Abstract
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. The semantic framework for Answer Set Programs is based on the use of stable model semantics. There are two characteristics intrinsically associated with the construction of stable models for answer set programs. Any member of an answer set is supported through facts and chains of rules and those members are in the answer set only if generated minimally in such a manner. These two characteristics, supportedness and minimality, provide the essence of stable models. Additionally, answer sets are implicitly partial and that partiality provides epistemic overtones to the interpretation of disjunctive rules and default negation. This paper is intended to shed light on these characteristics by defining a semantic framework for answer set programming based on an extended first-order Kleene logic with weak and strong negation. Additionally, a definition of strongly supported models is introduced, separate from the minimality assumption explicit in stable models. This is used to both clarify and generate alternative semantic interpretations for answer set programs with disjunctive rules in addition to answer set programs with constraint rules. An algorithm is provided for computing supported models and comparative complexity results between strongly supported and stable model generation are provided.
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 Project), the EU FP7 project SHERPA (grant agreement 600958), and Vinnova NFFP6 Project 2013-01206.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alviano, M., Faber, W., Leone, N., Perri, S., Pfeifer, G., Terracina, G.: The disjunctive datalog system DLV. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2010. LNCS, vol. 6702, pp. 282–301. Springer, Heidelberg (2011)
Baral, C.: Knowledge Representation, Reasoning, and Declarative Problem Solving. Cambridge University Press (2003)
Bonatti, P., Calimeri, F., Leone, N., Ricca, F.: Answer set programming. In: Dovier, A., Pontelli, E. (eds.) GULP. LNCS, vol. 6125, pp. 159–182. Springer, Heidelberg (2010)
Brewka, G.: Preferences, contexts and answer sets. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 22–22. Springer, Heidelberg (2007)
Brewka, G., Eiter, T., Truszczynski, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011)
Brewka, G., Niemelä, I., Truszczynski, M.: Answer set optimization. In: Gottlob, G., Walsh, T. (eds.) Proc. 18th IJCAI, pp. 867–872. Morgan Kaufmann (2003)
Cabalar, P., Pearce, D., Valverde, A.: A revised concept of safety for general answer set programs. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS, vol. 5753, pp. 58–70. Springer, Heidelberg (2009)
Denecker, M., Marek, V., Truszczynski, M.: Stable operators, well-founded fixpoints and applications in nnonmonotonic reasoning. In: Minker, J. (ed.) Logic-based Artificial Intelligence, pp. 127–144. Kluwer Academic Pub. (2000)
Eiter, T., Gottlob, G.: Complexity results for disjunctive logic programming and application to nonmonotonic logics. In: Miller, D. (ed.) Proceedings of the 1993 International Symposium on Logic Programming, pp. 266–278 (1993)
Fages, F.: A new fixpoint sematics for general logic programs compared with the well-founded and stable model semantics. New Generation Computing 9, 425–443 (1991)
Fages, F.: Consistency of Clark’s completion and existence of stable models. Methods of Logic in Computer Science 1, 51–60 (1994)
Fenstad, J.E.: Situations, Language and Logic. D. Reidel Publishing Company (1987)
Ferraris, P., Lifschitz, V.: On the minimality of stable models. In: Balduccini, M., Son, T.C. (eds.) Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning. LNCS, vol. 6565, pp. 64–73. Springer, Heidelberg (2011)
Fitting, M.: A Kripke-Kleene semantics for logic programs. J. Logic Programming 2(4), 295–312 (1985)
Fitting, M.: The family of stable models. J. Logic Programming 17(2/3&4), 197–225 (1993)
Gelfond, M., Kahl, Y.: Knowledge Representation, Reasoning, and the Design of Intelligent Agents - The Answer-Set Programming Approach. Cambridge University Press (2014)
Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R., Bowen, K. (eds.) Proc. of Int’l Logic Programming, pp. 1070–1080. MIT Press (1988)
Kleene, S.C.: On a notation for ordinal numbers. Symbolic Logic 3, 150–155 (1938)
Lifschitz, V.: Thirteen definitions of a stable model. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. LNCS, vol. 6300, pp. 488–503. Springer, Heidelberg (2010)
Lonc, Z., Truszczynski, M.: Computing minimal models, stable models and answer sets. TPLP 6(4), 395–449 (2006)
Pearce, D.: Equilibrium logic. Annals of Mathematics and AI 47(1-2), 3–41 (2006)
Przymusinski, T.: Stable semantics for disjunctive programs. New Generation Comput. 9(3/4), 401–424 (1991)
Shepherdson, J.C.: A sound and complete semantics for a version of negation as failure. Theoretical Computer Science 65(3), 343–371 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Doherty, P., Szałas, A. (2015). Stability, Supportedness, Minimality and Kleene Answer Set Programs. In: Eiter, T., Strass, H., Truszczyński, M., Woltran, S. (eds) Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation. Lecture Notes in Computer Science(), vol 9060. Springer, Cham. https://doi.org/10.1007/978-3-319-14726-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-14726-0_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14725-3
Online ISBN: 978-3-319-14726-0
eBook Packages: Computer ScienceComputer Science (R0)