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Characterizing and extending answer set semantics using possibility theory

Published online by Cambridge University Press:  10 January 2014

KIM BAUTERS ,
STEVEN SCHOCKAERT ,
MARTINE DE COCK  and
DIRK VERMEIR
KIM BAUTERS
Affiliation:
Department of Applied Mathematics, Computer Science and Statistics, Universiteit Gent Krijgslaan 281 (WE02), 9000 Gent, Belgium (e-mail: kim.bauters@gmail.com)
STEVEN SCHOCKAERT
Affiliation:
School of Computer Science & Informatics, Cardiff University 5 The Parade, Cardiff CF24 3AA, UK (e-mail: s.schockaert@cs.cardiff.ac.uk)
MARTINE DE COCK
Affiliation:
Department of Applied Mathematics, Computer Science and Statistics, Universiteit Gent Krijgslaan 281 (WE02), 9000 Gent, Belgium (e-mail: martine.decock@ugent.be)
DIRK VERMEIR
Affiliation:
Department of Computer Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium (e-mail: dirk.vermeir@vub.ac.be)
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Abstract

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Answer Set Programming (ASP) is a popular framework for modelling combinatorial problems. However, ASP cannot be used easily for reasoning about uncertain information. Possibilistic ASP (PASP) is an extension of ASP that combines possibilistic logic and ASP. In PASP a weight is associated with each rule, whereas this weight is interpreted as the certainty with which the conclusion can be established when the body is known to hold. As such, it allows us to model and reason about uncertain information in an intuitive way. In this paper we present new semantics for PASP in which rules are interpreted as constraints on possibility distributions. Special models of these constraints are then identified as possibilistic answer sets. In addition, since ASP is a special case of PASP in which all the rules are entirely certain, we obtain a new characterization of ASP in terms of constraints on possibility distributions. This allows us to uncover a new form of disjunction, called weak disjunction, that has not been previously considered in the literature. In addition to introducing and motivating the semantics of weak disjunction, we also pinpoint its computational complexity. In particular, while the complexity of most reasoning tasks coincides with standard disjunctive ASP, we find that brave reasoning for programs with weak disjunctions is easier.

Keywords

logic programminganswer set programmingpossibility theory
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 15 , Issue 1 , January 2015 , pp. 79 - 116
Copyright
Copyright © Cambridge University Press 2014 

