An FLP-Style Answer-Set Semantics for Abstract-Constraint Programs with Disjunctions

We introduce an answer-set semantics for abstract-constraint programs with disjunction in rule heads in the style of Faber, Leone, and Pfeifer (FLP). To this end, we extend the definition of an answer set for logic programs with aggregates in rule bodies using the usual FLP-reduct. Additionally, we also provide a characterisation of our semantics in terms of unfounded sets, likewise generalising the standard concept of an unfounded set. Our work is motivated by the desire to have simple and rule-based definitions of the semantics of an answer-set programming (ASP) language that is close to those implemented by the most prominent ASP solvers. The new definitions are intended as a theoretical device to allow for development methods and methodologies for ASP, e.g., debugging or testing techniques, that are general enough to work for different types of solvers. We use abstract constraints as an abstraction of literals whose truth values depend on subsets of an interpretation. This includes weight constraints, aggregates, and external atoms, which are frequently used in real-world answer-set programs. We compare the new semantics to previous semantics for abstract-constraint programs and show that they are equivalent to recent extensions of the FLP semantics to propositional and first-order theories when abstract-constraint programs are viewed as theories

Reconciling Well-Founded Semantics of DL-Programs and Aggregate Programs

Logic programs with aggregates and description logic programs (dl-programs) are two recent extensions to logic programming. In this paper, we study the relationships between these two classes of logic programs, under the well-founded semantics. The main result is that, under a satisfaction-preserving mapping from dl-atoms to aggregates, the well-founded semantics of dl-programs by Eiter et al., coincides with the well-founded semantics of aggregate programs, defined by Pelov et al. as the least fixpoint of a 3-valued immediate consequence operator under the ultimate approximating aggregate. This result enables an alternative definition of the same well-founded semantics for aggregate programs, in terms of the first principle of unfounded sets. Furthermore, the result can be applied, in a uniform manner, to define the well-founded semantics for dl-programs with aggregates, which agrees with the existing semantics when either dl-atoms or aggregates are absent

Exploiting Unfounded Sets for HEX-Program Evaluation

HEX programs extend logic programs with external computations through external atoms, whose answer sets are the minimal models of the Faber-Leone-Pfeifer-reduct. As already reasoning from Horn programs with nonmonotonic external atoms of polynomial complexity is on the second level of the polynomial hierarchy, answer set checking needs special attention; simply computing reducts and searching for smaller models does not scale well. We thus extend an approach based on unfounded sets to HEX and integrate it in a Conflict Driven Clause Learning framework for HEX program evaluation. It reduces the check to a search for unfounded sets, which is more efficiently implemented as a SAT problem. We give a basic encoding for HEX and show optimizations by additional clauses. Experiments show that the new approach significantly decreases runtime

On Elementary Loops of Logic Programs

Using the notion of an elementary loop, Gebser and Schaub (2005. Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR\u2705), 53–65) refined the theorem on loop formulas attributable to Lin and Zhao (2004) by considering loop formulas of elementary loops only. In this paper, we reformulate the definition of an elementary loop, extend it to disjunctive programs, and study several properties of elementary loops, including how maximal elementary loops are related to minimal unfounded sets. The results provide useful insights into the stable model semantics in terms of elementary loops. For a nondisjunctive program, using a graph-theoretic characterization of an elementary loop, we show that the problem of recognizing an elementary loop is tractable. On the other hand, we also show that the corresponding problem is coNP-complete for a disjunctive program. Based on the notion of an elementary loop, we present the class of Head-Elementary-loop-Free (HEF) programs, which strictly generalizes the class of Head-Cycle-Free (HCF) programs attributable to Ben-Eliyahu and Dechter (1994. Annals of Mathematics and Artificial Intelligence 12, 53–87). Like an HCF program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body

Relating Weight Constraint and Aggregate Programs: Semantics and Representation

Weight constraint and aggregate programs are among the most widely used logic programs with constraints. In this paper, we relate the semantics of these two classes of programs, namely the stable model semantics for weight constraint programs and the answer set semantics based on conditional satisfaction for aggregate programs. Both classes of programs are instances of logic programs with constraints, and in particular, the answer set semantics for aggregate programs can be applied to weight constraint programs. We show that the two semantics are closely related. First, we show that for a broad class of weight constraint programs, called strongly satisfiable programs, the two semantics coincide. When they disagree, a stable model admitted by the stable model semantics may be circularly justified. We show that the gap between the two semantics can be closed by transforming a weight constraint program to a strongly satisfiable one, so that no circular models may be generated under the current implementation of the stable model semantics. We further demonstrate the close relationship between the two semantics by formulating a transformation from weight constraint programs to logic programs with nested expressions which preserves the answer set semantics. Our study on the semantics leads to an investigation of a methodological issue, namely the possibility of compact representation of aggregate programs by weight constraint programs. We show that almost all standard aggregates can be encoded by weight constraints compactly. This makes it possible to compute the answer sets of aggregate programs using the ASP solvers for weight constraint programs. This approach is compared experimentally with the ones where aggregates are handled more explicitly, which show that the weight constraint encoding of aggregates enables a competitive approach to answer set computation for aggregate programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP), 2011. 30 page