Lifschitz and Turner introduced the notion of the splitting set and provided a method to divide a logic program into two parts. They showed that the task of computing the answer sets of the program can be converted into the tasks of computing the answer sets of these parts. However, the empty set and the set of all atoms are the only two splitting sets for many programs, then these programs cannot be divided by the splitting method. In this paper, we extend Lifschitz and Turner's splitting set theorem to allow the program to be split by an arbitrary set of atoms, while some new atoms may be introduced in the process. To illustrate the usefulness of the result, we show that for some typical programs the splitting process is efficient and the program simplification problem can be investigated using the concept of splitting.

This paper focuses on computing general first-order parallel and prioritized circumscription with varying constants. We propose linear translations from general first-order circumscription to first-order theories under stable model semantics over arbitrary structures, including Tr_v for parallel circumscription and Tr^s_v for conjunction of parallel circumscriptions (further for prioritized circumscription). To improve the efficiency, we give an optimization \Gamma_{\exists} to reduce logic programs in size when eliminating existential quantifiers during the translations. Based on these results, a general first-order circumscription solver, named cfo2lp, is developed by calling answer set programming (ASP) solvers. Using circuit diagnosis problem and extended stable marriage problem as benchmarks, we compare cfo2lp with a propositional circumscription solver circ2dlp and an ASP solver with complex optimization metasp on efficiency. Experimental results demonstrate that for problems represented by first-order circumscription naturally and intuitively, cfo2lp can compute all solutions over finite structures. We also apply our approach to description logics with circumscription and repairs in inconsistent databases, which can be handled effectively.