References

Banerjee, M. and Dubois, D. 2009. A simple modal logic for reasoning about revealed beliefs. In Proceedings of the 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU'09), 805–816.Google Scholar
Baral, C. 2003. Knowledge, Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge, UK.Google Scholar
Bauters, K., Schockaert, S., De Cock, M. and Vermeir, D. 2010. Possibilistic answer set programming revisited. In Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence (UAI).Google Scholar
Bauters, K., Schockaert, S., De Cock, M. and Vermeir, D. 2011. Weak and strong disjunction in possibilistic ASP. In Proceedings of the 11th International Conference on Scalable Uncertainty Management (SUM).Google Scholar
Benferhat, S., Dubois, D., Garcia, L. and Prade, H. 2002. On the transformation between possibilistic logic bases and possibilistic causal networks. International Journal of Approximate Reasoning 29, 2, 135–173.Google Scholar
Benferhat, S., Dubois, D. and Prade, H. 1992. Representing default rules in possibilistic logic. In Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR), 673–684.Google Scholar
Benferhat, S., Dubois, D. and Prade, H. 1997. Nonmonotonic reasoning, conditional objects and possibility theory. Artificial Intelligence 92, 1–2, 259–276.Google Scholar
Brewka, G. 2002. Logic programming with ordered disjunction. In Proceedings of the 18th National Conference on Artificial Intelligence (AAAI), 100–105.Google Scholar
Buccafurri, F., Faber, W. and Leone, N. 2002. Disjunctive logic programs with inheritance. Theory and Practice of Logic Programming 2, 3, 293–321.Google Scholar
Buccafurri, F., Leone, N. and Rullo, P. 2000. Enhancing disjunctive datalog by constraints. IEEE Transactions on Knowledge and Data Engineering 12, 5, 845–860.Google Scholar
Cai, J.-Y., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K. and Wechsung, G. 1988. The Boolean hierarchy I: Structural properties. SIAM Journal on Computing 17, 6, 1232–1252.Google Scholar
Damásio, C. V. and Pereira, L. M. 2001. Monotonic and residuated logic programs. In Proceedings of the 6th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU '01), 748–759.Google Scholar
Dubois, D., Lang, J. and Prade, H. 1991. Towards possibilistic logic programming. In Proceedings of the 8th International Conference on Logic Programming (ICLP), 581–595.Google Scholar
Dubois, D., Lang, J. and Prade, H. 1994. Possibilistic logic. In Handbook of Logic for Artificial Intelligence and Logic Programming, Gabbay, D. M., Hogger, C. J. and Robinson, J. A., Eds. Vol. 3. Oxford University Press, Oxford, UK, 439–513.Google Scholar
Dubois, D. and Prade, H. 1991. Epistemic entrenchment and possibilistic logic. Artificial Intelligence 50, 2, 223–239.Google Scholar
Dubois, D. and Prade, H. 1997. A synthetic view of belief revision with uncertain inputs in the framework of possibility theory. International Journal of Approximate Reasoning 17, 2–3, 295–324.Google Scholar
Dubois, D., Prade, H. and Schockaert, S. 2012. Stable models in generalized possibilistic logic. In Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR'12), 519–529.Google Scholar
Eiter, T. and Gottlob, G. 1993. Complexity results for disjunctive logic programming and application to non-monotonic logics. In Proceedings of the 1993 International Logic Programming Symposium (ILPS), 266–278.Google Scholar
Faber, W. and Woltran, S. 2009. Manifold answer-set programs for meta-reasoning. In LPNMR, Lecture Notes in Computer Science, Vol. 5753, Springer, New York, NY, 115–128.Google Scholar
Gelfond, M. 1987. On stratified autoepistemic theories. In Proceedings of the 6th National Conference on Artificial Intelligence (AAAI), 207–211.Google Scholar
Gelfond, M. 1991. Strong introspection. In Proceedings of the 9th National Conference on Artificial Intelligence (AAAI'91), 386–391.Google Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–385.Google Scholar
Gelfond, M. and Lifzchitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the 5th International Conference on Logic Programming (ICLP), 1081–1086.Google Scholar
Huth, M. and Ryan, M. 2004. Logic in Computer Science: Modelling and Reasoning About Systems. Cambridge University Press, Cambridge, UK.Google Scholar
Jaynes, E. 2003. Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, UK.Google Scholar
Lifschitz, V. 2010. Thirteen definitions of a stable model. In Fields of Logic and Computation, Lecture Notes in Computer Science, Vol. 6300, 488–503.Google Scholar
Lifschitz, V. and Schwarz, G. 1993. Extended logic programs as autoepistemic theories. In Proceedings of the 2nd International Workshop on Logic Programming and Nonmonotonic Reasoning, 101–114.Google Scholar
Loyer, Y. and Straccia, U. 2006. Epistemic foundation of stable model semantics. Theory and Practice of Logic Programming 6, 4, 355–393.Google Scholar
Marek, W. and Truszczyński, M. 1991. Autoepistemic logic. Journal of the ACM 38, 587–618.Google Scholar
Moore, R. 1985. Semantical considerations on non-monotonic logic. Artificial Intelligence 29, 1, 75–94.Google Scholar
Nguyen, L. A. 2005. On the complexity of fragments of modal logics. In Proceedings of the 5th International Conference on Advances in Modal Logic (AiML'05), 249–268.Google Scholar
Nicolas, P., Garcia, L., Stéphan, I. and Lefèvre, C. 2006. Possibilistic uncertainty handling for answer set programming. Annals of Mathematics and Artificial Intelligence 47, 1–2, 139–181.Google Scholar
Nieves, J. C. and Lindgren, H. 2012. Possibilistic nested logic programs. In Technical Communications of the 28th International Conference on Logic Programming (ICLP'12), 267–276.Google Scholar
Nieves, J. C., Osorio, M. and Cortés, U. 2013. Semantics for possibilistic disjunctive programs. Theory and Practice of Logic Programming 13, 1, 33–70.Google Scholar
Papadimitriou, C. 1994. Computational Complexity. Addison-Wesley, Boston, MA.Google Scholar
Pearce, D. 1997. A new logical characterization of stable models and answer sets. In Proceedings of the 2nd International Workshop on Non-Monotonic Extensions of Logic Programming (NMELP), Lecture Notes in Artificial Intelligence, Vol. 1216, 57–70.Google Scholar
Sakama, C. and Inoue, K. 1994. An alternative approach to the semantics of disjunctive logic programs and deductive databases. Journal of Automated Reasoning 13, 1, 145–172.Google Scholar
Truszczyński, M. 2011. Revisiting epistemic specifications. In Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning, Lecture Notes in Computer Science, Vol. 6565. Springer, Berlin, Germany, 315–333.Google Scholar
Vennekens, J., Verbaeten, S. and Bruynooghe, M. 2004. Logic programs with annotated disjunctions. In Proceedings of the 20th International Conference on Logic Programming (ICLP), Lecture Notes in Computer Science, Vol. 3132, 431–445.Google Scholar
Vlaeminck, H., Vennekens, J., Bruynooghe, M. and Denecker, M. 2012. Ordered epistemic logic: Semantics, complexity and applications. In Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR'12).Google Scholar
Zadeh, L. 1992. Fuzzy logic and the calculus of fuzzy if-then rules. In Proceedings of the 22nd IEEE International Symposium on Multiple-Valued Logic (ISMVL), 480–480.Google Scholar
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