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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In \(\mathsf {ASP}\), a literal is a propositional variable p or a strongly-negated propositional variable \({\sim }p\).
- 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.
We use ‘|’ (a bit informally) to separate the rules of a program in this paper.
- 4.
In \(\mathsf {ASP}\), constraints show their effect on programs by eliminating or keeping their answer sets.
- 5.
For the truth conditions of \(\mathsf {EHT}\), you can refer to [4].
- 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.
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.
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.
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.
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 \(\varPi \).
References
Baral, C., Gelfond, M.: Logic programming and knowledge representation. J. Log. Program. 19, 73–148 (1994)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2001)
Cabalar, P., Fandinno, J., Fariñas del Cerro, L.: Splitting epistemic logic programs. In: Proceedings of the 17th International Workshop on Nonmonotonic Reasoning, NMR 2018, Tempe, Arizona, USA, 27–29 October 2018 (2018)
Fariñas del Cerro, L., Herzig, A., Su, E.I.: Epistemic equilibrium logic. In: Yang, Q., Wooldridge, M. (eds.) Proceedings of the 24th International Joint Conference on Artificial Intelligence, pp. 2964–2970. AAAI Press (2015). http://ijcai.org/papers15/Abstracts/IJCAI15-419.html
Chen, J.: The generalized logic of only knowing (GOL) that covers the notion of epistemic specifications. J. Log. Comput. 7(2), 159–174 (1997)
Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases, Symposium on Logic and Data Bases, Centre d’études et de recherches de Toulouse, France. Advances in Data Base Theory, pp. 293–322. Plemum Press, New York (1977)
Gabbay, D.M.: What is negation as failure? In: Artikis, A., Craven, R., Kesim Çiçekli, N., Sadighi, B., Stathis, K. (eds.) Logic Programs, Norms and Action. LNCS (LNAI), vol. 7360, pp. 52–78. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29414-3_5
Gelfond, M.: Strong introspection. In: Dean, T.L., McKeown, K. (eds.) Proceedings of the 9th National Conference on Artificial Intelligence, Anaheim, CA, USA, 14–19 July 1991, vol, 1, pp. 386–391. AAAI Press/The MIT Press (1991)
Gelfond, M.: Logic programming and reasoning with incomplete information. Ann. Math. Artif. Intell. 12(1–2), 89–116 (1994)
Gelfond, M.: Answer sets. In: Handbook of Knowledge Representation, vol. 1, p. 285 (2008)
Gelfond, M.: New semantics for epistemic specifications. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS (LNAI), vol. 6645, pp. 260–265. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20895-9_29
Gelfond, M.: New definition of epistemic specifications. In: KR Seminar. Texas Tech University, 28 April 2011. (talk)
Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R.A., Bowen, K.A. (eds.) Logic Programming, Proceedings of the Fifth International Conference and Symposium, Seattle, Washington, USA, 15–19 August 1988, vol. 2, pp. 1070–1080. MIT Press (1988)
Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Gener. Comput. 9(3/4), 365–386 (1991)
Heyting, A.: Die formalen Regeln der intuitionistischen Logik. Sitzungsber. Preuss. Akad. Wiss. 42–71, 158–169 (1930)
Inoue, K., Sakama, C.: Negation as failure in the head. J. Log. Program. 35(1), 39–78 (1998)
Kahl, P., Watson, R., Balai, E., Gelfond, M., Zhang, Y.: The language of epistemic specifications (refined) including a prototype solver. J. Log. Comput. (2015)
Kahl, P.T.: Refining the semantics for epistemic logic programs. Ph.D. thesis, Texas Tech University, Department of Computer Science, Lubblock, TX, USA, May 2014
Kahl, P.T., Leclerc, A.P.: Epistemic logic programs with world view constraints. In: Palù, A.D., Tarau, P., Saeedloei, N., Fodor, P. (eds.) Technical Communications of the 34th International Conference on Logic Programming, ICLP 2018, Oxford, United Kingdom, 14–17 July 2018. OpenAccess Series in Informatics OASICS, vol. 64, pp. 1:1–1:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018). https://doi.org/10.4230/OASIcs.ICLP.2018.1
