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Epistemic Answer Set Programming

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Logics in Artificial Intelligence (JELIA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11468))

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Abstract

This paper introduces a new epistemic extension of answer set programming (\(\mathsf {ASP}\)) called epistemic ASP (\(\mathsf {E\text {-}\mathsf {ASP}}\)). Then, it compares \(\mathsf {E\text {-}\mathsf {ASP}}\) with existing approaches, showing the advantages and the novelties of the new semantics and discusses which formalisms provide more intuitive results: compared to Gelfond’s epistemic specifications (\(\mathsf {ES}\)), \(\mathsf {E\text {-}\mathsf {ASP}}\) defines a simpler, but sufficiently strong language. Its epistemic view semantics is a natural and more standard generalisation of \(\mathsf {ASP}\)’s original answer set semantics, so it allows for \(\mathsf {ASP}\)’s previous language extensions. Moreover, compared to all semantics proposals in the literature, epistemic view semantics facilitates understanding of the intuitive meaning of epistemic logic programs and solves unintended results discussed in the literature, especially for epistemic logic programs including constraints.

I want to thank Andreas Herzig, Luis Fariñas del Cerro, Michael Gelfond, Patrick Thor Kahl, Thomas Eiter, Yi-Dong Shen, Pedro Cabalar, and Jorge Fandinno for their research related to this paper and the anonymous reviewers for their valued comments on the drafts of this work.

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Notes

  1. 1.

    In \(\mathsf {ASP}\), a literal is a propositional variable p or a strongly-negated propositional variable \({\sim }p\).

  2. 2.

    The use of variables in \(\mathsf {ES}\) is understood as abbreviations for the collection of their ground instances. Thus, for simplicity, we restrict here the language \( \mathcal{L}_{\scriptscriptstyle {\mathsf {ES}}} \) to the propositional case.

  3. 3.

    We use ‘|’ (a bit informally) to separate the rules of a program in this paper.

  4. 4.

    In \(\mathsf {ASP}\), constraints show their effect on programs by eliminating or keeping their answer sets.

  5. 5.

    For the truth conditions of \(\mathsf {EHT}\), you can refer to [4].

  6. 6.

    However, as in Kahl’s approach, adding a constraint into a program may also give here unexpected results. For instance, take the eligibility program \(\varPi _G\) and a constraint \(\leftarrow i\). Then, the resulting \(\mathsf {EHT}\) theory \(\varPi _G ^* \cup \{ \lnot i \} \) has a unique AEEM \(\mathcal {T}_2 = \big \{ \{ h,e \} \big \} \), instead of having no AEEM.

  7. 7.

    However, this formalisation was then discovered to cause problems [12]. Consider \(\varPi =\big \{p \, \texttt {or} \,q ~|~{\sim }p \leftarrow \texttt {not} \,\!p\). Then, \(\texttt {AS}(\varPi )= \big \{ \{ p \} , \{ q,{\sim }p \} \big \} \), and it answers the query \({\sim }p?\) unknown (as it does not appear in both answer sets) while p is undetermined. This result is unintended.

  8. 8.

    The satisfaction relation \(\models _{\scriptscriptstyle {\mathsf {E\text {-}\mathsf {ASP}}}}\) of \(\mathsf {E\text {-}\mathsf {ASP}}\) is the same as the relation \(\models _{\scriptscriptstyle {\mathsf {ES}}}\) (see Sect. 2.2).

  9. 9.

    Singleton minimal models of a program \(\varPi \) are sometimes source of a problem in capturing intuitive results: for a singleton set, \( \mathsf {K} \, \! p\) and p are of no difference, as well as \( \texttt {not} \,\! \mathsf {K} \, \!p\) and \( \texttt {not} \,\! p\). Thus, an \(\mathsf {E\text {-}\mathsf {ASP}}\) program performs like an \(\mathsf {ASP}\) program, and we may obtain “unjustified” minimal models. For instance, in \(\underline{\varSigma }\), if we replace \( \texttt {not} \,\! \mathsf {K} \, \) with \( \texttt {not} \,\!\), the resulting \(\mathsf {ASP}\) program has the answer sets \( \{ p \} \) and \( \{ {\sim }q, r \} \). Note that \( \{ \{ p \} \} \) and \( \{ \{ {\sim }q, r \} \} \) are minimal models of \(\underline{\varSigma }\). We get a similar result if we change \( \mathsf {K} \, p\) with p in \(\underline{\varGamma }\). Thus, singleton sets do not allow us to quantify over all possible beliefs. In order to overcome this obstacle, we need to check the behaviour of singletons in an interplay with other minimal models by using an ordering.

  10. 10.

    Fact [in \(\mathsf {ASP}\)]: if \(A \in \texttt {AS}(\varPi )\), then every \(l \in A\) belongs to the head of one of the rules in \(\varPi \).

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  1. Department of Computer Engineering, Istinye University, Istanbul, Turkey

    Ezgi Iraz Su

  2. IRIT, University of Toulouse, Toulouse, France

    Ezgi Iraz Su

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  1. University of Calabria, Rende, Italy

    Francesco Calimeri

  2. University of Calabria, Rende, Italy

    Nicola Leone

  3. Mathematics and Computer Science, University of Calabria, Rende, Italy

    Marco Manna

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Su, E.I. (2019). Epistemic Answer Set Programming. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_40

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