Unfounded Sets and Well-Founded Semantics of Answer Set Programs with Aggregates

Logic programs with aggregates (LP) are one of the major linguistic extensions to Logic Programming (LP). In this work, we propose a generalization of the notions of unfounded set and well-founded semantics for programs with monotone and antimonotone aggregates (LPm,a programs). In particular, we present a new notion of unfounded set for LPm,a programs, which is a sound generalization of the original definition for standard (aggregate-free) LP. On this basis, we define a well-founded operator for LPm,a programs, the fixpoint of which is called well-founded model (or well-founded semantics) for LPm,a programs. The most important properties of unfounded sets and the well-founded semantics for standard LP are retained by this generalization, notably existence and uniqueness of the well-founded model, together with a strong relationship to the answer set semantics for LPm,a programs. We show that one of the D̃-well-founded semantics, defined by Pelov, Denecker, and Bruynooghe for a broader class of aggregates using approximating operators, coincides with the well-founded model as defined in this work on LPm,a programs. We also discuss some complexity issues, most importantly we give a formal proof of tractable computation of the well-founded model for LPm,a programs. Moreover, we prove that for general LP programs, which may contain aggregates that are neither monotone nor antimonotone, deciding satisfaction of aggregate expressions with respect to partial interpretations is coNP-complete. As a consequence, a well-founded semantics for general LP programs that allows for tractable computation is unlikely to exist, which justifies the restriction on LPm,a programs. Finally, we present a prototype system extending DLV, which supports the well-founded semantics for LPm,a programs, at the time of writing the only implemented system that does so. Experiments with this prototype show significant computational advantages of aggregate constructs over equivalent aggregate-free encodings.