Determining inference semantics for disjunctive logic programs.

In a seminal paper, Gelfond and Lifschitz [34] introduced simple disjunctive logic programs, where in rule heads the disjunction operator “|” is used to express incomplete information, and defined the answer set semantics (called GL-semantics for short) based on a program transformation (called GL-reduct) and the minimal model requirement. Our observations reveal that the requirement of the GL-semantics, i.e., an answer set should be a minimal model of rules of the GL-reduct, may sometimes be too strong a condition and exclude some answer sets that would be reasonably acceptable. To address this, we present an alternative, more permissive answer set semantics, called the determining inference (DI) semantics. Specifically, we introduce a head selection function to formalize the operator | and define answer sets as follows: (i) Given an interpretation I and a selection function sel, we transform a disjunctive program Π into a normal program ΠselI, called a disjunctive program reduct; (ii) given a base answer set semantics X for normal programs, we define I to be a candidate answer set of Π w.r.t. X if I is an answer set of ΠselI under X; and (iii) we define I to be an answer set of Π w.r.t. X if I is a minimal candidate answer set. The DI-semantics is general and applicable to extend any answer set semantics X for normal programs to disjunctive programs. By replacing X with the GLnlp-semantics defined by Gelfond and Lifschitz [33], we induce a DI-semantics for simple disjunctive programs, and by replacing X with the well-justified semantics defined by Shen et al. [65], we further induce a DI-semantics for general disjunctive programs. We also establish a novel characterization of the GL-semantics in terms of a disjunctive program reduct, which reveals the essential difference of the DI-semantics from the GL-semantics and leads us to giving a satisfactory solution to the open problem presented by Hitzler and Seda [36] about characterizing split normal derivatives of a simple disjunctive program Π such that answer sets of the normal derivatives are answer sets of Π under the GL-semantics. Finally we give computational complexity results; in particular we show that in the propositional case deciding whether a simple disjunctive program Π has some DI-answer set is NP-complete. This is in contrast to the GL-semantics and equivalent formulations such as the FLP-semantics [24], where deciding whether Π has some answer set is Σ2p-complete, while brave and cautious reasoning are Σ2p- and Π2p-complete, respectively, for both GL- and DI-answer sets. For general disjunctive programs with compound formulas as building blocks, the complexity of brave and cautious reasoning increases under DI-semantics by one level of the polynomial hierarchy, which thus offers higher problem solving capacity.