gap-pkg-kbmag 1.5.11

KBMAG (pronounced Kay-bee-mag) stands for Knuth-Bendix on Monoids, and Automatic Groups. It is a stand-alone package written in C, for use under UNIX, with an interface to GAP. There are interfaces for the use of KBMAG with finitely presented groups, monoids and semigroups defined within GAP. The package also contains a collection of routines for manipulating finite state automata, which can be accessed via the GAP interface.

The overall objective of KBMAG is to construct a normal form for the elements of a finitely presented group G in terms of the given generators together with a word reduction algorithm for calculating the normal form representation of an element in G, given as a word in the generators. If this can be achieved, then it is also possible to enumerate the words in normal form up to a given length, and to determine the order of the group, by counting the number of words in normal form. In most serious applications, this will be infinite, since finite groups are (with some exceptions) usually handled better by Todd-Coxeter related methods. In fact a finite state automaton W is calculated that accepts precisely the language of words in the group generators that are in normal form, and W is used for the enumeration and counting functions. It is possible to inspect W directly if required; for example, it is often possible to use W to determine whether an element in G has finite or infinite order.

The normal form for an element g in G is defined to be the least word in the group generators (and their inverses) that represents G, with respect to a specified ordering on the set of all words in the group generators.

KBMAG offers two possible means of achieving these objectives. The first is to apply the Knuth-Bendix algorithm to the group presentation, with one of the available orderings on words, and hope that the algorithm will complete with a finite confluent presentation. (If the group is finite, then it is guaranteed to complete eventually but, like the Todd-Coxeter procedure, it may take a long time, or require more space than is available.) The second is to use the automatic group program. This also uses the Knuth-Bendix procedure as one component of the algorithm, but it aims to compute certain finite state automata rather than to obtain a finite confluent rewriting system, and it completes successfully on many examples for which such a finite system does not exist. In the current implementation, its use is restricted to the shortlex ordering on words. That is, words are ordered first by increasing length, and then words of equal length are ordered lexicographically, using the specified ordering of the generators.

The GAP4 version of KBMAG also offers extensive facilities for finding confluent presentations and finding automatic structures relative to a specified finitely generated subgroup of the group G. Finally, there is a collection of functions for manipulating finite state automata that may be of independent interest.




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Last modified: Mon Apr 29 16:57:37 MDT 2024 by Jerry James