X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=cps.mdwn;h=35d0680c2319da37047c65f5702dd47465baad3f;hp=73edf0d0bd24778e1bfe9c4cda35b259cc146f42;hb=aa47742f7ab6e132d2afe3dd6703855bfaeb7ecf;hpb=84044a390060ed446864f14556ece128d48e53d8 diff --git a/cps.mdwn b/cps.mdwn index 73edf0d0..35d0680c 100644 --- a/cps.mdwn +++ b/cps.mdwn @@ -9,8 +9,7 @@ A lucid discussion of evaluation order in the context of the lambda calculus can be found here: [Sestoft: Demonstrating Lambda Calculus Reduction](http://www.itu.dk/~sestoft/papers/mfps2001-sestoft.pdf). Sestoft also provides a lovely on-line lambda evaluator: -[Sestoft: Lambda calculus reduction workbench] -(http://www.itu.dk/~sestoft/lamreduce/index.html), +[Sestoft: Lambda calculus reduction workbench](http://www.itu.dk/~sestoft/lamreduce/index.html), which allows you to select multiple evaluation strategies, and to see reductions happen step by step. @@ -63,7 +62,7 @@ And we never get the recursion off the ground. Using a Continuation Passing Style transform to control order of evaluation --------------------------------------------------------------------------- -We'll exhibit and explore the technique of transforming a lambda term +We'll present a technique for controlling evaluation order by transforming a lambda term using a Continuation Passing Style transform (CPS), then we'll explore what the CPS is doing, and how. @@ -71,29 +70,31 @@ In order for the CPS to work, we have to adopt a new restriction on beta reduction: beta reduction does not occur underneath a lambda. That is, `(\x.y)z` reduces to `z`, but `\w.(\x.y)z` does not, because the `\w` protects the redex in the body from reduction. +(A redex is a subform ...(\xM)N..., i.e., something that can be the +target of reduction.) Start with a simple form that has two different reduction paths: -reducing the leftmost lambda first: `(\x.y)((\x.z)w) ~~> y' +reducing the leftmost lambda first: `(\x.y)((\x.z)w) ~~> y` -reducing the rightmost lambda first: `(\x.y)((\x.z)w) ~~> (x.y)z ~~> y' +reducing the rightmost lambda first: `(\x.y)((\x.z)w) ~~> (\x.y)z ~~> y` After using the following call-by-name CPS transform---and assuming that we never evaluate redexes protected by a lambda---only the first reduction path will be available: we will have gained control over the order in which beta reductions are allowed to be performed. -Here's the CPS transform: +Here's the CPS transform defined: - [x] => x - [\xM] => \k.k(\x[M]) - [MN] => \k.[M](\m.m[N]k) + [x] = x + [\xM] = \k.k(\x[M]) + [MN] = \k.[M](\m.m[N]k) Here's the result of applying the transform to our problem term: - [(\x.y)((\x.z)w)] - \k.[\x.y](\m.m[(\x.z)w]k) - \k.(\k.k(\x.[y]))(\m.m(\k.[\x.z](\m.m[w]k))k) + [(\x.y)((\x.z)w)] = + \k.[\x.y](\m.m[(\x.z)w]k) = + \k.(\k.k(\x.[y]))(\m.m(\k.[\x.z](\m.m[w]k))k) = \k.(\k.k(\x.y))(\m.m(\k.(\k.k(\x.z))(\m.mwk))k) Because the initial `\k` protects the entire transformed term, @@ -101,11 +102,14 @@ we can't perform any reductions. In order to see the computation unfold, we have to apply the transformed term to a trivial continuation, usually the identity function `I = \x.x`. - [(\x.y)((\x.z)w)] I - \k.[\x.y](\m.m[(\x.z)w]k) I - [\x.y](\m.m[(\x.z)w] I) + [(\x.y)((\x.z)w)] I = + (\k.[\x.y](\m.m[(\x.z)w]k)) I + * + [\x.y](\m.m[(\x.z)w] I) = (\k.k(\x.y))(\m.m[(\x.z)w] I) + * * (\x.y)[(\x.z)w] I + * y I The application to `I` unlocks the leftmost functor. Because that @@ -114,55 +118,65 @@ CPS transform of the argument. Compare with a call-by-value xform: - => \k.kx - <\aM> => \k.k(\a) - => \k.(\m.(\n.mnk)) + {x} = \k.kx + {\aM} = \k.k(\a{M}) + {MN} = \k.{M}(\m.{N}(\n.mnk)) This time the reduction unfolds in a different manner: - <(\x.y)((\x.z)w)> I - (\k.