how does rotate_simp actually work?

[Ross Duncan]

Hello,
Apologies for spamming the list so much recently. I’m using quanto for actual work at the moment, so I’m turning up a lot of issues. This email is not about an issue per se, I’m just trying to understand rotate_simp.

The main question is: when is it convergent?

I’m assuming that !-box matching is effectively non-deterministic, so the rewrite sequence that rotate_simp produces isn’t determined by the graph you start from.

The reason I ask is that I’m starting with highly symmetrical graphs, and rotate_simp is producing “normal forms” which are not symmetrical. I see no reason why that normal form should be preferred to (e.g.) it’s mirror image. All the graphs in question are undirected, and have only angle-free red-green vertices. I seem to recall reading somewhere that rotate_simp always finds the normal form for bialgebras… does directedness make a difference?

More generally, is there some intuition for what the strategy does? I’ve looked at the rotate rule, and I have the feeling that it is just shuffling edges around t random and waiting for one of the simplifications to do something useful. Is that fair?

Last, and not really related: is there some documentation on how to write simprocs?

Cheers!

-r

[Aleks Kissinger]

No, theres a bit more to it than that. It should always terminate with
a "pseudo-normal" form for any phase-free diagram. I say "should"
because its not implemented in a particularly efficient way so it
chokes on bigger graphs. The idea is that the "rotate" rule can always
be used to reduce the arity of an "interior" green spider (i.e. a
spider which is not connected to a boundary). Once the arity goes to
1, it can be eliminated by means of a copy rule. A "pseudo-normal"
form is a diagram that has no interior green spiders. They are not
unique, and are in fact in 1-to-1 correspondence with sets of binary
vectors. Two pseudo-normal forms are equal when they span the same
subspace. These subspaces are finite, so you can get an "honest"
normal form by blowing them up to the whole subspace, but this isn't
practical in general.

Here's an unpublished note about this worked out by Pawel & I a couple
of years ago:

http://www.cs.ru.nl/A.Kissinger/papers/phase-free-nf.pdf

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[Ross Duncan]

Hi Aleks — I finally got round to reading this note. Thanks very much, it’s most helpful!
-r