X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=cps.mdwn;h=6668b48ce1bcde7c63aed5a13b113c376e4bc884;hp=f869b7956f4f86e6a95309301c07d3fff8207075;hb=339a5442b568742f36ddab4a44f77cfb26b609a2;hpb=54360af5dafc49cca9e484ecdc74c26c4adfb65c;ds=sidebyside

diff --git a/cps.mdwn b/cps.mdwn
index f869b795..6668b48c 100644
--- a/cps.mdwn
+++ b/cps.mdwn
@@ -69,16 +69,16 @@ 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 `\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
+`\u.z`, because the `\u` 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 leftmost lambda first: `(\x.y)((\x.z)u)  ~~> y`
 
-reducing the rightmost lambda first: `(\x.y)((\x.z)w)  ~~> (\x.y)z ~~> y`
+reducing the rightmost lambda first: `(\x.y)((\x.z)u)  ~~> (\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
@@ -91,7 +91,7 @@ Here's the CPS transform defined:
     [\xM] = \k.k(\x[M])
     [MN] = \k.[M](\m.m[N]k)
 
-Here's the result of applying the transform to our problem term:
+Here's the result of applying the transform to our simple example:
 
     [(\x.y)((\x.z)u)] =
     \k.[\x.y](\m.m[(\x.z)u]k) =
@@ -109,13 +109,15 @@ trivial continuation, usually the identity function `I = \x.x`.
     [\x.y](\m.m[(\x.z)u] I) =
     (\k.k(\x.y))(\m.m[(\x.z)u] I)
      *           *
-    (\x.y)[(\x.z)u] I
+    (\x.y)[(\x.z)u] I           --A--
      *
     y I
 
 The application to `I` unlocks the leftmost functor.  Because that
-functor (`\x.y`) throws away its argument, we never need to expand the
-CPS transform of the argument.
+functor (`\x.y`) throws away its argument (consider the reduction in the
+line marked (A)), we never need to expand the
+CPS transform of the argument.  This means that we never bother to
+reduce redexes inside the argument.
 
 Compare with a call-by-value xform:
 
@@ -125,7 +127,7 @@ Compare with a call-by-value xform:
 
 This time the reduction unfolds in a different manner:
 
-    {(\x.y)((\x.z)w)} I =
+    {(\x.y)((\x.z)u)} I =
     (\k.{\x.y}(\m.{(\x.z)u}(\n.mnk))) I
      *
     {\x.y}(\m.{(\x.z)u}(\n.mnI)) =
@@ -140,7 +142,7 @@ This time the reduction unfolds in a different manner:
     {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)
+    (\x.{z})u(\n.(\x.{y})nI)       --A--
      *
     {z}(\n.(\x.{y})nI) =
     (\k.kz)(\n.(\x.{y})nI)
@@ -152,6 +154,9 @@ This time the reduction unfolds in a different manner:
      *
     I y
 
+In this case, the argument does get evaluated: consider the reduction
+in the line marked (A).
+
 Both xforms make the following guarantee: as long as redexes
 underneath a lambda are never evaluated, there will be at most one
 reduction available at any step in the evaluation.
@@ -216,9 +221,9 @@ for functional types a->b, (a->b)' = ((a' -> σ) -> σ) -> (b' -> &sig
 
     Terms                            Types
 
-    [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
+    [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
@@ -227,14 +232,14 @@ the following types:
     M:a->b
     N:a
     MN:b 
-    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
+    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
 
 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