Automatic Generation of Some Results in Finite Algebra

This is a report of the application of the Model Generation Theorem Prover developed at ICOT to problems in the theory of finite quasigroups. Several of the problems were pre­ viously open. In this paper, we discuss our the­ orem proving methods, related to those of the existing provers SATCHMO (Manthey, Bry) and OTTER (McCune), and note how parallel processing on the ICOT Parallel Inference Ma­ chines was used to obtain high speeds. We then present and discuss our machine-aided investi­ gation of seven problems concerning the exis­ tence of types of quasigroup. The field of finite algebra is rich in problems suitable for computational treatment. In particular, questions about the existence of structures of given finite sizes may usefully be approached using techniques for theo­ rem proving or for constraint satisfaction. Perhaps the best known recent result of this sort is that of Lam, Thiel and Swiercz [Lam et al, 1989] showing that there is no projective plane of order 10. The present paper reports further research in the same tradition in which our tool was the ICOT theorem prover MGTP and in which we were able to obtain new existence and nonexistence the­ orems for certain interesting classes of quasigroup. It should be noted that MGTP is a general-purpose theo­ rem prover not at all designed with quasigroups in mind. Hence this application of Artif icial Intelligence research to open mathematical problems was rather unexpected. We welcome i t , however, as we feel that automated rea­ soning has reached the point where it can and should address the needs of researchers in other disciplines. 1 F i n i t e d o m a i n search ing Where a first order theory has finite models, a reasonable way to find some is to fix on a particular finite domain of objects and to search for interpretations of the func­ tion and predicate symbols as functions and predicates on that domain, constrained to make all of the axioms of the theory true. Finite model generation may thus become a constraint satisfaction problem of the familiar sort, and may yield to the familiar techniques approprite to such problems. To illustrate with a deliberately triv­ ial example, consider the theory of semigroups. This has only one axiom, an equation and has, of course, models of every cardinality. To find the semigroups of order 3 (for instance) we fix the do­ main as consisting of three objects; the numbers 1, 2 and 3 will do nicely. Now we need to determine the value of x o y for each choice of x and y from the domain, giv­ ing 9 cases to determine and 3 possible values for each case. Evidently there are 3 or 19683 possible vectors of 9 values in this search space. The "good" ones are those which make true every ground instance of the associa­ tivity axiom such as So each bad vector contains a set of (at most) four values which suf­ fice to refute some such formula. For example, the four assignments together force thus violating as­ sociativity. Therefore the disjunction is one of the ground negative constraints of cardinality 4 which all good vectors must obey. There are many methods of enumerating the vectors which do obey such a set of constraints. We used a back­ tracking search technique based on automated deduction with a clausal representation of the problem, as detailed below, but there is no reason why other styles of search such as that embodied in arc consistency or path con­ sistency algorithms should not be employed to much the same effect. Our concern in this paper is to report the method which was successful in practice, not to prove that it is the best method possible. Whatever techniques are used, some general heuristics are in order. Firstly, whenever one of the ground neg­ ative constraints, as instanced above, is used to refute an attempt to build a model, it should be remembered somehow and the search control should ensure that it 52 Automated Reasoning never again gets incorporated into a partial model can­ didate. The search wil l typically backtrack many times, but it should never do so twice for the same reason. Sec­ ondly, it obviously pays to minimize the furcation of the search tree. Hence where there is a choice as to which cell is to be given a value next, choose one with the smallest number of possible values (given the partial structure al­ ready in place). An early version of our program MGTP failed to do this and constructed over 52 million branches on problem QG5.7 below; after the heuristic was added, it branched just 9 ways on the same problem! Thirdly, as is well known, most algebraic search problems permit early detection of some or all isomorphisms and by at­ tending to these we may frequently cut down the search space by some orders of magnitude. In the cases treated below, we used a simple initial restriction of the problem to avoid searching a great many isomorphic subspaces. Naturally, we do not claim originality for any of these general search heuristics. Indeed, they are rather wellworn. Nonetheless, we wish to draw attention to them since they were essential to the success of our experi­ ments and since it is still common to find such obvious points overlooked. 