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Subsections
IsAutomaticGroup(F: parameters) : GrpFP -> BoolElt, GrpAtc
Internally a monoid presentation P of the group F is constructed. By
default the generators of P are taken to be
g1, (g1) - 1, ..., gn, (gn) - 1 where g1, ..., gn are
the generators of F. The relations of
P are taken to be the relations of F. The trivial
relations between the generators and their inverses are also added.
The word ordering is the short-lex ordering.
The Knuth--Bendix completion procedure for monoids is now run on P
to calculate the word difference automata corresponding to the generated
equations, which are then used to calculate the finite state automata
associated with a short-lex automatic group. In successful cases these
automata are proved correct in the final step.
If the procedure succeeds the result will be an automatic group, G,
containing four automata. These are the first and second word-difference
machines, the word acceptor, and the word multiplier. The form
AutomaticGroup returns an automatic group while the form
IsAutomaticGroup returns the boolean value true and
the automatic group. If the procedure fails, the first form does
not return a value while the second returns the boolean value false.
For simple examples, the algorithms work quickly, and do not require
much space. For more difficult examples, the algorithms are often capable of
completing successfully, but they can sometimes be expensive in terms of
time and space requirements.
Another point to be borne in mind is that the algorithms sometimes produce
temporary disk files which the user does not normally see (because they are
automatically removed after use), but can occasionally be very large.
These files are stored in the /tmp directory. If you interrupt
a running automatic group calculation you must remove these temporary
files yourself.
As the Knuth--Bendix procedure will more often than not run forever,
some conditions must be specified under which it will stop. These take the
form of limits that are placed on certain variables, such as the number of
reduction relations. If any of these limits are exceeded during a run of the
completion procedure it will fail, returning a non-confluent automatic group.
The optimal values for these limits varies from example to example. Some
of these limits may be specified by setting parameters (see the next section).
In particular, if a first attempt to compute the automatic structure of a group
fails, it should be run again with the parameter Large set to true.
We construct the automatic structure for the fundamental group of the torus.
Since a generator ordering is not specified, the default generator ordering,
[ a, a - 1, b, b - 1, c, c - 1, d, d - 1], is used.
> FG<a,b,c,d> := FreeGroup(4);
> F := quo< FG | a^-1*b^-1*a*b=d^-1*c^-1*d*c>;
> f, G := IsAutomaticGroup(F);
Running Knuth-Bendix with the following parameter values
MaxRelations = 200
MaxStates = 0
TidyInt = 20
MaxWdiffs = 512
HaltingFactor = 100
MinTime = 5
#Halting with 118 equations.
#First word-difference machine with 33 states computed.
#Second word-difference machine with 33 states computed.
#System is confluent, or halting factor condition holds.
#Word-acceptor with 36 states computed.
#General multiplier with 104 states computed.
#Validity test on general multiplier succeeded.
#General length-2 multiplier with 220 states computed.
#Checking inverse and short relations.
#Checking relation: _8*_6*_7*_5 = _2*_4*_1*_3
#Axiom checking succeeded.
> G;
An automatic group.
Generator Ordering = [ a, a^-1, b, b^-1, c, c^-1, d, d^-1 ]
The second word difference machine has 33 states.
The word acceptor has 36 states.
In this section we describe the various parameters used to control the
execution of the procedures employed to determine the automatic structure.
IsAutomaticGroup(F: parameters) : GrpFP -> BoolElt, GrpAtc
Attempt to construct an automatic structure for the finitely presented
group F (see the main entry). We now present details of the various
parameters used to control the execution of the procedures.
Large: BoolElt Default: false
If Large is set to true large hash tables are used
internally. Also the Knuth--Bendix algorithm is run with larger
parameters, specifically TidyInt is set to 500, MaxRelns is set to 262144, MaxStates is set to unlimited, HaltingFactor is set to 100, MinTime is set to 20 and ConfNum is set to 0. It is advisable to use this option only after
having first tried without it, since it will result in much longer
execution times for easy examples.
MaxRelations: RngIntElt Default: 200
Limit the maximum number of reduction equations to MaxRelations.
TidyInt: RngIntElt Default: 20
After finding n new reduction equations, the completion procedure
interrupts the main process of looking for overlaps, to tidy up the
existing set of equations. This will eliminate any redundant
equations performing some reductions on their left and right hand
sides to make the set as compact as possible. (The point is that
equations discovered later often make older equations redundant or
too long.) The word-differences arising from the equations are
calculated after each such tidying and the number reported if verbose
printing is on. The best strategy in general is to try a small value
of TidyInt first and, if that is not successful, try increasing
it. Large values such as 1000 work best in really difficult
examples.
GeneratorOrder: SeqEnum Default:
Give an ordering for the generators of P. This ordering affects the
ordering of words in the alphabet. If not specified, the ordering
defaults to [g1, (g1) - 1, ..., gn, (gn) - 1] where g1, ..., gn are the generators of F.
MaxWordDiffs: RngIntElt Default:
Limit the maximum number of word differences to MaxWordDiffs.
The default behaviour is to increase the number of allowed word
differences dynamically as required, and so usually one does not need
to set this option.
