not underneath lambdas.
Plotkin did important early work with CPS transforms (around 1975), and they are now a staple of academic computer science.
Here's another interesting fact about these transforms. Compare the translations for variables and applications in the call-by-value transform:
[x] --> \k. k x
[M N] --> \k. [M] (\m. [N] (\n. m n k))
To the implementations we proposed for `unit` and `bind` when developing a Continuation monads, for example [here](/list_monad_as_continuation_monad). I'll relabel some of the variable names to help the comparison:
let cont_unit x = fun k -> k x
let cont_bind N M = fun k -> N (fun n -> M n k)
The transform for `x` is just `cont_unit x`! And the transform for `M N` is, though not here exactly the same as `cont_bind N M`, quite reminiscent of it. (I don't yet know whether there's an easy and satisfying explanation of why these two diverge as they do.)
Doing CPS transforms by hand is very cumbersome. (Try it.) But you can leverage our lambda evaluator to help you out. Here's how to do it. From here on out, we'll be working with and extending the call-by-value CPS transform set out above:
let var = \x (\k. k x) in
let lam = \x_body (\k. k (\x. x_body x)) in
let app = \m n. (\k. m (\m. n (\n. m n k))) in
...
Then if you want to use [x], you'd write `var x`. If you want to use [\x. body], you'd write `lam (\x. BODY)`, where `BODY` is whatever [body] amounts to. If you want to use [m n], you'd write `app M N`, where M and N are whatever [m] and [n] amount to.
To play around with this, you'll also want to help yourself to some primitives already in CPS form. (You won't want to rebuild everything again from scratch.) For a unary function like `succ`, you can take its primitive CPS analogue [succ] to be `\u. u (\a k. k (succ a))` (where `succ` in this expansion is the familiar non-CPS form of `succ`). Then for example:
[succ x]
= \k. [succ] (\m. [x] (\n. m n k))
~~> ...
~~> \k. k (succ x)
Or, using the lambda evaluator, that is:
...
let op1 = \op. \u. u (\a k. k (op a)) in
app (op1 succ) (var x)
~~> \k. k (succ x)
Some other handy tools:
let app2 = \a b c. app (app a b) c in
let app3 = \a b c d. app (app (app a b) c) d in
let op2 = \op. \u. u (\a v. v (\b k. k (op a b))) in
let op3 = \op. \u. u (\a v. v (\b w. w (\c k. k (op a b c)))) in
...
Then, for instance, [plus x y] would be rendered in the lambda evaluator as:
app2 (op2 plus) (var x) (var y)
~~> \k. k (plus x y)
To finish off a CPS computation, you have to supply it with an "initial" or "outermost" continuation. (This is somewhat like "running" a monadic computation.) Usually you'll give the identity function, representing that nothing further happens to the continuation-expecting value.
If the program you're working with is already in CPS form, then some elegant and powerful computational patterns become available, as we've been seeing. But it's tedious to convert to and work in fully-explicit CPS form. Usually you'll just want to be using the power of continuations at some few points in your program. It'd be nice if we had some way to make use of those patterns without having to convert our code explicitly into CPS form.
Callcc
======
Well, we can. Consider the space of lambda formulas. Consider their image under a CPS transform. There will be many well-formed lambda expressions not in that image---that is, expressions that aren't anybody's CPS transform. Some of these will be useful levers in the CPS patterns we want to make use of. We can think of them as being the CPS transforms of some new syntax in the original language. For example, the expression `callcc` is explained as a new bit of syntax having some of that otherwise unclaimed CPS real-estate. The meaning of the new syntax can be understood in terms of how the CPS transform we specify for it behaves, when the whole language is in CPS form.
I won't give the CPS transform for `callcc` itself, but instead for the complex form:
[callcc (\k. body)] = \outk. (\k. [body] outk) (\v localk. outk v)
The behavior of `callcc` is this. The whole expression `callcc (\k. body)`, call it C, is being evaluated in a context, call it E[\_]. When we convert to CPS form, the continuation of this occurrence of C will be bound to the variable `outk`. What happens then is that we bind the expression `\v localk. outk v` to the variable `k` and evaluate [body], passing through to it the existing continuation `outk`. Now if `body` is just, for example, `x`, then its CPS transform [x] will be `\j. j x` and this will accept the continuation `outk` and feed it `x`, and we'll continue on with nothing unusual occurring. If on the other hand `body` makes use of the variable `k`, what happens then? For example, suppose `body` includes `foo (k v)`. In the reduction of the CPS transform `[foo (k v)]`, `v` will be passed to `k` which as we said is now `\v localk. outk v`. The continuation of that application---what is scheduled to happen to `k v` after it's evaluated and `foo` gets access to it---will be bound next to `localk`. But notice that this `localk` is discarded. The computation goes on without it. Instead, it just continues evaluating `outk v`, where as we said `outk` is the outside continuation E[\_] of the whole `callcc (\k. body)` invocation.
