From 54360af5dafc49cca9e484ecdc74c26c4adfb65c Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Fri, 10 Jun 2011 14:52:19 -0400 Subject: [PATCH] changes --- cps.mdwn | 131 ++++++++++++++++++++++++++++++++++++--------------------------- 1 file changed, 74 insertions(+), 57 deletions(-) diff --git a/cps.mdwn b/cps.mdwn index 35d0680c..f869b795 100644 --- a/cps.mdwn +++ b/cps.mdwn @@ -18,8 +18,8 @@ Evaluation order matters We've seen this many times. For instance, consider the following reductions. It will be convenient to use the abbreviation `w = -\x.xx`. I'll indicate which lambda is about to be reduced with a * -underneath: +\x.xx`. I'll +indicate which lambda is about to be reduced with a * underneath: 
 (\x.y)(ww)
@@ -68,10 +68,11 @@ what the CPS is doing, and how.
 
 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.)
+That is, `(\x.y)z` reduces to `z`, but `\u.(\x.y)z` does not reduce to
+`\w.z`, because the `\w` protects the redex in the body from
+reduction.  (In this context, a redex is a part of a term that matches
+the pattern `...((\xM)N)...`, i.e., something that can potentially be
+the target of beta reduction.)
 
 Start with a simple form that has two different reduction paths:
 
@@ -92,23 +93,23 @@ Here's the CPS transform defined:
 
 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) =
-    \k.(\k.k(\x.y))(\m.m(\k.(\k.k(\x.z))(\m.mwk))k)
+    [(\x.y)((\x.z)u)] =
+    \k.[\x.y](\m.m[(\x.z)u]k) =
+    \k.(\k.k(\x.[y]))(\m.m(\k.[\x.z](\m.m[u]k))k) =
+    \k.(\k.k(\x.y))(\m.m(\k.(\k.k(\x.z))(\m.muk))k)
 
-Because the initial `\k` protects the entire transformed term, 
-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`.
+Because the initial `\k` protects (i.e., takes scope over) the entire
+transformed term, we can't perform any reductions.  In order to watch
+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)((\x.z)u)] I =
+    (\k.[\x.y](\m.m[(\x.z)u]k)) I
      *
-    [\x.y](\m.m[(\x.z)w] I) =
-    (\k.k(\x.y))(\m.m[(\x.z)w] I)
+    [\x.y](\m.m[(\x.z)u] I) =
+    (\k.k(\x.y))(\m.m[(\x.z)u] I)
      *           *
-    (\x.y)[(\x.z)w] I
+    (\x.y)[(\x.z)u] I
      *
     y I
 
@@ -125,21 +126,21 @@ Compare with a call-by-value xform:
 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
+    (\k.{\x.y}(\m.{(\x.z)u}(\n.mnk))) I
      *
-    {\x.y}(\m.{(\x.z)w}(\n.mnI)) =
-    (\k.k(\x.{y}))(\m.{(\x.z)w}(\n.mnI))
+    {\x.y}(\m.{(\x.z)u}(\n.mnI)) =
+    (\k.k(\x.{y}))(\m.{(\x.z)u}(\n.mnI))
      *             *
-    {(\x.z)w}(\n.(\x.{y})nI) =
-    (\k.{\x.z}(\m.{w}(\n.mnk)))(\n.(\x.{y})nI)
+    {(\x.z)u}(\n.(\x.{y})nI) =
+    (\k.{\x.z}(\m.{u}(\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)))
+    {\x.z}(\m.{u}(\n.mn(\n.(\x.{y})nI))) =
+    (\k.k(\x.{z}))(\m.{u}(\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))
+    {u}(\n.(\x.{z})n(\n.(\x.{y})nI)) =
+    (\k.ku)(\n.(\x.{z})n(\n.(\x.{y})nI))
      *      *
-    (\x.{z})w(\n.(\x.{y})nI)
+    (\x.{z})u(\n.(\x.{y})nI)
      *
     {z}(\n.(\x.{y})nI) =
     (\k.kz)(\n.(\x.{y})nI)
@@ -156,27 +157,40 @@ underneath a lambda are never evaluated, there will be at most one
 reduction available at any step in the evaluation.
 That is, all choice is removed from the evaluation process.
 
+Now let's verify that the CBN CPS avoids the infinite reduction path
+discussed above (remember that `w = \x.xx`):
+
+    [(\x.y)(ww)] I =
+    (\k.[\x.y](\m.m[ww]k)) I
+     *
+    [\x.y](\m.m[ww]I) =
+    (\k.k(\x.y))(\m.m[ww]I)
+     *             *
+    (\x.y)[ww]I
+     *
+    y I
+
+
 Questions and exercises:
 
-1. Why is the CBN xform for variables `[x] = x' instead of something
+1. Prove that {(\x.y)(ww)} does not terminate.
+
+2. 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
+3. Write an Ocaml function that takes a lambda term and returns a
 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 
-`{MN} = \k.{N}(\n.{M}(\m.mnk))`?
+4. The discussion above talks about the "leftmost" redex, or the
+"rightmost".  But these words apply accurately only in a special set
+of terms.  Characterize the order of evaluation for CBN (likewise, for
+CBV) more completely and carefully.
 
-4. What happens when the application rules for the CPS xforms are changed to
+5. What happens (in terms of evaluation order) when the application
+rule for CBV CPS is changed to `{MN} = \k.{N}(\n.{M}(\m.mnk))`?
 
-
-   [MN] = \k.{M}(\m.m{N}k)
-   {MN} = \k.[M](\m.[N](\n.mnk))
-

 
 Thinking through the types
 --------------------------
@@ -196,15 +210,15 @@ 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' -> o) -> o) -> (b' -> o) -> o.
+for functional types a->b, (a->b)' = ((a' -> σ) -> σ) -> (b' -> σ) -> σ.
 
     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'->σ)->σ
+    [\xM] = \k.k(\x[M])              [a->b] = ((a->b)'->σ)->σ
+    [MN] = \k.[M](\m.m[N]k)          [b] = (b'->σ)->σ
 
 Remember that types associate to the right.  Let's work through the
 application xform and make sure the types are consistent.  We'll have
@@ -213,16 +227,19 @@ the following types:
     M:a->b
     N:a
     MN:b 
-    k:b'->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'->o)->o)->(b'->o)->o)->o)->o
-    [M](\m.m[N]k):o
-    [MN]:(b'->o)->o
-
-Note that even though the transform uses the same symbol for the
-translation of a variable, in general it will have a different type in
-the transformed term.
-
+    k:b'->σ
+    [N]:(a'->σ)->σ
+    m:((a'->σ)->σ)->(b'->σ)->σ
+    m[N]:(b'->σ)->σ
+    m[N]k:σ 
+    [M]:((a->b)'->σ)->σ = ((((a'->σ)->σ)->(b'->σ)->σ)->σ)->σ
+    [M](\m.m[N]k):σ
+    [MN]:(b'->σ)->σ
+
+Be aware that even though the transform uses the same symbol for the
+translation of a variable (i.e., `[x] = x`), in general the variable
+in the transformed term will have a different type than in the source
+term.
+
+Excercise: what should the function ' be for the CBV xform?  Hint: 
+see the Meyer and Wand abstract linked above for the answer.
-- 
2.11.0