X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=cps.mdwn;h=a3f0459515172dda78972e342b03cf9805f2ff88;hp=259f412ebcdec5c3fb18fdd033db683b35c63cf1;hb=d12897af7d3a9b1946a084b0680a2bbb1fb1e57a;hpb=260cb10c9b6fab43f660ccaaf709418cd66fdc50;ds=inline diff --git a/cps.mdwn b/cps.mdwn index 259f412e..a3f04595 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)
@@ -68,45 +68,49 @@ 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 beta 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:
 
 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)
-    \k.(\k.k(\x.y))(\m.m(\k.(\k.k(\x.z))(\m.mwk))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`.
-
-    [(\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
+    [(\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 (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)u)] I =
+    (\k.[\x.y](\m.m[(\x.z)u]k)) I
+     *
+    [\x.y](\m.m[(\x.z)u] I) =
+    (\k.k(\x.y))(\m.m[(\x.z)u] I)
+     *           *
+    (\x.y)[(\x.z)u] I
+     *
     y I
 
 The application to `I` unlocks the leftmost functor.  Because that
@@ -115,28 +119,37 @@ 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)u}(\n.mnk))) I
+     *
+    {\x.y}(\m.{(\x.z)u}(\n.mnI)) =
+    (\k.k(\x.{y}))(\m.{(\x.z)u}(\n.mnI))
+     *             *
+    {(\x.z)u}(\n.(\x.{y})nI) =
+    (\k.{\x.z}(\m.{u}(\n.mnk)))(\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)))
+     *             *
+    {u}(\n.(\x.{z})n(\n.(\x.{y})nI)) =
+    (\k.ku)(\n.(\x.{z})n(\n.(\x.{y})nI))
+     *      *
+    (\x.{z})u(\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
@@ -144,25 +157,39 @@ 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.
 
-Questions and excercises:
+Now let's verify that the CBN CPS avoids the infinite reduction path
+discussed above (remember that `w = \x.xx`):
 
-1. Why is the CBN xform for variables `[x] = x' instead of something
+    [(\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. 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 `
-= \k.[N](\n.[M](\m.mnk))`?
-
-4. What happens when the application rules for the CPS xforms are changed to
+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.
 
-    [MN] = \k.(\m.mk)
-     = \k.[M](\m.[N](\n.mnk))
+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))`?
 
 
 Thinking through the types
@@ -176,22 +203,22 @@ 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 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' -> 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'->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
@@ -201,15 +228,18 @@ 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
 
-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.
+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.