Answer Sets For Logic Programs Research Articles

Witnesses for Answer Sets of Logic Programs

In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.

Boolean Functions with Ordered Domains in Answer Set Programming

Boolean functions in Answer Set Programming have proven a useful modelling tool. They are usually specified by means of aggregates or external atoms. A crucial step in computing answer sets for logic programs containing Boolean functions is verifying whether partial interpretations satisfy a Boolean function for all possible values of its undefined atoms. In this paper, we develop a new methodology for showing when such checks can be done in deterministic polynomial time. This provides a unifying view on all currently known polynomial-time decidability results, and furthermore identifies promising new classes that go well beyond the state of the art. Our main technique consists of using an ordering on the atoms to significantly reduce the necessary number of model checks. For many standard aggregates, we show how this ordering can be automatically obtained.

Conflict-driven answer set solving: From theory to practice

We introduce an approach to computing answer sets of logic programs, based on concepts successfully applied in Satisfiability (SAT) checking. The idea is to view inferences in Answer Set Programming (ASP) as unit propagation on nogoods. This provides us with a uniform constraint-based framework capturing diverse inferences encountered in ASP solving. Moreover, our approach allows us to apply advanced solving techniques from the area of SAT. As a result, we present the first full-fledged algorithmic framework for native conflict-driven ASP solving. Our approach is implemented in the ASP solver clasp that has demonstrated its competitiveness and versatility by winning first places at various solver contests.

Abstract answer set solvers with backjumping and learning

AbstractNieuwenhuis et al. (2006. Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). Journal of the ACM 53(6), 937977 showed how to describe enhancements of the Davis–Putnam–Logemann–Loveland algorithm using transition systems, instead of pseudocode. We design a similar framework for several algorithms that generate answer sets for logic programs: smodels, smodelscc, asp-sat with Learning (cmodels), and a newly designed and implemented algorithm sup. This approach to describe answer set solvers makes it easier to prove their correctness, to compare them, and to design new systems.

Splitting Computation of Answer Set Program and Its Application on E-service

As a primary means for representing and reasoning about knowledge, Answer Set Programming (ASP) has been applying in many areas such as planning, decision making, fault diagnosing and increasingly prevalent e-service. Based on the stable model semantics of logic programming, ASP can be used to solve various combinatorial search problems by finding the answer sets of logic programs which declaratively describe the problems. It's not an easy task to compute answer sets of a logic program using Gelfond and Lifschitz's definition directly. In this paper, we show some results on characterization of answer sets of a logic program with constraints, and propose a way to split a program into several non-intersecting parts step by step, thus the computation of answer sets for every subprogram becomes relatively easy. To instantiate our splitting computation theory, an example about personalized product configuration in e-retailing is given to show the effectiveness of our method.

Experimental Analysis of Graph-based Answer Set Computation over Parallel and Distributed Architectures

This article presents a distributed version of the adjSolver algorithm for computing the answer sets of logic programs. adjSolver operates a classical branch-and-bound structure; its intrinsic parallelism is exploited to control, with a centralized architecture, the delegation of promising search subspaces to distributed handling agents. adjSolver has been implemented and tested on a Beowulf platform, using MPI message passing. The communication overhead was minimized by adopting a compact representation of the data exchanged among agents and by reusing previously-computed partial solutions.

Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.

A characterization of answer sets for logic programs

Checking if a program has an answer set, and if so, compute its answer sets are just some of the important problems in answer set logic programming. Solving these problems using Gelfond and Lifschitz’s original definition of answer sets is not an easy task. Alternative characterizations of answer sets for nested logic programs by Erdem and Lifschitz, Lee and Lifschitz, and You et al. are based on the completion semantics and various notions of tightness. However, the notion of tightness is a local notion in the sense that for different answer sets there are, in general, different level mappings capturing their tightness. This makes it hard to be used in the design of algorithms for computing answer sets. This paper proposes a characterization of answer sets based on sets of generating rules. From this characterization new algorithms are derived for computing answer sets and for performing some other reasoning tasks. As an application of the characterization a sufficient and necessary condition for the equivalence between answer set semantics and completion semantics has been proven, and a basic theorem is shown on computing answer sets for nested logic programs based on an extended notion of loop formulas. These results on tightness and loop formulas are more general than that in You and Lin’s work.

Graphs and colorings for answer set programming

We investigate the usage of rule dependency graphs and their colorings for characterizing and computing answer sets of logic programs. This approach provides us with insights into the interplay between rules when inducing answer sets. We start with different characterizations of answer sets in terms of totally colored dependency graphs that differ in graph-theoretical aspects. We then develop a series of operational characterizations of answer sets in terms of operators on partial colorings. In analogy to the notion of a derivation in proof theory, our operational characterizations are expressed as (non-deterministically formed) sequences of colorings, turning an uncolored graph into a totally colored one. In this way, we obtain an operational framework in which different combinations of operators result in different formal properties. Among others, we identify the basic strategy employed by the noMoRe system and justify its algorithmic approach. Furthermore, we distinguish operations corresponding to Fitting's operator as well as to well-founded semantics.

Applications of intuitionistic logic in Answer Set Programming

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.