Semantics of disjunctive programs with monotone aggregates - an operator-based approach

All major semantics of normal logic programs and normal logic programs with aggregates can be described as fixpoints of the one-step provability operator or of operators that can be derived from it. No such systematic operator-based approach to semantics of disjunctive logic programs has been developed so far. This paper is the first step in this direction. We formalize the concept of one-step-provability for disjunctive logic programs by means of non-deterministic operators on the lattice of interpretations. We establish characterizations of models, minimal models, supported models and stable models of disjunctive logic programs in terms of pre-fixpoints and fixpoints of non-deterministic immediateconsequence operators and their extensions to the four-valued setting. We develop our results for programs in propositional language extended with monotone aggregate atoms. For the most part, our concepts, results and proof techniques are algebraic, which opens a possibility for further generalizations to the abstract algebraic setting of non-deterministic operators on complete lattices.