Model generation and state generation for disjunctive logic programs

This paper investigates two fixpoint approaches for minimal model reasoning with disjunctive logic programs P. The first one, called model generation, is based on an operator TPINT defined on sets of Herbrand interpretations whose least fixpoint is logically equivalent to the set of minimal Herbrand models of the program. The second approach, called state generation, uses a fixpoint operation TPs based on hyperresolution. It operates on disjunctive Herbrand states, and its least fixpoint 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 PPs and model generation by TPINT. Then we investigate the problem of continuity of the two operators TPs and TPINT. It is known that the operator TPs is continuous, and so it reaches its least fixpoint in at most ω iterations. On the other hand, the question of whether TPINT is continuous has been open. We show by a counterexample that TPINT is not continuous. Nevertheless, we prove that it converges towards its least fixpoint in at most ω iterations, too, as follows from the relationship that we show exists between hyperresolution and model generation. We define an iterative version of TPINT that computes the perfect model semantics of stratified disjunctive logic programs. On each stratum of the program, this operator converges in at most ω iterations. Model generations for the stable semantics and the partial stable semantics are respectively achieved by using this iterative operator together with the evidential transformation and the 3-S transformation.