Fuzzy Aggregates in Fuzzy Answer Set Programming

Fuzzy answer set programming [Saad, 2010; Saad, 2009; Subrahmanian, 1994] is a declarative framework for representing and reasoning about knowledge in fuzzy environments. However, the unavailability of fuzzy aggregates in disjunctive fuzzy logic programs, DFLP, with fuzzy answer set semantics [Saad, 2010] prohibits the natural and concise representation of many interesting problems. In this paper, we extend DFLP to allow arbitrary fuzzy aggregates. We define fuzzy answer set semantics for DFLP with arbitrary fuzzy aggregates including monotone, antimonotone, and nonmonotone fuzzy aggregates. We show that the proposed fuzzy answer set semantics subsumes both the original fuzzy answer set semantics of DFLP [Saad, 2010] and the classical answer set semantics of classical disjunctive logic programs with classical aggregates [Faber et al., 2010], and consequently subsumes the classical answer set semantics of classical disjunctive logic programs [Gelfond and Lifschitz, 1991]. We show that the proposed fuzzy answer sets of DFLP with fuzzy aggregates are minimal fuzzy models and hence incomparable, which is an important property for nonmonotonic fuzzy reasoning.