cps & contification

[Matthew Fluet]

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> I have one proposal for further simplification.  I think it is easier to avoid
> defining A' entirely.  Instead, define a predicate Reach: Fun -> Bool so that
> Reach(f) iff there is a path of calls from fm to f.  Then, instead of writing
> A'(f) = Unknown, write Reach(f) and instead of writing A'(f) = Uncalled, write
> not(Reach(f)). 

I like this.  I agree that the reasoning is a little easier.

> I find it easier to reason directly in terms of reach than going through A'.
> Following is a reworking of the framework.  I also simplified the proof of
> Theorem 3 and combined Lemmas 4 and 5.

> Theorem 3: For all f in Fun, there is a path from Root to f in G.
> Proof:
> Either Reach(f) or not(Reach(f).
> 
> If Reach(f), then there is a path of calls from fm to f.  Every g in Fun on
> this path also has Reach(g).  Hence every call on the path corresponds to an
> edge in Edge by fourth or fifth clause.  Since (Root, fm) in Edge, there is a
> path from Root to f in G. 
> 
> If not(Reach(f)), then (Root, f) in Edge by third clause.  Hence there is a path
> from Root to f in G.

In the Reach(f) case, we also have to note that (Root, j) in Edge for all
j in Jump, because some edges of the form (j, g) might be included by the
fifth clause.


> I guess you're working on the implementation as we speak?

Here it is.  I did it as a replacement of the existing new analysis, but
if we really wanted to do comparisons with the two-phase analysis, it
should be simple enough.  


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