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An Entailment Procedure for Kleene Answer Set Programs

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Multi-disciplinary Trends in Artificial Intelligence (MIWAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10053))

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Abstract

Classical Answer Set Programming is a widely known knowledge representation framework based on the logic programming paradigm that has been extensively studied in the past decades. Semantic theories for classical answer sets are implicitly three-valued in nature, yet with few exceptions, computing classical answer sets is based on translations into classical logic and the use of SAT solving techniques. In this paper, we introduce a variation of Kleene three-valued logic with strong connectives, \({\text {R}_3}\), and then provide a sound and complete proof procedure for \({\text {R}_3}\) based on the use of signed tableaux. We then define a restriction on the syntax of \({\text {R}_3}\) to characterize Kleene ASPs. Strongly-supported models, which are a subset of \({\text {R}_3}\) models are then defined to characterize the semantics of Kleene ASPs. A filtering technique on tableaux for \({\text {R}_3}\) is then introduced which provides a sound and complete tableau-based proof technique for Kleene ASPs. We then show a translation and semantic correspondence between Classical ASPs and Kleene ASPs, where answer sets for normal classical ASPs are equivalent to strongly-supported models. This implies that the proof technique introduced can be used for classical normal ASPs as well as Kleene ASPs. The relation between non-normal classical and Kleene ASPs is also considered.

This work is partially supported by the Swedish Research Council (VR) Linnaeus Center CADICS, the ELLIIT network organization for Information and Communication Technology, the Swedish Foundation for Strategic Research (CUAS, SymbiCloud Project), and Vinnova NFFP6 Project 2013-01206.

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Notes

  1. 1.

    For clarity, we restrict the results to the propositional case, but they are easily generalized to the first-order case with certain restrictions.

  2. 2.

    Strong negation, conjunction and disjunction have also been used in [26].

  3. 3.

    The implication connective, \(\rightarrow _s\,\), is in fact equivalent to that used in [35].

  4. 4.

    This signed approach to semantic tableaux for multi-valued logic was discovered independently, by [6, 18] who generalized the technique for a larger class of multi-valued logics. The approach of [6] was used as a basis for [28].

  5. 5.

    In understanding the theorems, recall that the syntax for ASP\(^{K}\) programs and classical ASP programs is identical.

  6. 6.

    Note that both \(\{p\}\) and \(\{\lnot q, p\}\) are \({\text {R}_3}\) models for the considered formula, so the fact that q is entailed by \(\varPi \) cannot be proved using only rules provided in Sect. 3.1.

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Authors and Affiliations

  1. Department of Computer and Information Science, Linköping University, SE-581 83, Linköping, Sweden

    Patrick Doherty & Andrzej Szałas

  2. Institute of Informatics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Szałas

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  1. Patrick Doherty
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Correspondence to Andrzej Szałas .

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Editors and Affiliations

  1. Mahasarakham University , Maha Sarakham, Thailand

    Chattrakul Sombattheera

  2. Harz University of Applied Sciences , Wernigerode, Germany

    Frieder Stolzenburg

  3. University of Science and Technology , Hong Kong, China

    Fangzhen Lin

  4. Macquarie University , North Ryde, New South Wales, Australia

    Abhaya Nayak

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Doherty, P., Szałas, A. (2016). An Entailment Procedure for Kleene Answer Set Programs. In: Sombattheera, C., Stolzenburg, F., Lin, F., Nayak, A. (eds) Multi-disciplinary Trends in Artificial Intelligence. MIWAI 2016. Lecture Notes in Computer Science(), vol 10053. Springer, Cham. https://doi.org/10.1007/978-3-319-49397-8_3

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