We introduce a formal proof system based on tableau methods for analyzing computations made in Answer Set Programming (ASP). Our approach furnishes declarative and fine-grained instruments for characterizing operations as well as strategies of ASP-solvers. First, the granulation is detailed enough to capture the variety of propagation and choice operations of algorithms used for ASP; this also includes SAT-based approaches. Second, it is general enough to encompass the various strategies pursued by existing ASP-solvers. This provides us with a uniform framework for identifying and comparing fundamental properties of algorithms. Third, the approach allows us to investigate the proof complexity of algorithms for ASP, depending on choice operations. We show that exponentially different best-case computations can be obtained for different ASP-solvers. Finally, our approach is flexible enough to integrate new inference patterns, so to study their relation to existing ones. As a result, we obtain a novel approach to unfounded set handling based on loops, being applicable to non-SAT-based solvers. Furthermore, we identify backward propagation operations for unfounded sets.

We elaborate upon techniques for unfounded set computations by building upon the concept of loops. This is driven by the desire to minimize redundant computations in solvers for Answer Set Programming. We begin by investigating the relationship between unfounded sets and loops in the context of partial assignments. In particular, we show that subset-minimal unfounded sets correspond to active elementary loops. Consequentially, we provide a new loop-oriented approach along with an algorithm for computing unfounded sets. Unlike traditional techniques that compute greatest unfounded sets, we aim at computing small unfounded sets and rather let propagation (and iteration) handle greatest unfounded sets. This approach reflects the computation of unfounded sets employed in the nomore++ system. Beyond that, we provide an algorithm for identifying active elementary loops within unfounded sets. This can be used by SATbased solvers, like assat, cmodels, or pbmodels, for optimizing the elimination of invalid candidate models.

Abstract. We propose a model to manage the distributed computation of answer sets within a general framework. This design incorporates a variety of software and hardware architectures and allows its easy use with a diverse cadre of computational elements. Starting from a generic algorithmic scheme, we develop a platform for distributed answer set computation, describe its current state of implementation, and give some experimental results. 1

Abstract. We present a new answer set solver nomore++. Distinguishing features include its treatment of heads and bodies equitably as computational objects and a new hybrid lookahead. nomore++ is close to being competitive with stateof-the-art answer set solvers, as demonstrated by selected experimental results. 1

Abstract. We introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in [Gebser&Schaub, ICLP 2006], with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for clause sets, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial length proof of a family of normal logic programs {Πn} for which ASP Tableaux has exponential length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the length of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving. 1

Abstract. Computational approaches to Satisfiability Checking (SAT) and Answer Set Programming (ASP) have many aspects in common. In fact, the basic algorithms of ASP solvers are very similar to the Davis-Logemann-Loveland procedure (DLL) for SAT. The major difference lies in the inference rules, which are more complex in ASP. In this paper, we provide a generic framework, based on concepts from Constraint Processing (CSP), which allows us to view ASP inferences as forms of unit propagation. We develop declarative characterizations of ASP solvers nomore++ and smodels in terms of constraints. By putting ASP solving into a common context with SAT and CSP, we shed new light on ASP solving techniques and their relationships to neighboring fields. 1

Abstract. This paper presents the concept of parallelisation of a solver for Answer Set Programming (ASP). While there already exist some approaches to parallel ASP solving, there was a lack of a parallel version of the powerful clasp solver. We implemented a parallel version of clasp based on message-passing. Experimental results on Blue Gene P/L indicate the potential of such an approach.

This paper develops automated testing and debugging techniques for answer set solver development. We describe a flexible grammar-based black-box ASP fuzz testing tool which is able to reveal various defects such as unsound and incomplete behavior, i.e. invalid answer sets and inability to find existing solutions, in state-of-the-art answer set solver implementations. Moreover, we develop delta debugging techniques for shrinking failureinducing inputs on which solvers exhibit defective behavior. In particular, we develop a delta debugging algorithm in the context of answer set solving, and evaluate two different elimination strategies for the algorithm.

Abstract In this paper, we study the relation among Answer Set Programming (ASP) systems from a computational point of view. We consider smodels, dlv, and cmodels ASP systems based on stable model semantics, the first two being native ASP systems and the last being a SAT-based system. We first show that smodels, dlv, andcmodels explore search trees with the same branching nodes (assuming, of course, a same branching heuristic) on the class of tight logic programs. Leveraging on the fact that SAT-based systems rely on the deeply studied Davis–Logemann– Loveland (dll) algorithm, we derive new complexity results for the ASP procedures. We also show that on nontight programs the SAT-based systems are computationally different from native procedures, and the latter have computational advantages. Moreover, we show that native procedures can guarantee the “correctness ” of a reported solution when reaching the leaves of the search trees (i.e., no stability check is needed), while this is not the case for SAT-based procedures on nontight programs. A similar advantage holds for dlv in comparison with smodels if the “well-founded” operator is disabled and only Fitting’s operator is used for negative inferences. We finally study the “cost ” of achieving such advantages and comment on to what extent the results presented extend to other systems. A preliminary version of a part of the results presented in this work has been published in [25].

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