Complexity results for answer set programming with bounded predicate arities and implications

Eiter, Thomas; Faber, Wolfgang; Fink, Michael; Woltran, Stefan

doi:10.1007/s10472-008-9086-5

Cited by 44 publications

(56 citation statements)

References 58 publications

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“…These rewritings make heavy use of large rules, which can be used to encode NP-checks. For disjunctive ASP, if the predicate arity is bounded, it becomes possible to solve problems of up to Σ 3 P -hardness with this approach, since the consistency problem for disjunctive ASP with bounded predicate arity is itself Σ 3 P -complete (Eiter et al 2007). Because existing grounders are slow to ground large rules, we then present an extended rule decomposition algorithm to make our encodings solvable in practice.…”

Section: Resultsmentioning

confidence: 99%

“…It can be checked that Π Φ has an answer set iff Φ is unsatisfiable. A different reduction algorithm for QBFs of the form (2) has been proposed in (Eiter et al 2007), which in fact makes use of the paradigm proposed in Section 3. Let Φ be a QBF as before.…”

Section: Rewriting Qsat Into Logic Programsmentioning

confidence: 99%

“…Such rules can, in general, encode NP-hard checks, even when predicate arities are bounded (this follows immediately from the NP-hardness of answering conjunctive queries over databases (Chandra and Merlin 1977)). As shown by Eiter et al (2007), the combined complexity of solving arbitrary ASP programs of bounded arity increases to Σ 3 P , but drops back to Σ 2 P for normal (i.e. non-disjunctive) programs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The power of non-ground rules in Answer Set Programming

Bichler¹,

Morak²,

Woltran³

2016

Theory and Practice of Logic Programming

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Answer set programming (ASP) is a well-established logic programming language that offers an intuitive, declarative syntax for problem solving. In its traditional application, a fixed ASP program for a given problem is designed and the actual instance of the problem is fed into the program as a set of facts This approach typically results in programs with comparably short and simple rules However, as is known from complexity analysis, such an approach limits the expressive power of ASP; in fact, an entire NP-check can be encoded into a single large rule body of bounded arity that performs both a guess and a check within the same rule Here, we propose a novel paradigm for encoding hard problems in ASP by making explicit use of large rules which depend on the actual instance of the problem. We illustrate how this new encoding paradigm can be used, providing examples of problems from the first, second, and even third level of the polynomial hierarchy As state-of-the-art solvers are tuned towards short rules, rule decomposition is a key technique in the practical realization of our approach We also provide some preliminary benchmarks which indicate that giving up the convenient way of specifying a fixed program can lead to a significant speed-up. This paper is under consideration for acceptance in TPLP.

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Section: Resultsmentioning

confidence: 99%

Section: Rewriting Qsat Into Logic Programsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The power of non-ground rules in Answer Set Programming

Bichler¹,

Morak²,

Woltran³

2016

Theory and Practice of Logic Programming

Self Cite

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show abstract

“…Under BPA, the model guess has polynomial size, and thus the whole countermodel check is feasible in polynomial space; PSpace-hardness is inherited from evaluation of a given FOL formula over a finite structure. For the ASP X languages, see [4,5].…”

Section: Computational Complexitymentioning

confidence: 99%

“…Assuming that I 2n+1 is a no-instance, the specifications will correspond, i.e., reasoning problem for ASP ¬ resp. ASP ∨,¬ programs: given a set of facts F , a program P and a fact α, does there exist some F ⊆ F such that P ∪ F |= α (where |= is cautious consequence); this problem is, as shown with slight extensions of the respective proofs for cautious consequence [4,5], complete for NP O . Finally, the results for REPAIR remain unchanged if only strict repairs are considered as CORR has lower complexity.…”

Section: Computational Complexitymentioning

confidence: 99%

Inconsistency Management for Traffic Regulations: Formalization and Complexity Results

Beck

Eiter

Krennwallner

2012

Logics in Artificial Intelligence

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Abstract. Smart Cities is a vision driven by the availability of governmental data that fosters many challenging applications. One of them is the management of inconsistent traffic regulations, i.e., the handling of inconsistent traffic signs and measures in urban areas such as wrong sign posting, or errors in data acquisition in traffic sign administration software. We investigate such inconsistent traffic scenarios and formally model traffic regulations using a logic-based approach for traffic signs and measures, and logical theories describe emerging conflicts on a graph-based street model. Founded on this model, we consider major reasoning tasks including consistency testing, diagnosis, and repair, and we analyze their computational complexity for different logical representation formalisms. Our results provide a basis for an ongoing implementation of the approach.

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Abstraction for Non-ground Answer Set Programs

Saribatur

Schüller

Eiter

2019

Logics in Artificial Intelligence

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ion is an important technique utilized by humans in model building and problem solving, in order to figure out key elements and relevant details of a world of interest. This naturally has led to investigations of using abstraction in AI and Computer Science to simplify problems, especially in the design of intelligent agents and automated problem solving. By omitting details, scenarios are reduced to ones that are easier to deal with and to understand, where further details are added back only when they matter. Despite the fact that abstraction is a powerful technique, it has not been considered much in the context of nonmonotonic knowledge representation and reasoning, and specifically not in Answer Set Programming (ASP), apart from some related simplification methods. In this work, we introduce a notion for abstracting from the domain of an ASP program such that the domain size shrinks while the set of answer sets (i.e., models) of the program is over-approximated. To achieve the latter, the program is transformed into an abstract program over the abstract domain while preserving the structure of the rules. We show in elaboration how this can be also achieved for single or multiple sub-domains (sorts) of a domain, and in case of structured domains like grid environments in which structure should be preserved. Furthermore, we introduce an abstraction-&-refinement methodology that makes it possible to start with an initial abstraction and to achieve automatically an abstraction with an associated abstract answer set that matches an answer set of the original program, provided that the program is satisfiable. Experiments based on prototypical implementations reveal the potential of the approach for problem analysis, by its ability to focus on the parts of the program that cause unsatisfiability and by achieving concrete abstract answer sets that merely reflect relevant details. This makes domain abstraction an interesting topic of research whose further use in important areas like Explainable AI remains to be explored.

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Complexity results for answer set programming with bounded predicate arities and implications

Cited by 44 publications

References 58 publications

The power of non-ground rules in Answer Set Programming

The power of non-ground rules in Answer Set Programming

Inconsistency Management for Traffic Regulations: Formalization and Complexity Results

Abstraction for Non-ground Answer Set Programs

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