[GAP Forum] "Knuth-Bendix Rewriting Confluent" hangs

Derek Holt dfh at maths.warwick.ac.uk
Mon May 3 18:41:14 BST 2004

On Mon, May 03, 2004 at 09:59:20AM -0600, Alexander Hulpke wrote:
> Dear GAP-Forum,
> Robert Eckert wrote:
> > Forcing Knuth-Bendix to run asks if this is in "the swamp", when, alas,
> > the swamp is bottomless.  Holt's KBMag package says, effectively, "let's
> > just map the top ten meters of the swamp".  Pardon me if I am asking a
> > "duh, of course!" or "duh, of course not!" question, but is there a way of
> > force-feeding a set of rules?  "Forget what Knuth-Bendix says looks
> > good, you will use THESE rules and like it"?  The problem may be that K-B
> While I'm not 100% certain on this interpretation, I understand yoiur
> question as asking whether you can prescribe your own rewriting rules.

I was also uncertain about what was being asked. I guess KBMAG should have an
option to essentially do nothing and to use just the existing rules for
reduction - I have occasionally wanted to do that myself. But you can
essentially achieve that aim by setting the maximum number of rewrite rules
allowed to be something less than the initial number. It might, however,
insist on tidying up your rule set slightly - for example, the rewriting
algorithm in KBMAG gets confused if two different rewriting rules have the
same left hand side.

> This is possible (see the manual chapter on rewriting systems). However in
> this case:
> - you will have to ensure confluence on your own. (which can be
>   nontrivial).
> - Unless you substitute your rewriting system (for this you would have to
>   change the library code), you will have to call `ReducedForm' with monoid
>   elements and cannot simply do group arithmetic.
> > wants to find the shortening possibilities, where actually a rewriting
> > that makes most things longer is what some purposes need.

A number of orderings of words are available in KBMAG, some of which will
result in rewriting systems which can make words longer. For example, in the
wreath product orderings (which are the ones on which reduction to normal
form in polycyclic groups is based) generators have levels, and one
occurrence of a generator at a higher level will be regarded as larger (in
the ordering) than any number of occurrencs of lower level generators.

But again, I don't know whether this is the kind of thing that you
are looking for!

Dere Holt.

> > If I rewrite in an ugly set of generators, I can deploy a filtration,
> > pure-F4 = top-F4 semidirect product by pure-C3, which is top-C3 by
> > pure-C2, which is top-C2 by pure-A1, which is free on letter-1-squared.
> > For each high-grade generator G and lower-grade g there is exactly one
> > relation like
> > G g = g G' G" G G"- G'-
> > moving g left and replacing G with its conjugate by a varying number (zero
> What you are doing effectively is to work in some quotient group. GAP can
> get you some of these (see e.g. `SolvableQuotient'). 
> In principle you could try to generalize existing code to work even in a
> profinite quotient. The problem then becomes to know when to stop evaluating
> to prove (non)-equality of elements.
> Best wishes,
>    Alexander Hulpke
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