2 The program: M G T P 2.1 O t t e r , Satchmo and M G T P Otter [McCune, 1990] is a very efficient first order theo­ rem prover. It can be seen as computing the closure of a set of axioms under selected rules of inference. It works with two set of clauses called the Usable Set (U) and set of support (SOS). In each step it moves a clause C from SOS to U, generates immediate consequences of C in combination with with members of U and stores them in SOS. Essential to its strategy is the avoidance of possible regeneration of redundant consequences mainly by rejecting subsumable clauses instead of storing them (forward subsumption). It may also apply backward subsumption, whereby newly kept clauses are used to sim­ plify the existing U and SOS. It gives the option of vari­ ous rules including resolution and some of its more pow­ erful relatives such as hyperresolution and unit-resulting resolution, as well as a wide variety of equality reasoning facilities such as forms of paramodulation and demodu­ lation. Satchmo [Manthy and Bry, 1988] can be seen as a spe­ cialized theorem prover for solving only range restricted problems, using the important technique of case split­ t ing. Range restricted problems are those which assure that all derived positive clauses are ground. Case split­ t ing, as is familiar, is a matter of treating clauses of a certain type (usually, as in Satchmo, positive ground clauses) by assuming each of their literals in turn in order to reason by cases. It is thus fundamental to reasoning in the style of semantic tableaux. Given range restrictedness, case splitt ing is safe from the problem of com­ mon variable handling between the split literals. Like the Prolog Technology Theorem Prover described in [Stickel, 1988] Satchmo takes advantage of Prolog's optimization techniques by compiling clauses for runtime efficiency. MGTP (Model Generation Theorem Prover) is writ­ ten in the parallel logic programming language K L 1 . There are two versions of the program: MGTP/G for range-restricted problems and M G T P / N for (Horn) nonground problems. See [Fujita tt a/, 1992] for a descrip­ tion. The basic algorithm of MGTP/G is equivalent to that of Satchmo, while that of M G T P / N is based on that of Otter. Both versions run sequentially on the PSI workstations and also on the P IM (Parallel Inference Ma­ chine) developed at IGOT. For the algebraic problems considered in the present paper, only MGTP/G was re­ quired. Satchmo's technique of compiling clauses into a logic programming language is adopted by MGTP nat­ urally but not trivially. How M G T P / G uses the power of KL1 to maintain efficiency [Fuchi, 1990; Fujita and Hasegawa, 199l]is outside the scope of this paper. This basic efficiency, is the one of the important sources of MGTP's success, even though without heuristics there is no hope of managing the combinatorial explosion. 2.2 P r o b l e m representa t ion and heur is t ics As in most artificial intelligence applications, heuris­ tics appear to be very important in the attack on dif­ ficult model generation problems. Al l advanced uses of Otter, such as solving the very difficult condensed de­ tachment and related problems reported in [Wos et a/, 1990], use weighting mechanisms for picking the next clause from SOS. The simplest method of assigning weights to clauses is to count the constituent literals, and the next simplest is to count symbols. The latter is Otter's default. More elaborate weighting techniques are of considerable interest, but are not the focus of the present paper. Heuristics for MGTP are in a sense similar to those for the Otter. We used a weighting function of the number of literals in each clause, corresponding to the number of ways the search tree wil l branch. A unit clause has lowest weight of all. We then simply choose to split a clause of lower weight rather than one of higher. Clearly, this is an implementation of the general principle alluded to in the last section, of minimising furcation of the search tree. In counting literals for this purpose, we omit any that could directly produce the empty clause by clashing with a negative clause. We also omit any literal L such that there exist clauses in the set U, since we can look far enough ahead to see that such L could be resolved away and wil l therefore generate only a dead branch. Of course such weighting methods may destroy com­ pleteness if the domain of the problem is infinite, but there are many ways to escape from this. For example, Otter has the strategy of moderating its weight-directed search by letting every nth clause come from a breadth first search. For finite domain problems such as n-queen problems and the Peiletier and Rudnicki problems, how­ ever, a weight strategy such as ours does not endanger completeness. Moreover it prunes an extremely large Fujita, Slaney, and Bennett 53