In a seminal paper, Gelfond and Lifschitz [34] introduced simple disjunctive logic programs, where in rule heads the disjunction operator "vertical bar" 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 vertical bar and define answer sets as follows: (i) Given an interpretation I and a selection function sel, we transform a disjunctive program Pi into a normal program Pi(I)(sel), called a disjunctive program reduct; (ii) given a base answer set semantics chi for normal programs, we define I to be a candidate answer set of Pi w.r.t. chi if I is an answer set of Pi(I)(sel) under chi; and (iii) we define I to be an answer set of Pi w.r.t. chi if I is a minimal candidate answer set. The DI-semantics is general and applicable to extend any answer set semantics chi for normal programs to disjunctive programs. By replacing chi with the GL(nlp)-semantics defined by Gelfond and Lifschitz [33], we induce a DI-semantics for simple disjunctive programs, and by replacing chi 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 Pi such that answer sets of the normal derivatives are answer sets of Pi 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 Pi 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 Pi has some answer set is Sigma(p)(2)-complete, while brave and cautious reasoning are Sigma(p)(2)- and Pi(p)(2)-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. (C) 2019 The Authors. Published by Elsevier B.V.