So in other words, since the continuation in which `foo` was to be applied to the value of `k v` was discarded, that application never gets evaluated. We escape from that whole block of code.
It's important to understand that `callcc` binds `k` to a pipe into the continuation as still then installed. Not just to a function that performs the same computation as the context E[\_] does---that has the same normal form and extension. But rather, a pipe into E[\_] *in its continuation-playing role*. This is manifested by the fact that when `k v` finishes evaluating, that value is not delivered to `foo` for the computation to proceed. Instead, when `k v` finishes evaluating, the program will then be done. Not because of some "stop here" block attached to `k`, but rather because of what it is that `k` represents. Walking through the explanation above several times may help you understand this better.
So too will examples. We'll give some examples, and show you how to try them out in a variety of formats:
1. using the lambda evaluator to check how the CPS transforms reduce
To do this, you can use the following helper function:
let callcc = \k_body. \outk. (\k. (k_body k) outk) (\v localk. outk v) in
...
Used like this: [callcc (\k. body)] = `callcc (\k. BODY)`, where `BODY` is [body].
2. using a `callcc` operation on our Continuation monad
This is implemented like this:
let callcc body = fun outk -> body (fun v localk -> outk v) outk
3. `callcc` was originally introduced in Scheme. There it's written `call/cc` and is an abbreviation of `call-with-current-continuation`. Instead of the somewhat bulky form:
(call/cc (lambda (k) ...))
I prefer instead to use the lighter, and equivalent, shorthand:
(let/cc k ...)
Callcc examples
---------------
First, here are two examples in Scheme:
(+ 100 (let/cc k (+ 10 1)))
|-----------------|
This binds the continuation `outk` of the underlined expression to `k`, then computes `(+ 10 1)` and delivers that to `outk` in the normal way (not through `k`). No unusual behavior. It evaluates to `111`.
What if we do instead:
(+ 100 (let/cc k (+ 10 (k 1))))
|---------------------|
This time, during the evaluation of `(+ 10 (k 1))`, we supply `1` to `k`. So then the local continuation, which delivers the value up to `(+ 10 [_])` and so on, is discarded. Instead `1` gets supplied to the outer continuation in place when `let/cc` was invoked. That will be `(+ 100 [_])`. When `(+ 100 1)` is evaluated, there's no more of the computation left to evaluate. So the answer here is `101`.
You are not restricted to calling a bound continuation only once, nor are you restricted to calling it only inside of the `call/cc` (or `let/cc`) block. For example, you can do this:
(let ([p (let/cc k (cons 1 k))])
(cons (car p) ((cdr p) (cons 2 (lambda (x) x)))))
; evaluates to '(2 2 . #)
What happens here? First, we capture the continuation where `p` is about to be assigned a value. Inside the `let/cc` block, we create a pair consisting of `1` and the captured continuation. This pair is bound to p. We then proceed to extract the components of the pair. The head (`car`) goes into the start of a tuple we're building up. To get the next piece of the tuple, we extract the second component of `p` (this is the bound continuation `k`) and we apply it to a pair consisting of `2` and the identity function. Supplying arguments to `k` takes us back to the point where `p` is about to be assigned a value. The tuple we had formerly been building, starting with `1`, will no longer be accessible because we didn't bring along with us any way to refer to it, and we'll never get back to the context where we supplied an argument to `k`. Now `p` gets assigned not the result of `(let/cc k (cons 1 k))` again, but instead, the new pair that we provided: `'(2 . #)`. Again we proceed to build up a tuple: we take the first element `2`, then we take the second element (now the identity function), and feed it a pair `'(2 . #)`, and since it's an argument to the identity procedure that's also the result. So our final result is a nested pair, whose first element is `2` and whose second element is the pair `'(2 . #)`. Racket displays this nested pair like this:
'(2 2 . #)
Ok, so now let's see how to perform these same computations via CPS.
