From 75d46539f289007a9983110854425c3477e85406 Mon Sep 17 00:00:00 2001
From: Chris Barker <barker@kappa.linguistics.fas.nyu.edu>
Date: Wed, 8 Jun 2011 14:16:39 -0400
Subject: [PATCH] added little file on cpsX

---
 cps.mdwn | 125 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 125 insertions(+)
 create mode 100644 cps.mdwn

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+Gaining control over order of evaluation
+----------------------------------------
+
+We know that evaluation order matters.  We're beginning to learn how
+to gain some control over order of evaluation (think of Jim's abort handler).
+We continue to reason about order of evaluation.
+
+A superbly clear and lucid discussion can be found here:
+[Sestoft: Demonstrating Lambda Calculus Reduction](http://tinyurl.com/27nd3ub).
+Sestoft also provides a really lovely lambda evaluator, 
+which allows you to select multiple evaluation strategies, 
+and to see reductions happen step by step:
+[Sestoft's lambda reduction webpage](http://ellemose.dina.kvl.dk/~sestoft/lamreduce/lamframes.html).
+
+
+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:
+
+<pre>
+(\x.y)(ww)
+ *
+y
+</pre>
+
+Done!  We have a normal form.  But if we reduce using a different
+strategy, things go wrong:
+
+<pre>
+(\x.y)(ww) =
+(\x.y)((\x.xx)w) =
+        *
+(\x.y)(ww) =
+(\x.y)((\x.xx)w) =
+        *
+(\x.y)(ww) 
+</pre>
+
+Etc.  
+
+As a second reminder of when evaluation order matters, consider using
+`Y = \f.(\h.f(hh))(\h.f(hh))` as a fixed point combinator to define a recursive function:
+
+<pre>
+Y (\f n. blah) =
+(\f.(\h.f(hh))(\h.f(hh))) (\f n. blah) 
+     *
+(\f.f((\h.f(hh))(\h.f(hh)))) (\f n. blah) 
+       *
+(\f.f(f((\h.f(hh))(\h.f(hh))))) (\f n. blah) 
+         *
+(\f.f(f(f((\h.f(hh))(\h.f(hh)))))) (\f n. blah) 
+</pre>
+
+And we never get the recursion off the ground.
+
+
+Restricting evaluation: call by name, call by value
+---------------------------------------------------
+
+One way to begin to gain some control is by adjusting our notion of
+beta reduction.  This strategy has some of the flavor of adding types,
+but is actually a part of the untyped calculus.
+
+In order to have a precise discussion, we'll need some vocabulary.
+We've been talking about normal form (lambda terms that can't be
+further reduced), and leftmost evaluation (the reduction strategy of
+always choosing the left most reducible lambda); we'll need to refine
+both of those concepts.
+
+Kinds of normal form:
+---------------------
+
+Recall that there are three kinds of lambda term.  Let `a` be an
+arbitrary variable, and let `M` and `N` be arbitrary terms:
+
+<pre>
+The pure untyped lambda calculus (again):
+
+             Form     Examples  
+Variable     a        x, y, z
+Abstract     \aM      \x.x, \x.y, \x.\y.y
+Application  MN       (x x), ((\x.x) y), ((\x.x)(\y.y))
+</pre>
+
+It will be helpful to define a *redex*, which is a lambda term of
+the form `((\aM) N)`.  That is, a redex is a form for which beta
+reduction is defined (ignoring, as usual, the manouvering required in
+order to avoid variable collision).
+
+It will also be helpful to have names for the two components of an
+application `(M N)`: we'll call `M` the *functor*, and `N` the
+*argument*.  Note that functor position can be occupied by a variable,
+since `x` is in the functor position of the term `(x y)`.
+
+Ok, with that vocabulary, we can distinguish four different types of 
+normal form:
+
+
+
+
+
+
+
+Take Plotkin's CBN CPS:
+
+[x] ~~> x
+[\xM] ~~> \k.k(\x[M])
+[MN] ~~> \k.[M](\m.m[N]k)
+
+let w = \x.xx in
+
+[(\xy)(ww)] ~~>
+\k.[\xy](\m.m[ww]k) ~~>
+\k.[\xy](\m.m(\k.[w](\m.m[w]k))k) ~~>
+\k.[\xy](\m.m(\k.(\k.k(\x[xx]))(\m.m[w]k))k) ~~> beta*
+\k.[\xy](\m.m(\k.(\x[xx])[w]k)k) ~~>
+\k.[\xy](\m.m(\k.(\x(\k.[x](\m.m[x]k)))[w]k)k) ~~>
+\k.[\xy](\m.m(\k.(\x(\k.x(\m.mxk)))[w]k)k) ~~> beta
+\k.[\xy](\m.m(\k.[w](\m.m[w]k))k) --- same as second line!
+
-- 
2.11.0