Modifications

[lambda.git] / cps.mdwn

diff --git a/cps.mdwn b/cps.mdwn
index e1f7f58..35d0680 100644 (file)
--- a/cps.mdwn
+++ b/cps.mdwn
@@ -71,30 +71,30 @@ 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
 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.)
+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`
 
 
 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.
 
 
 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:
 
 
 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, 
     \k.(\k.k(\x.y))(\m.m(\k.(\k.k(\x.z))(\m.mwk))k)
 
 Because the initial `\k` protects the entire transformed term, 


@@ -102,11 +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`.
 
 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)
     (\k.k(\x.y))(\m.m[(\x.z)w] I)

+     *           *

     (\x.y)[(\x.z)w] I
     (\x.y)[(\x.z)w] I

+     *

     y I
 
 The application to `I` unlocks the leftmost functor.  Because that
     y I
 
 The application to `I` unlocks the leftmost functor.  Because that


@@ -115,28 +118,37 @@ CPS transform of the argument.
 
 Compare with a call-by-value xform:
 
 
 Compare with a call-by-value xform:
 

-    <x> => \k.kx
-    <\aM> => \k.k(\a<M>)
-    <MN> => \k.<M>(\m.<N>(\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:
 
 
 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.<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
+    {(\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
     (\k.ky)I

+     *

     I y
 
 Both xforms make the following guarantee: as long as redexes
     I y
 
 Both xforms make the following guarantee: as long as redexes


@@ -144,7 +156,7 @@ 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.
 
 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?  
 
 1. Why is the CBN xform for variables `[x] = x' instead of something
 involving kappas?  


@@ -157,13 +169,14 @@ CPS-xformed lambda term.  You can use the following data declaration:
 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 
 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))`?
+`{MN} = \k.{N}(\n.{M}(\m.mnk))`?

 
 4. What happens when the application rules for the CPS xforms are changed to
 
 
 4. What happens when the application rules for the CPS xforms are changed to
 

-   [MN] = \k.<M>(\m.m<N>k)
-   <MN> = \k.[M](\m.[N](\n.mnk))
-
+<pre>
+   [MN] = \k.{M}(\m.m{N}k)
+   {MN} = \k.[M](\m.[N](\n.mnk))
+</pre>

 
 Thinking through the types
 --------------------------
 
 Thinking through the types
 --------------------------


@@ -176,22 +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
 
 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 &rho; --> &sigma; for some
 choice of &rho;.
 
 We'll need an ancilliary function ': for any ground type a, a' = a;
 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 &rho; --> &sigma; for some
 choice of &rho;.
 
 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
 
 
     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
 
 Remember that types associate to the right.  Let's work through the
 application xform and make sure the types are consistent.  We'll have


@@ -201,11 +214,11 @@ the following types:
     N:a
     MN:b 
     k:b'->o
     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[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
 
     [M](\m.m[N]k):o
     [MN]:(b'->o)->o