In the lambda evaluator:
let var = \x (\k. k x) in
let lam = \x_body (\k. k (\x. x_body x)) in
let app = \m n. (\k. m (\m. n (\n. m n k))) in
let app2 = \a b c. app (app a b) c in
let app3 = \a b c d. app (app (app a b) c) d in
let op1 = \op. \u. u (\a k. k (op a)) in
let op2 = \op. \u. u (\a v. v (\b k. k (op a b))) in
let op3 = \op. \u. u (\a v. v (\b w. w (\c k. k (op a b c)))) in
let callcc = \k_body. \outk. (\k. (k_body k) outk) (\v localk. outk v) in
; (+ 100 (let/cc k (+ 10 1))) ~~> 111
app2 (op2 plus) (var hundred) (callcc (\k. app2 (op2 plus) (var ten) (var one)))
; evaluates to \k. k (plus hundred (plus ten one))
Next:
; (+ 100 (let/cc k (+ 10 (k 1)))) ~~> 101
app2 (op2 plus) (var hundred) (callcc (\k. app2 (op2 plus) (var ten) (app (var k) (var one))))
; evaluates to \k. k (plus hundred one)
We won't try to do the third example in this framework.
Finally, using the Continuation monad from our OCaml monad library. We begin:
# #use "path/to/monads.ml"
# module C = Continuation_monad;;
Now what we want to do is something like this:
# C.(run0 (100 + callcc (fun k -> 10 + 1)));;
`run0` is a special function in the Continuation monad that runs a value of that monad using the identity function as its initial continuation. The above expression won't type-check, for several reasons. First, we're trying to add 100 to `callcc (...)` but the latter is a `Continuation.m` value, not an `int`. So we have to do this instead:
# C.(run0 (callcc (fun k -> 10 + 1) >>= fun i -> 100 + i));;
Except that's still no good, because `10 + 1` and `100 + i` are of type `int`, but their context demands Continuation monadic values. So we have to throw in some `unit`s:
# C.(run0 (callcc (fun k -> unit (10 + 1)) >>= fun i -> unit (100 + i)));;
- : int = 111
This works and as you can see, delivers the same answer `111` that we got by the other methods.
Next we try:
# C.(run0 (callcc (fun k -> unit (10 + (k 1))) >>= fun i -> unit (100 + i)));;
That won't work because `k 1` doesn't have type `int`, but we're trying to add it to `10`. So we have to do instead:
# C.(run0 (callcc (fun k -> k 1 >>= fun j -> unit (10 + j)) >>= fun i -> unit (100 + i)));;
- : int = 101
This also works and as you can see, delivers the expected answer `101`.
The third example is more difficult to make work with the monadic library, because its types are tricky. I was able to get this to work, which uses OCaml's "polymorphic variants." These are generally more relaxed about typing. There may be a version that works with regular OCaml types, but I haven't yet been able to identify it. Here's what does work:
# C.(run0 (callcc (fun k -> unit (1,`Box k)) >>= fun (p1,`Box p2) -> p2 (2,`Box unit) >>= fun p2' -> unit (p1,p2')));;
- : int * (int * [ `Box of 'b -> ('a, 'b) C.m ] as 'b) as 'a =
(2, (2, `Box ))
Delimited control operators
===========================
`callcc` is what's known as an *undelimited control operator*. That is, the continuations `outk` that get bound to our `k`s behave as though they include all the code from the `call/cc ...` out to *and including* the end of the program.
Often times it's more useful to use a different pattern, where we instead capture only the code from the invocation of our control operator out to a certain boundary, not including the end of the program. These are called *delimited control operators*. A variety of the latter have been formulated.
The most well-behaved from where we're coming from is the pair `reset` and `shift`. `reset` sets the boundary, and `shift` binds the continuation from the position where it's invoked out to that boundary.
It works like this:
(1) outer code
------- reset -------
| (2) |
| +----shift k ---+ |
| | (3) | |
| | | |
| | | |
| +---------------+ |
| (4) |
+-------------------+
(5) more outer code
First, the code in position (1) runs. Ignore position (2) for the moment. When we hit the `shift k`, the continuation between the `shift` and the `reset` will be captured and bound to `k`. Then the code in position (3) will run, with `k` so bound. The code in position (4) will never run, unless it's invoked through `k`. If the code in position (3) just ends with a regular value, and doesn't apply `k`, then the value returned by (3) is passed to (5) and the computation continues.
