FACTORES DE LA ALIMENTACION

Intuitively, the Knuth-Bendix completion procedure for strings can be seen as T l, F2 and (to a lesser extent) T4. T l, the adding in of a new relator to the group, is similar to deducing a critical pair (and then ordering it) in completion. T2, the deleting of a relator which can be proved true by the other relators in the presentation, is equivalent to an equation or rewrite

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rale being rewritten and then deleted. If x is a generator, w is a word in which x does not

occur and a completion procedure generates a rewrite rule x w, then we can think of

X as being ‘rewritten out’. When the rewrite system and set of equations are normalised

with respect to æ -4 w, then the only occurrence of x will be in a; —)■ u?. The lexicographic

recursive path ordering (see Sims [113] and Dershowitz [38]), orders the rale this way when there is a precedence on the generators such that

æ > { generators occurring in w } This is equivalent to Tietze transformation T4.

We will first give an example to justify our approach. The example comes from Jantzen [66], who showed that the group in question had an infinite convergent rewriting system given one set of generators, and a finite convergent rewriting system given another.

Example 7.3 Let G = ( a , If one attempts to complete 1}

with the lexicographic recursive path ordering with precedence h > a one finds that there

are infinitely many rales of the form 6 *6*( o * 6 ) * * u * o —^ (a * 6)*. It is important to note here that this is a monoid presentation for the group; i.e. we are not using any of the natural inverse relations. An obvious choice of substring to ‘rewrite out’ would seem to be a * 6, which occurs infinitely many times. Suppose we added a =h 6 -> c into the rewriting system. Running the completion procedure one gets the following ‘new’ rewriting system:

R' — {c * 6 —>■ 6 * 6 * n, n * 6 -4 c}

i.e. this rewriting system is infinite. It is also possible to see that the ‘same’ critical pairs are being calculated. So to choose a ‘useful’ substring to ‘rewrite out’ of the presentation, it would seem wise to check the overlaps which are creating the critical pairs. In this case the overlap of interest is

6 * 6 * (n * } (a*

So we need a way to ‘destroy’ these critical pairs; i.e. choose a string which will stop

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choice is 6 * 6 = c. Completing {a*6*6*a = 1,6*6 = c} with the lexicographie recursive

path ordering with precedence b > a > c, one obtains the finite rewriting system S:

5 = { 6 * 6 - - ) ' C , 6 * c —>-c*6, a * c - > c * a , c * a * u —) - e , 6 * a * a —>-a*a*6}

Of course, this heuristic could change a finite system into an infinite one. A (ground) com­ plete rewrite system is so useful, however, that it is often wortli exploring this possibility. It should be noted at this point that adding in the generator is the Tietze transformation T3. Thus it follows straight from Theorem 7.2 that the presentation with the newly defined generator is isomorphic to the original group.

Needham [98] needs ways to determine when an infinite family of rewrite rules are being created by overlaps. Possible rules for deciding when an infinite family of rewrite rules is being generated are stated. We wish, though, to define a heuristic, and so the rules we can use to introduce new generators are much weaker than the rules needed in the work of Needham [98].

We now give an informal description of the procedure which we are using. This depends upon being able to analyse how the infinite families of critical pairs are being generated, and being able to ‘extend’ a reduction ordering > in which there is a precedence on the gener­ ators. We will analyse families of overlaps in order to determine where infinite families of critical pairs originate. We simply choose a substring of an overlap, which, if it is renamed, will ensure that this overlap will not occur again. So in Example 7.3 there is an infinite family of critical pairs 6 * 6 * ( a * 6 ) ^ * a * a = (a *6)* and an infinite family of overlaps o * 6 * 6 * 6 * ( a * 6 ) ^ * o * a . Thus the overlap will not occur again if we choose to ‘rewrite out’ the substring 6*6. In order to introduce a generator, a new constant c is defined and c is put at the bottom of the precedence in the reduction ordering. Thus the rale is oriented 6 * 6 —> c, and the substring 6*6 is rewritten out of the presentation. This is called breaking the overlap.

The procedure which is used will now be described.

1. given a set of equations and a reduction ordering which includes a precedence on the operators, start the completion procedure.

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3. continue with completion procedure

We can think of all of the equations which define potential new generators as always having existed, but there are times when modifying the ordering so that they are then ‘of use’ makes sense. If the rewrite rules are convergent on a set of generators, the rest of the potential new generators can be added at the top of the precedence in the reduction ordering. This will mean that they will be oriented

new generator > word in ‘already defined’ generators

and automatically this rewriting system will be convergent, as the left hand sides of these rewrite rules do not overlap with the left hand side of any other rewrite rules. Hence, in a sense, the finiteness of a rewriting system depends entirely upon the ‘choice of ordering’. Example 7.4 We will now look at a particularly interesting family of examples from Martin [91]. The groups in question have the following presentation for A: > 1:

Hk = ^ a, 6 I 6^^ = 1, aba = h^^~^,abb = bba ^

H k has 2 finite rewriting systems, and the following k - 1 infinite rewriting systems. The

infinite rewriting systems are defined as follows for 1 < p < A; — 1 and 1 < t< o o :

Bp = abab —>■ 1, baba 1, abb —> bba, aba^b ba*ba, b^a^ba -> a^~^b,

jRi arises when using the recursive path ordering with b > a. There are also k — I infinite convergent rewrite systems Rpwhich have the words in the above presentation reversed. A preliminary analysis of the rules of Rp shows that there are the following critical pairs (where cp{li, I2) denotes that there is a critical pair between rale h ~¥ ri and rale I2 -> rg):

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1. cp{abb, bba^*~^^b) = bba^ha —> a^b

2. cpibaba, aba^b) = bba*ba —>• a*b

3. cp{abab, aba%) = bba^ba —>■

4. cp{abb,b^^^~'^^a^b) =

So a first new generator to introduce may be baa ~ d. Making the operator precedence

b > a > d with the lexicographic recursive path ordering, one can complete Hk to a finite,

convergent system:

C =

add —)■ dda,

ada -> ddd,

b -> d'^^^~^^a^d}

Future work will concentrate on investigating the idea of adding in generators in a more theoretical framework. In particular the work of Bachmair [5] in proof orderings will be used to prove theoretical results about such a system.