Towards a Compact Knowledge Representation by Ar Models

An AR model (= atomic representation of a Herbrand structure) is given through a set A = fA 1 ; : : : ; A n g of atoms over some Herbrand universe H with the following intended meaning: a ground atom over H evaluates to true, ii it is an instance of some atom A i 2 A. The applicability of AR models to automated model building has been demonstrated by recent publications in this eld (cf. FL 96]). Moreover, AR models are also a useful formalism for knowledge representation in general. For the practical work with AR models, the complexity of algorithms for evaluating atoms and arbitrary clauses is decisive. Actually, both problems are shown to be coNP complete in GP98]. The complexity considerations in GP98] and the algorithms presented in Pic 98] reveal, that the number of ground instances represented by an atom set A (or by a single atom) has only little innuence on the complexity of evaluating atoms and clauses, while the form of A is crucial. In particular, it is advantageous to have a small number of general atoms in A rather than a large number of speciic atoms. The purpose of this work is to investigate 2 desirable properties of AR models, namely "most-generality" and "minimality". Their positive eeect on the complexity of evaluating atoms and clauses is clear from the above cited papers. What still has to be investigated is the cost of testing these properties and, hence, of compiling an arbitrary AR model into an equivalent one with these properties. The main result of this work is the p 2-completeness proof for testing whether a given AR model is most general. Furthermore, "most-generality" is contrasted with "minimality" and it is sketched that the latter one has a similar complexity.