So it's as though the middle box---the (2) and (4) region---is by default not evaluated. This code is instead bound to `k`, and it's up to other code whether and when to apply `k` to any argument. If `k` is applied to an argument, then what happens? Well it will be as if that were the argument supplied by (3) only now that argument does go to the code (4) syntactically enclosing (3). When (4) is finished, that value also goes to (5) (just as (3)'s value did when it ended with a regular value). `k` can be applied repeatedly, and every time the computation will traverse that same path from (4) into (5).
I set (2) aside a moment ago. The story we just told is a bit too simple because the code in (2) needs to be evaluated because some of it may be relied on in (3).
For instance, in Scheme this:
(require racket/control)
(reset
(let ([x 1])
(+ 10 (shift k x))))
will return 1. The `(let ([x 1]) ...` part is evaluated, but the `(+ 10 ...` part is not.
Notice we had to preface the Scheme code with `(require racket/control)`. You don't have to do anything special to use `call/cc` or `let/cc`; but to use the other control operators we'll discuss you do have to include that preface in Racket.
This pattern should look somewhat familiar. Recall from our discussion of aborts, and repeated at the top of this page:
let foo x =
+---try begin----------------+
| (if x = 1 then 10 |
| else abort 20 |
| ) + 100 |
+---end----------------------+
in (foo 2) + 1000;;
The box is working like a reset. The `abort` is implemented with a `shift`. Earlier, we refactored our code into a more CPS form:
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;;
`snapshot` here corresponds to the code outside the `reset`. `continue_normally` is the middle block of code, between the `shift` and its surrounding `reset`. This is what gets bound to the `k` in our `shift`. The `if...` statement is inside a `shift`. Notice there that we invoke the bound continuation to "continue normally". We just invoke the outer continuation, saved in `snapshot` when we placed the `reset`, to skip the "continue normally" code and immediately abort to outside the box.
Using `shift` and `reset` operators in OCaml, this would look like this:
#require "delimcc";;
let p = Delimcc.new_prompt ();;
let reset = Delimcc.push_prompt p;;
let shift = Delimcc.shift p;;
let abort = Delimcc.abort p;;
let foo x =
reset(fun () ->
shift(fun continue ->
if x = 1 then continue 10
else 20
) + 100
)
in foo 2 + 1000;;
- : int = 1020
If instead at the end we did `... foo 1 + 1000`, we'd get the result `1110`.
The above OCaml code won't work out of the box; you have to compile and install a special library that Oleg wrote. We discuss it on our [translation page](/translating_between_ocaml_scheme_and_haskell). If you can't get it working, then you can play around with `shift` and `reset` in Scheme instead. Or in the Continuation monad. Or using CPS transforms of your code, with the help of the lambda evaluator.
The relevant CPS transforms will be performed by these helper functions:
let reset = \body. \outk. outk (body (\i i)) in
let shift = \k_body. \midk. (\k. (k_body k) (\i i)) (\a localk. localk (midk a)) in
let abort = \body. \midk. body (\i i) in
...
You use these like so:
* [reset m] is `reset M` where `M` is [m]
* [shift k M] is `shift (\k. M)` where `M` is [m]
* and [abort M] is `abort M` where `M` is [m]
There are also `reset` and `shift` and `abort` operations in the Continuation monad in our OCaml [[monad library]]. You can check the code for details.
As we said, there are many varieties of delimited continuations. Another common pair is `prompt` and `control`. There's no difference in meaning between `prompt` and `reset`; it's just that people tend to say `reset` when talking about `shift` and `prompt` when talking about `control`. `control` acts subtly differently from `shift`. In the uses you're likely to make as you're just learning about continuations, you won't see any difference. If you'll do more research in this vicinity, you'll soon enough learn about the differences.
(You can start by reading [the Racket docs](http://docs.racket-lang.org/reference/cont.html?q=shift&q=do#%28part._.Classical_.Control_.Operators%29).)
Ken Shan has done terrific work exploring the relations of `shift` and `control` and other control operators to each other.
In collecting these CPS transforms and implementing the monadic versions, we've been helped by Ken and by Oleg and by these papers:
* Danvy and Filinski, "Representing control: a study of the CPS transformation" (1992)
* Sabry, "Note on axiomatizing the semantics of control operators" (1996)
Examples of shift/reset/abort
-----------------------------
Here are some more examples of using delimited control operators. We present each example three ways: first a Scheme formulation; then we compute the same result using CPS and the lambda evaluator; then we do the same using the Continuation monad in OCaml. (We don't demonstrate the use of Oleg's delimcc library.)
