Abstract not found

Abstract. Finite model search for first-order logic theories is complementary to theorem proving. Systems like Falcon, SEM and FMSET use the known LNH (Least Number Heuristic) heuristic to eliminate some trivial symmetries. Such symmetries are worthy, but their exploitation is limited to the first levels of the model search tree, since they disappear as soon as the first cells have been interpreted. The symmetry property is well-studied in propositional logic and CSPs, but only few trivial results on this are known on model generation in first-order logic. We study in this paper both an ordering strategy that selects the next terms to be interpreted and a more general notion of symmetry for finite model search in first-order logic. We give an efficient detection method for such symmetry and show its combination with the trivial one used by LNH and LNHO heuristics. This increases the efficiency of finite model search generation. The method SEM with and without both the function ordering and symmetry detection is experimented on several interesting mathematical problems to show the advantage of reasoning by symmetry and the function ordering. 1

Introduction Few systems for finite model enumeration use pruning techniques based on the notion of isomorphism between finite interpretations (see [2],[1]), known to correspond to elementary equivalence of interpretations. These pruning techniques have two important restrictions: only special isomorphisms are considered, and only the last interpretation which has not been eliminated this way is tested for isomorphism with new candidate interpretations, which may be isomorphic with previous interpretations. The first point is due to the fact that there are many isomorphisms (n! for an interpretation of size n), isomorphism testing is time consuming, and it would be pointless to spend more time on such tests than would require the evaluation of the formula on the candidate interpretation. This is even more likely to happen were all the previous interpretations to be tested for isomorphism. Hence only a very limited amount of isomorphism testing m

this paper---though rather simple from a model building point of view---show once more the usefulness of model builders in studying mathematical theories. In the near future, we shall continue our investigation of von Plato's axiomatization of geometry, using the model building systems developed into the framework of the Atinf project.