Extending the Well-Founded and Valid Semantics for Aggregation

We present a very general technique for deening semantics for programs that use aggregation. We use the technique to extend the well-founded semantics and the valid semantics, both of which were designed to provide semantics for programs with negation, to handle programs that contain possibly recursive use of aggregation. The generalization is based on a simple but powerful idea of aggregation on three-valued multisets. The use of three-valued multisets makes our extended well-founded semantics, which we call aggregate-well-founded semantics, more intuitive than the extension of well-founded models by Van Gelder Van92]. Our semantics and Van Gelder's semantics agree on many programs, and on others our semantics provides results that Van Gelder says are intuitive and desirable, but that his semantics does not provide. The extended valid semantics, which we call the aggregate-valid semantics, generalizes the intuition behind several semantics presented earlier for aggregation; for instance, our semantics properly subsumes the semantics deened by Ganguly et al. GGZ91] for cost-monotonic programs. The rst author, on behalf of all those alphabetic-order challenged, led a successful crusade to have the names in reverse alphabetical order. The order does not reeect on the degree of contribution by the various authors. Extensions planned for the future include height and weight ordering of authors. y All communication may be addressed to the rst author.