Example 1:
; (+ 1000 (+ 100 (abort 11))) ~~> 11
app2 (op2 plus) (var thousand)
(app2 (op2 plus) (var hundred) (abort (var eleven)))
# Continuation_monad.(run0(
abort 11 >>= fun i ->
unit (100+i) >>= fun j ->
unit (1000+j)));;
- : int = 11
When no `reset` is specified, there's understood to be an implicit one surrounding the entire computation (but unlike in the case of `callcc`, you still can't capture up to *and including* the end of the computation). So it makes no difference if we say instead:
# Continuation_monad.(run0(
reset (
abort 11 >>= fun i ->
unit (100+i) >>= fun j ->
unit (1000+j))));;
- : int = 11
Example 2:
; (+ 1000 (reset (+ 100 (abort 11)))) ~~> 1011
app2 (op2 plus) (var thousand)
(reset (app2 (op2 plus) (var hundred) (abort (var eleven))))
# Continuation_monad.(run0(
reset (
abort 11 >>= fun i ->
unit (100+i)
) >>= fun j ->
unit (1000+j)));;
- : int = 1011
Example 3:
; (+ 1000 (reset (+ 100 (shift k (+ 10 1))))) ~~> 1011
app2 (op2 plus) (var thousand)
(reset (app2 (op2 plus) (var hundred)
(shift (\k. ((op2 plus) (var ten) (var one))))))
Continuation_monad.(
let v = reset (
let u = shift (fun k -> unit (10 + 1))
in u >>= fun x -> unit (100 + x)
) in let w = v >>= fun x -> unit (1000 + x)
in run0 w);;
- : int = 1011
Example 4:
; (+ 1000 (reset (+ 100 (shift k (k (+ 10 1)))))) ~~> 1111
app2 (op2 plus) (var thousand)
(reset (app2 (op2 plus) (var hundred)
(shift (\k. (app (var k) ((op2 plus) (var ten) (var one)))))))
Continuation_monad.(
let v = reset (
let u = shift (fun k -> k (10 :: [1]))
in u >>= fun x -> unit (100 :: x)
) in let w = v >>= fun x -> unit (1000 :: x)
in run0 w);;
- : int list = [1000; 100; 10; 1]
To demonstrate the different adding order between Examples 4 and 5, we use `::` in the OCaml version instead of `+`. Here is Example 5:
; (+ 1000 (reset (+ 100 (shift k (+ 10 (k 1)))))) ~~> 1111 but added differently
app2 (op2 plus) (var thousand)
(reset (app2 (op2 plus) (var hundred)
(shift (\k. ((op2 plus) (var ten) (app (var k) (var one)))))))
Continuation_monad.(let v = reset (
let u = shift (fun k -> k [1] >>= fun x -> unit (10 :: x))
in u >>= fun x -> unit (100 :: x)
) in let w = v >>= fun x -> unit (1000 :: x)
in run0 w);;
- : int list = [1000; 10; 100; 1]
Example 6:
; (+ 100 ((reset (+ 10 (shift k k))) 1)) ~~> 111
app2 (op2 plus) (var hundred)
(app (reset (app2 (op2 plus) (var ten)
(shift (\k. (var k))))) (var one))
(* not sure if this example can be typed as-is in OCaml... this is the best I an do at the moment... *)
# type 'x either = Left of (int -> ('x,'x either) Continuation_monad.m) | Right of int;;
# Continuation_monad.(let v = reset (
shift (fun k -> unit (Left k)) >>= fun i -> unit (Right (10+i))
) in let w = v >>= fun (Left k) ->
k 1 >>= fun (Right i) ->
unit (100+i)
in run0 w);;
- : int = 111
Example 7:
; (+ 100 (reset (+ 10 (shift k (k (k 1)))))) ~~> 121
app2 (op2 plus) (var hundred)
(reset (app2 (op2 plus) (var ten)
(shift (\k. app (var k) (app (var k) (var one))))))
Continuation_monad.(let v = reset (
let u = shift (fun k -> k 1 >>= fun x -> k x)
in u >>= fun x -> unit (10 + x)
) in let w = v >>= fun x -> unit (100 + x)
in run0 w)
- : int = 121