&KBMAG; (pronounced ``Kay-bee-mag'') stands for
<E>Knuth--Bendix on Monoids, and Automatic Groups</E>.
It is a stand-alone package written in `C', for use under UNIX,
with an interface to &GAP;.
Chapters <Ref Chap="chap-rws"/> and <Ref Chap="chap-cosets"/>
describe its use as an external library from within &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.
Chapter <Ref Chap="chap-standalone"/>
lists the functions in the stand-alone package.
<P/>
To use this package effectively, some knowledge of the underlying
theory and algorithms is advisable.
The Knuth-Bendix algorithm is described in various places in the literature.
Good general references that deal with the applications to groups and monoids
are <Cite Key="LeC86"/> and the first few chapters of <Cite Key="Sims94"/>.
For the theory of automatic groups see the multi-author book
<Cite Key="ECHLPT92"/>.
The algorithms employed by &KBMAG; are described more specifically in
<Cite Key="EHR91"/> and <Cite Key="Holt94"/>.
<P/>
The manual for the stand-alone &KBMAG; package (which can be found in the
<File>standalone/doc</File> directory of the package) provides more detailed
information on the external `C' programs that are called from &GAP;.
<P/>
Suppose that <M>G</M> is a finitely presented semigroup, monoid or group
defined as a quotient of the free structure <M>F</M>.
The overall objective of &KBMAG; is to construct a normal form for
the elements of <M>G</M> in terms of the generators of <M>F</M>,
together with a word reduction algorithm for calculating
the normal form representative of an element in <M>G</M>, given by a word
in the generators of <M>F</M>.
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 <M>G</M>, by counting the number of words in
normal form. In most serious applications, this will be infinite,
since (for example) finite groups are (with some exceptions) usually
handled better by Todd-Coxeter related methods.
In fact a finite state automaton <M>W</M>
is calculated that accepts precisely the language of words in the
monoid generators of <M>F</M> that are in normal form, and <M>W</M> is used for the enumeration and counting functions.
<P/>
The normal form of an element <M>g \in G</M> is defined to be the least
word in the generators of <M>F</M> (and their inverses) that represents
<M>g</M>, with respect to a specified ordering on the set of all
words in the generators of <M>F</M>.
The available orderings are described in section <Ref Sect="sec-set-order"/>.
<P/>
&KBMAG; offers two possible means of achieving these objectives.
The first is to apply the Knuth-Bendix algorithm to the
presentation, with one of the available orderings on words, and hope
that the algorithm will complete with a finite confluent presentation.
(If <M>G</M> 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,
which is only applicable to groups (not to monoids or semigroups).
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 stand-alone 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. However, there are now some &GAP; procedures available
in the package written by Sarah Rees that enable it be used also for the
&wtlex; ordering, and the &wreathprod; ordering.
See section <Ref Sect="sec-set-order"/> for further details of these orderings.
<P/>
For both of the above procedures, the first step is to create a &GAP;
object known as a <E>Knuth-Bendix rewriting system</E> <M>R</M>
from the finitely presented structure <M>G</M>.
There are functions available that can be used to
specify the input parameters for the external programs, such as the
ordering on words to be used by the Knuth-Bendix procedure. One of the
two external programs is then run on <M>R</M>. If successful, it updates
<M>R</M>, which can then be used to reduce words in
the generators of <M>F</M> to normal form, and to count and enumerate the
words in normal form.
<P/>
There are also now some routines available for performing
corresponding operations with the cosets of a specified subgroup
<M>H</M> of the group <M>G</M>. (These are not currently available for semigroups or
monoids.) The words in normal form then correspond to minimal
representatives under the ordering of the system of the right
cosets of <M>H</M> in <M>G</M>.
If successful, the index of <M>H</M> in <M>G</M> can be determined.
The Knuth-Bendix routines also allow a confluent
rewriting system for <M>H</M> to be computed, whereas the automatic groups
routines allow a presentation of <M>H</M> to be computed (although not yet on
a user-specified generating set).
<P/>
In the descriptions of the functions that follow, it is important to
distinguish between irreducible words, and words in normal form. As
already stated, a word is in normal form if it is the least word under
the ordering of the rewriting system that defines a particular group element or coset. So there is always a unique word in normal form
for each group element or coset, and it is determined by the group
generators and the
ordering on words in the group generators. A word in a rewriting
system is said to be <E>irreducible</E> if it does not contain the left hand
side of any of the reduction rules in the system as a subword. Words
in normal form are always irreducible, but the converse is true if and
only if the rewriting system is confluent. The automatic groups
programs provide a method of reducing words to normal form without
obtaining a finite confluent rewriting system (which may not even
exist).
<P/>
Various levels of diagnostic output from the &GAP; procedures can
be turned on by
setting the Info variable <C>InfoRWS</C> to <M>1, 2</M> or <M>3</M>.
<P/>
In the descriptions that follow functions declared in the main &GAP;
library, for which additional methods are implemented, are referred to
as <E>library functions</E>.
<P/>
</Chapter>
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