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Answer Set Programming from a Logical Point of View

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Abstract

In this paper we provide an introductory explanation of the underlying semantics of answer set programming in terms of equilibrium logic. Rather than a thorough formal presentation of this formalism and its properties, we emphasize the intuitive meaning of its main logical definitions, explaining their effect on some example programs. We also overview some of the main extensions and relations to other logical approaches.

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Notes

  1. Notice that \(\lnot { fire}\) is, read as “we have no evidence on fire,” is either certain (when indeed \({ fire}\) is false) or false (when some evidence exists, i.e. \({ fire}\) is certain or weakly true).

  2. These sets are informally known as “here” and “there”. The reader familar with possible worlds semantics may think of them as the atoms verified at two worlds, ‘here’ and ‘there’.

  3. See [6], and also [7] for more detailed information and references.

  4. Using HT-logic we can only test the equivalence of ground rules and programs. In practice it makes sense to apply a first-order, quantified version of HT that captures the strong equivalence of programs with variables. For details, see [9, 10]. The complexity of strong equivalence for non-ground programs is analysed in [11].

  5. It is slightly unfortunate that [19] called it “classical” negation.

  6. Notice that the meaning of such a relation is still co-determined by the two components \(\mathcal {T}\) and \(\Pi\) because normally not all models of \(\mathcal {T}\) will be able to be enriched into equilibrium models of \(\mathcal {T} \cup \Pi\).

  7. The main references are as follows. For the complexity of satisfiability in many-valued logics such as HT, see [38]. For the complexity of reasoning tasks associated with disjunctive logic programs, see [39]. For the strong equivalence of logic programs, see [37] and also independently some results of [40] and [41]. For the full details of the reduction method sketched here, see [37, 42].

  8. Ie. logics captured semantically by possible worlds frames with a reflexive accessibility relation.

  9. See [48].

  10. Another modal embedding of HT can be found in [54].

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Authors and Affiliations

  1. University of A Coruña, A Coruña, Spain

    Pedro Cabalar

  2. Universidad Politécnica de Madrid, Madrid, Spain

    David Pearce

  3. Universidad de Málaga, Málaga, Spain

    Agustín Valverde

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  1. Pedro Cabalar
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  2. David Pearce
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Correspondence to Agustín Valverde.

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Partial support from Mineco (TIN2017-84453-P and TIN2015-70266-C2), Xunta de Galicia (GPC ED431B 2016/035 and 2016-2019 ED431G/01, CITIC), European Regional Development Fund (ERDF), UPM RP151046021.

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Cabalar, P., Pearce, D. & Valverde, A. Answer Set Programming from a Logical Point of View. Künstl Intell 32, 109–118 (2018). https://doi.org/10.1007/s13218-018-0547-7

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