6 Group Theory and Automated Deduction
7.4 Tietze Transformations and Group Presentation Simplification
Automated group presentation simplification programs are a tool much used in computa tional group theory to simplify the very large presentations produced by some algorithms, permitting further human or machine-assisted investigation. Typically, presentations which
require simplification are the result of finding a presentation of a subgroup H of finite index
in a finitely presented group G via the Reidemeister-Schreier procedure. This procedure
tends to produce presentations which have many redundant generators and relations. A simplification of such a presentation is essential if one wants to apply other computational techniques to it. No precise definition of ‘simplification’ is given, but that usually accepted in practice is the following:
1. a reduction in the number of generators,
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3. a reduction in the number of relators.
This corresponds roughly to an increase in the practicability of applying well-known tools (such as coset enumeration or the nilpotent quotient algorithm) to the presentation. It should
be noted that there is really no satisfactory definition of simplification for the presentation
of a finitely presented group. A presentation may have many generators, many relations and a great overall length, but it may also have the property that it is a finite convergent rewriting system. This means that it is possible to ‘read off’ important structural information about the group. The standard tools from group theory, e.g. the coset enumeration, nilpotent quotient algorithms, may not be useful in studying such a presentation.
Applying a simplification procedure to a presentation is generally much less time-consuming than subsequent analysis of the presentation, so that even small additional simplifications at significant costs in simplification time are worthwhile. Sims [112] proposes a presentation simplification program using Knuth-Bendix completion. In general, however, this approach is not widely used. This is because the presentations are simplified with respect to a given term ordering, and not the three criteria mentioned above.
Group presentation simplification programs (see e.g. Havas et al [54]) tend to be based on
the Tietze Transformations T2, T3 and T4. This can be seen in large group theory packages
such as GAP [109]. T2 and TA are done automatically by these programs, but T3 is used
interactively.
We propose an additional heuristic for a Tietze transfoimation program which uses the idea of the ‘critical pair’ from Knuth-Bendix completion. When the presentation simplification program can do no more, we convert the relators into a set of rewrite rules and calculate a set of critical pairs associated with those rewrite rules. This is then converted back to the language of relators and the resulting presentation is again simplified. This will help in the automatic simplification of a given presentation. As a justification for this approach, we have the following example.
Example 7.5
We investigate the example F(2, 5) which was referred to in Chapter6.
We state the presentation again for ease of reference:7 TIETZE TRANSFORMATIONS
The GAP group presentation simplification program eliminates b, d and e and returns the
presentation:
G = (^a,c \ a~^ * * c * a~^, c* a~^* * a * c~^* a
^
Feeding this into a Knuth-Bendix program for strings (see Holt [60]) with the recursive
path ordering with the precedence c > a, and completing the rewriting system we obtain
the following set of rewrite rules:
Rewrite Rules = {c“ ^ - > a * a * a * a * a * a , c ~ > a * a * a * a * a ,
a * a * a * a * a * a * a * a * a * a * a —> 1} The resulting simplified presentation is
G' = ^ a ] ^
Of course, the presentation could have been fed directly to a Knuth-Bendix program; exper imental data (see Chapter 6), however, shows that the approach of simplifying the present ation first is more efficient. There is also a question of which critical pairs to calculate. The critical pairs which are added into the presentation are defined below.
Definition 7.6
Given a presentation G = {X \ R ), with BA, AC~^€
R, the relator BC is called a relator critical pairinG.
Lemma 7.7
Given a presentation G — {X \ R), any relator critical pair of relatorsri =
BA,
T
2=
AC~^ £ R is a relator which holds in GProof- It is possible to form the overlap BAC~^, which can be rewritten in two ways to
obtain B — C~^. Therefore BCis a relator. □
This is equivalent to a critical pair in Knuth-Bendix completion, and adding the critical pair to Ris an instance of the Tietze transformation T l. For experimental purposes, we chose to convert each of the relators r Ç. Rinto a rewrite rule and take critical pairs in the usual way:
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a number of orderings can be chosen and different orderings will correspond to different sets of relator critical pairs being computed. There is also a question of symmetrising the rules by multiplying through with the inverses of the generators which occur. Experiments, however, suggest that this can make the rewrite system unduly large. We choose to look at
the rewrite rules r e for each r E B and calculate the critical pairs of these rales.
We have the following six step algorithm; given a presentation G = ( X | B ), we simplify the presentation as follows:
1. Simplify the presentation with a standard group presentation simplification program to obtain Gi = ( Xi | Bi >
2. Convert each r E i?i into a rewrite rale r —> e 3. Calculate all critical pairs of the rewrite rales from 2 4. Take the union of the rales and equations from 2 and 3
5. For each equation and rewrite rale in 4, convert to relators: e.g. ri —>■ rg becomes '^1(^2)“ ^ and f3 = f4 becomes r3(r4)“ ^; thus obtain a set of relators Bg
6. Simplify presentation Gg = (X i | Bg )
In practice it is useful to iterate (2) — (6) a number of times in order to obtain a reasonably simplified presentation. We now give some results as to how the new heuristic works when used in conjunction with the GAP presentation simplification program. We call our present ation simplification heuristic IT-KB: this was implemented in GAP in order to be able to use the efficient presentation simplification program which is implemented in it. The above algorithm is used and the best presentation after 3 iterations is chosen. This is compared to the automatically derived GAP simplified presentation; these results appeal* in Table 6. The experiments were executed on a SPARC IPX with 64Mb of memory, and the cpu times are in seconds.
The presentations considered occur in Havas et al [54]. The Reidemeister-Schreier al gorithm is used to generate a presentation for the subgroup H(n ) = (a, 6^) of index j2n+3| (for n = 5, ±7, ±8, ±10) in the group G(n) =
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Table 6: Comparing Automatic Group Presentation Simplification Programs
gen number of generators rel number of relators len sum of lengths of relators cpu cpu time in seconds
Original GAP TT-KB
Group gen rel len gen rel len cpu gen rel len cpu
B(5) 14 26 238 2 3 37 0.9 2 3 37 1.5 B(7) 18 34 378 4 20 1,189 2 2 5 112 30 B (-7 ) 12 22 228 2 12 613 1.3 2 12 598 11 B(8) 20 38 466 4 22 1,095 3 3 22 3,258 68 77(-8) 14 26 294 4 16 474 1.3 4 16 474 9 F(10) 24 46 648 2 24 8,391 16 2 24 8,391 575 B (-IO ) 18 34 450 4 20 818 2.3 4 21 725 21.3
The results in Table 6 show that the TT-KB approach does work in the area in which present ation simplification programs aie used. The gains are modest, and the CPU times longer, but, as remarked above, this may well be worthwhile in practice. This method has the advantage that it can be varied to use the expertise built up both in group theory, where presentation simplification algorithms have been designed, and also the theoretical com puter science community which has expertise in Knuth-Bendix completion.
8 CONCLUSIONS AND FUTURE WORK