Model Generation and State Generation 32

This paper investigates two xpoint approaches for minimal model reasoning with disjunctive logic programs P. The rst one, called model generation 5], is based on an operator T INT P deened on sets of Herbrand interpretations , whose least xpoint is logically equivalent to the set of minimal Herbrand models of the program. The second approach, called state generation 16], uses a xpoint operator T s P based on hyperresolution. It operates on disjunctive Herbrand states and its least xpoint is the set of logical consequences of P, the so{called minimal model state of the program. We establish a useful relationship between hyperresolution by T s P and model generation by T INT P. Then we investigate the problem of continuity of the two operators T s P and T INT P. It is known that the operator T s P is continuous 16], and so it reaches its least xpoint in at most ! iterations. On the other hand, the question of whether T INT P is continuous has been open. We show by a counterexample that T INT P is not continuous. Nevertheless, we prove that it converges towards its least xpoint in at most ! iterations too, as follows from the relationship that we show exists between hyperresolution and model generation. We deene an iterative version of T INT P that computes the perfect model semantics of stratiied disjunctive logic programs. On each stratum of the program, this operator converges in at most ! iterations. Model generations for the stable semantics and the partial stable semantics 7, 20] are respectively achieved by using this iterative operator together with the evidential transformation 4] and the 3{S transformation 22].