Kahl, P.T., Leclerc, A.P., Son, T.C.: A parallel memory-efficient epistemic logic program solver: harder, better, faster. CoRR abs/1608.06910 (2016). http://arxiv.org/abs/1608.06910
Kowalski, R.A.: Logic programming. In: Siekmann, J.H. (ed.) Computational Logic, Handbook of the History of Logic, vol. 9, pp. 523–569. Elsevier (2014). https://doi.org/10.1016/B978-0-444-51624-4.50012-5
Leclerc, A.P., Kahl, P.T.: A survey of advances in epistemic logic program solvers. abs/1809.07141 (2018). http://arxiv.org/abs/1809.07141. (Also in the Proceedings of the 11th International Workshop on Answer Set Programming and other Computer Paradigms, ASPOCP 2018, Oxford, UK, 18 July 2018)
Lifschitz, V., Tang, L.R., Turner, H.: Nested expressions in logic programs. Ann. Math. Artif. Intell. 25(3–4), 369–389 (1999)
Marek, V.W., Truszczynski, M.: Stable models and an alternative logic programming paradigm. CoRR cs.LO/9809032 (1998). http://arxiv.org/abs/cs.LO/9809032
Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Ann. Math. Artif. Intell. 25(3–4), 241–273 (1999). https://doi.org/10.1023/A:1018930122475
Pearce, D.: A new logical characterisation of stable models and answer sets. In: Dix, J., Pereira, L.M., Przymusinski, T.C. (eds.) NMELP 1996. LNCS, vol. 1216, pp. 57–70. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0023801
Pearce, D.: Equilibrium logic. Ann. Math. Artif. Intell. 47(1–2), 3–41 (2006)
Przymusinski, T.C.: On the relationship between logic programming and nonmonotonic reasoning. In: Shrobe, H.E., Mitchell, T.M., Smith, R.G. (eds.) Proceedings of the 7th National Conference on Artificial Intelligence, St. Paul, MN, USA, 21–26 August 1988, pp. 444–448. AAAI Press/The MIT Press (1988). http://www.aaai.org/Library/AAAI/1988/aaai88-078.php
Shen, Y., Eiter, T.: Evaluating epistemic negation in answer set programming. Artif. Intell. 237, 115–135 (2016). https://doi.org/10.1016/j.artint.2016.04.004
Shen, Y., Eiter, T.: Evaluating epistemic negation in answer set programming (extended abstract). In: Sierra, C. (ed.) Proceedings of the 26th International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, 19–25 August 2017, pp. 5060–5064. ijcai.org (2017). https://doi.org/10.24963/ijcai.2017/722
Son, T.C., Le, T., Kahl, P.T., Leclerc, A.P.: On computing world views of epistemic logic programs. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, 19–25 August 2017, pp. 1269–1275 (2017). https://doi.org/10.24963/ijcai.2017/176
Stalnaker, R.: What is a nonmonotonic consequence relation? Fundam. Inform. 21(1/2), 7–21 (1994)
Su, E.I.: Extensions of equilibrium logic by modal concepts. (Extensions de la logique d’équilibre par des concepts modaux). Ph.D. thesis, Institut de Recherche en Informatique de Toulouse, France (2015). https://tel.archives-ouvertes.fr/tel-01636791
Truszczyński, M.: Revisiting epistemic specifications. In: Balduccini, M., Son, T.C. (eds.) Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning. LNCS (LNAI), vol. 6565, pp. 315–333. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20832-4_20
Wang, K., Zhang, Y.: Nested epistemic logic programs. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 279–290. Springer, Heidelberg (2005). https://doi.org/10.1007/11546207_22
Watson, R.: A splitting set theorem for epistemic specifications. CoRR cs.AI/0003038 (2000). http://arxiv.org/abs/cs.AI/0003038
Zhang, Y.: Updating epistemic logic programs. J. Log. Comput. 19(2), 405–423 (2009). https://doi.org/10.1093/logcom/exn100
Zhang, Y., Zhang, Y.: Epistemic specifications and conformant planning. In: Barták, R., McCluskey, T.L., Pontelli, E. (eds.) Proceedings of the 2017 Workshop on Knowledge-Based Techniques for Problem Solving and Reasoning (KnowProS 2017) (2017)
Zhang, Z.: Introspecting preferences in answer set programming. In: Palù, A.D., Tarau, P., Saeedloei, N., Fodor, P. (eds.) Technical Communications of the 34th International Conference on Logic Programming, ICLP 2018, 14–17 July 2018, Oxford, United Kingdom. OASICS, vol. 64, pp. 3:1–3:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018). https://doi.org/10.4230/OASIcs.ICLP.2018.3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-19570-0_40
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19569-4
Online ISBN: 978-3-030-19570-0
eBook Packages: Computer ScienceComputer Science (R0)