Head-Elementary-Set-Free Logic Programs

Abstract

Abstract. The recently proposed notion of an elementary set yielded a refinement of the theorem on loop formulas, telling us that the stable models of a disjunctive logic program can be characterized by the loop formulas of its elementary sets. Based on the notion of an elementary set, we propose the notion of head-elementary-set-free (HEF) programs, a more general class of disjunctive programs than head-cycle-free (HCF) programs proposed by Ben-Eliyahu and Dechter, that can still be turned into nondisjunctive programs in polynomial time and space by "shifting" the head atoms into the body. We show several properties of HEF programs that generalize earlier results on HCF programs. Given an HEF program, we provide an algorithm for finding an elementary set whose loop formula is not satisfied, which has a potential for improving stable model computation by answer set solvers.

On the complexity of identifying head-elementary-set-free programs

“…We have proved here that the problem at hand is coNP-complete, hereby providing an answer to a question left open in (Gebser et al 2007). This, basically negative, result leaves open the further problem of singling out a polynomial-time recognizable fragment of DLP, generalizing over HCF programs, while sharing their nice computational characteristics.…”

“…Definition 3 (Head-Elementary-Set-Free Program [Gebser et al 2007] ) Let P be a disjunctive program. P is HEF if for each rule B, F → H in P, there is no elementary set E for P such that |E ∩ H| > 1.…”

“…For instance, the disjunctive datalog system (DLV) engine (Leone et al 2006) takes advantage of identifying head-cycle-free (HCF) (sub)programs (Ben-Eliyahu and Dechter 1994;Ben-Eliyahu-Zohary and Palopoli 1997) in resolving disjunctive logic programs under the stable model semantics. Head-elementaryset-free (HEF) programs were recently introduced in (Gebser et al 2007) as a strict generalization of HCF programs featuring the same nice properties of that smaller class. In detail, likewise HCF programs, HEF programs can be turned into equivalent nondisjunctive programs in polynomial time and space by shifting.…”

“…As such, HEF programs can be regarded as "easy" disjunctive programs, since they actually denote syntactic variants of nondisjunctive coding. This fact has several formal consequences, which are precisely accounted for in (Gebser et al 2007). Just for an example, while checking for a disjunctive program to have a stable model is Σ P 2 -complete in general, it is NP-complete for HEF programs.…”

“…It was left as an open problem in (Gebser et al 2007) to establish the complexity of identifying HEF programs. This note solves the open problem by showing that the problem is complete for coNP.…”