Characterizations of the Disjunctive Stable . . .

(Extended abstract appeared in: Extended stable semantics for normal and disjunctive logic programs. Unfold/fold transformation of general logic programs for the well-founded semantics. 21 deenitions. In BD96a, BD96b] a rigorous description of STABLE, WFS and their disjunctive counterparts based on certain connuent calculi of transformations is given. Finally DS96] extends these transformations to programs with variables by using constraint logic programming techniques. ACKNOWLEDGEMENTS We are indebted to Li-Yan Yuan, F. Miguel Dionisio and to two anonymous referees for helpful comments on a draft of this paper. 20 Yuan noted that the stable semantics on the smaller class of all programs without an odd number of negative edges through negation satisses Relevance. We believe that this class can not only be extended using our idea of deleting non-genuine cycles through negation, but it also represents the maximal such class. More formally, we have the following Conjecture 4.1. (No Semantics for Genuine Non-Stratiied Programs) Let SEM be a non-trivial semantics satisfying GPPE, Elimination of Tautolo-gies, Elimination of Contradictions and Relevance. Then SEM is only deened on the class of all programs P such that res(P) contains no cycles with an odd number of negative edges. There is no semantics beyond this class. Note that if we cancel the Elimination of Contradictions then there are semantics deened on the whole class of programs, e.g. the static semantics of Przymusinski ((Prz95]) or the D-WFS introduced by the authors ((BD94b, BD95c]). We believe our conjecture to be true both for disjunctive and non-disjunctive programs. In the latter case already the wellfounded semantics WFS satisses all properties except Elimination of Contradictions. We think that the last two results and our conjecture show us that if we leave the class of stratiied disjunctive programs then a semantics should be based on three-valued models, i.e. Elimination of Contradictions should be given up. 5. CONCLUSIONS In this paper, we have shown that partial evaluation is an interesting property. It not only holds for various semantics but it also characterizes these semantics together with some other weak transformation conditions. GPPE is a powerful principle. Together with Elimination of Tautologies it enables us to deene a normal form of a program (Lemma 4.1). Both properties are suucient to ensure that a semantics only selects minimal two-valued models for positive disjunctive programs (Theorem 4.1). Together with Elimination of Contradictions (resp. the assumption that a semantics is based on weakly supported models), we …