<\x.y>(\m.<(\x.z)w>(\n.mnk))) I - <\x.y>(\m.<(\x.z)w>(\n.mnI)) - (\k.k(\x.))(\m.<(\x.z)w>(\n.mnI)) - <(\x.z)w>(\n.(\x.)nI) - (\k.<\x.z>(\m.(\n.mnk)))(\n.(\x.)nI) - <\x.z>(\m.(\n.mn(\n.(\x.)nI))) - (\k.k(\x.))(\m.(\n.mn(\n.(\x.)nI))) - (\n.(\x.)n(\n.(\x.)nI)) - (\k.kw)(\n.(\x.)n(\n.(\x.)nI)) - (\x.)w(\n.(\x.)nI) - (\n.(\x.)nI) - (\k.kz)(\n.(\x.)nI) - (\x.)zI - I + {(\x.y)((\x.z)w)} I = + (\k.{\x.y}(\m.{(\x.z)w}(\n.mnk))) I + * + {\x.y}(\m.{(\x.z)w}(\n.mnI)) = + (\k.k(\x.{y}))(\m.{(\x.z)w}(\n.mnI)) + * * + {(\x.z)w}(\n.(\x.{y})nI) = + (\k.{\x.z}(\m.{w}(\n.mnk)))(\n.(\x.{y})nI) + * + {\x.z}(\m.{w}(\n.mn(\n.(\x.{y})nI))) = + (\k.k(\x.{z}))(\m.{w}(\n.mn(\n.(\x.{y})nI))) + * * + {w}(\n.(\x.{z})n(\n.(\x.{y})nI)) = + (\k.kw)(\n.(\x.{z})n(\n.(\x.{y})nI)) + * * + (\x.{z})w(\n.(\x.{y})nI) + * + {z}(\n.(\x.{y})nI) = + (\k.kz)(\n.(\x.{y})nI) + * * + (\x.{y})zI + * + {y}I = (\k.ky)I + * I y Both xforms make the following guarantee: as long as redexes underneath a lambda are never evaluated, there will be at most one -reduction avaialble at any step in the evaluation. +reduction available at any step in the evaluation. That is, all choice is removed from the evaluation process. -Questions and excercises: +Questions and exercises: 1. Why is the CBN xform for variables `[x] = x' instead of something involving kappas? 2. Write an Ocaml function that takes a lambda term and returns a -CPS-xformed lambda term. +CPS-xformed lambda term. You can use the following data declaration: type form = Var of char | Abs of char * form | App of form * form;; 3. What happens (in terms of evaluation order) when the application rule for CBN CPS is changed to `[MN] = \k.[N](\n.[M]nk)`? Likewise, -What happens when the application rule for CBV CPS is changed to ` -= \k.[N](\n.[M](\m.mnk))'? +What happens when the application rule for CBV CPS is changed to +`{MN} = \k.{N}(\n.{M}(\m.mnk))`? 4. What happens when the application rules for the CPS xforms are changed to - [MN] = \k.(\m.mk) - = \k.[M](\m.[N](\n.mnk)) - +
+   [MN] = \k.{M}(\m.m{N}k)
+   {MN} = \k.[M](\m.[N](\n.mnk))
+
Thinking through the types -------------------------- @@ -175,22 +189,22 @@ well-typed. But what will the type of the transformed term be? The transformed terms all have the form `\k.blah`. The rule for the CBN xform of a variable appears to be an exception, but instead of -writing `[x] => x`, we can write `[x] => \k.xk`, which is +writing `[x] = x`, we can write `[x] = \k.xk`, which is eta-equivalent. The `k`'s are continuations: functions from something -to a result. Let's use $sigma; as the result type. The each `k` in -the transform will be a function of type `ρ --> σ` for some +to a result. Let's use σ as the result type. The each `k` in +the transform will be a function of type ρ --> σ for some choice of ρ. We'll need an ancilliary function ': for any ground type a, a' = a; -for functional types a->b, (a->b)' = a' -> (b' -> o) -> o. +for functional types a->b, (a->b)' = ((a' -> o) -> o) -> (b' -> o) -> o. Call by name transform Terms Types - [x] => \k.xk [a] => (a'->o)->o - [\xM] => \k.k(\x[M]) [a->b] => ((a->b)'->o)->o - [MN] => \k.[M](\m.m[N]k) [b] => (b'->o)->o + [x] = \k.xk [a] = (a'->o)->o + [\xM] = \k.k(\x[M]) [a->b] = ((a->b)'->o)->o + [MN] = \k.[M](\m.m[N]k) [b] = (b'->o)->o Remember that types associate to the right. Let's work through the application xform and make sure the types are consistent. We'll have @@ -200,11 +214,11 @@ the following types: N:a MN:b k:b'->o - [N]:a' - m:a'->(b'->o)->o + [N]:(a'->o)->o + m:((a'->o)->o)->(b'->o)->o m[N]:(b'->o)->o m[N]k:o - [M]:((a->b)'->o)->o = ((a'->(b'->o)->o)->o)->o + [M]:((a->b)'->o)->o = ((((a'->o)->o)->(b'->o)->o)->o)->o [M](\m.m[N]k):o [MN]:(b'->o)->o