HaltingFactor: RngIntElt Default: 100
MinTime: RngIntElt Default: 5
These options are experimental halting options. HaltingFactor
is a positive integer representing a percentage. After each tidying
it is checked whether both the number of equations and the number of
states have increased by more than HaltingFactor percent since
the number of word-differences was last less than what it is now. If
so the program halts. A sensible value seems to be 100, but
occasionally a larger value is necessary. If the MinTime
option is also set then halting only occurs if at least MinTime
seconds of cpu-time have elapsed altogether. This is sometimes
necessary to prevent very early premature halting. It is not very
satisfactory, because of course the cpu-time depends heavily on the
particular computer being used, but no reasonable alternative has
been found yet.
Set the verbose printing level for the Knuth--Bendix completion algorithm.
Setting this level allows a user to control how much extra information on
the progress of the algorithm is printed.
Currently the legal values for v are 0 to 3 inclusive. Setting
v to 0 corresponds to the `-silent' option of KBMAG in which no extra output
is printed. Setting v to 2 corresponds to the `-v' (verbose) option of
KBMAG in which a small amount of extra output is printed.
Setting v to 3 corresponds to
the `-vv' (very verbose) option of KBMAG in which
a huge amount of diagnostic information is printed.
We attempt to construct an automatic structure for one of Listing's
knot groups.
> F := Group< d, f | f*d*f^-1*d*f*d^-1*f^-1*d*f^-1*d^-1=1>;
> b, G := IsAutomaticGroup(F);
Running Knuth-Bendix with the following parameter values
MaxRelations = 200
MaxStates = 0
TidyInt = 20
MaxWdiffs = 512
HaltingFactor = 100
MinTime = 5
#Maximum number of equations exceeded.
#Halting with 195 equations.
#First word-difference machine with 45 states computed.
#Second word-difference machine with 53 states computed.
> b;
false;
So this attempt has failed. We run the IsAutomaticGroup function
again setting Large to true. This time we succeed.
> f, G := IsAutomaticGroup(F : Large := true);
Running Knuth-Bendix with the following parameter values
MaxRelations = 262144
MaxStates = 0
TidyInt = 500
MaxWdiffs = 512
HaltingFactor = 100
MinTime = 5
#Halting with 3055 equations.
#First word-difference machine with 49 states computed.
#Second word-difference machine with 61 states computed.
#System is confluent, or halting factor condition holds.
#Word-acceptor with 101 states computed.
#General multiplier with 497 states computed.
#Multiplier incorrect with generator number 3.
#General multiplier with 509 states computed.
#Multiplier incorrect with generator number 3.
#General multiplier with 521 states computed.
#Multiplier incorrect with generator number 3.
#General multiplier with 525 states computed.
#Validity test on general multiplier succeeded.
#General length-2 multiplier with 835 states computed.
#Checking inverse and short relations.
#Checking relation: _3*_1*_4*_1*_3 = _1*_3*_2*_3*_1
#Axiom checking succeeded.
> G;
An automatic group.
Generator Ordering = [ d, d^-1, f, f^-1 ]
The second word difference machine has 89 states.
The word acceptor has 101 states.
We construct the automatic group corresponding to the fundamental
group of the trefoil knot. A generator order is specified.
> F<a, b> := Group< a, b | b*a^-1*b=a^-1*b*a^-1>;
> f, G := IsAutomaticGroup(F: GeneratorOrder := [a,a^-1, b, b^-1]);
Running Knuth-Bendix with the following parameter values
MaxRelations = 200
MaxStates = 0
TidyInt = 20
MaxWdiffs = 512
HaltingFactor = 100
MinTime = 5
#Halting with 83 equations.
#First word-difference machine with 15 states computed.
#Second word-difference machine with 17 states computed.
#System is confluent, or halting factor condition holds.
#Word-acceptor with 15 states computed.
#General multiplier with 67 states computed.
#Multiplier incorrect with generator number 4.
#General multiplier with 71 states computed.
#Validity test on general multiplier succeeded.
#General length-2 multiplier with 361 states computed.
#Checking inverse and short relations.
#Checking relation: _3*_2*_3 = _2*_3*_2
#Axiom checking succeeded.
> G;
An automatic group.
Generator Ordering = [ a, a^-1, b, b^-1 ]
The second word difference machine has 21 states.
The word acceptor has 15 states.
The functions in this group provide access to basic information stored for a
rewrite group G.
The i-th defining generator for G. The integer i must lie in the
range [ - r, r], where r is the number of
group G.
A sequence containing the defining generators for G.
Ngens(G) : GrpRWS -> RngIntElt
The number of defining generators for G.
We illustrate the access operations using the
Von Dyck (2,3,5) group (isomorphic to A5).
> F<a,b> := FreeGroup(2);
> Q := quo< F | a*a=1, b*b=b^-1, a*b^-1*a*b^-1*a=b*a*b*a*b>;
> f, G<a,b> := IsAutomaticGroup(Q);
> G;
An automatic group.
Generator Ordering = [ a, a^-1, b, b^-1 ]
The second word difference machine has 33 states.
The word acceptor has 28 states.
> print G.1*G.2;
a * b
> print Generators(G);
[ a, b ]
> print Ngens(G);
2
> rels := Relations(G);
> print rels[1];
Q.2 * Q.2^-1 = Id(Q)
> print rels[2];
Q.2^-1 * Q.2 = Id(Q)
> print rels[3];
Q.1^2 = Id(Q)
> print rels[4];
Q.2^2 = Q.2^-1
> print Nrels(G);
18
> print Ordering(G);
ShortLex
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