From 260cb10c9b6fab43f660ccaaf709418cd66fdc50 Mon Sep 17 00:00:00 2001
From: Chris Barker
Date: Thu, 9 Jun 2011 17:22:50 -0400
Subject: [PATCH] changes
---
cps.mdwn | 21 +++++++++++----------
1 file changed, 11 insertions(+), 10 deletions(-)
diff --git a/cps.mdwn b/cps.mdwn
index 73edf0d0..259f412e 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.
@@ -71,12 +70,14 @@ 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.)
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
@@ -140,7 +141,7 @@ This time the reduction unfolds in a different manner:
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:
@@ -149,14 +150,14 @@ Questions and excercises:
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))'?
+= \k.[N](\n.[M](\m.mnk))`?
4. What happens when the application rules for the CPS xforms are changed to
@@ -177,8 +178,8 @@ 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
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;
--
2.11.0