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Reducing fuzzy answer set programming to model finding in fuzzy logics

Published online by Cambridge University Press:  21 June 2011

JEROEN JANSSEN ,
DIRK VERMEIR ,
STEVEN SCHOCKAERT  and
MARTINE DE COCK
JEROEN JANSSEN
Affiliation:
Department of Computer Science, Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, Belgium (e-mail: jeroen.janssen@vub.ac.be, dirk.vermeir@vub.ac.be)
DIRK VERMEIR
Affiliation:
Department of Computer Science, Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, Belgium (e-mail: jeroen.janssen@vub.ac.be, dirk.vermeir@vub.ac.be)
STEVEN SCHOCKAERT
Affiliation:
Department of Applied Mathematics and Computer Science, Universiteit Gent Krijgslaan 281, 9000 Ghent, Belgium (e-mail: steven.schockaert@ugent.be)
MARTINE DE COCK
Affiliation:
Institute of Technology, University of Washington 1900 Commerce Street, Tacoma, WA 98402, USA (e-mail: mdecock@u.washington.edu)
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Abstract

In recent years, answer set programming (ASP) has been extended to deal with multivalued predicates. The resulting formalismsallow for the modeling of continuous problems as elegantly as ASP allows for the modeling of discrete problems, by combining thestable model semantics underlying ASP with fuzzy logics. However, contrary to the case of classical ASP where manyefficient solvers have been constructed, to date there is no efficient fuzzy ASP solver. A well-knowntechnique for classical ASP consists of translating an ASP program P to a propositional theory whose models exactlycorrespond to the answer sets of P. In this paper, we show how this idea can be extended to fuzzy ASP, paving the wayto implement efficient fuzzy ASP solvers that can take advantage of existing fuzzy logic reasoners.

Keywords

fuzzy logicanswer set programmingASSAT
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 12 , Issue 6 , November 2012 , pp. 811 - 842
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
Copyright © Cambridge University Press 2011

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References

Alsinet, T., Godo, L. and Sandri, S. 2002. Two formalisms of extended possibilistic logic programming with context-dependent fuzzy unification: A comparative description. Electronic Notes in Theoretical Computer Science 66, 5, 1–21.Google Scholar
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A. and Protasi, M. 1999. Complexity and Approximation. Springer-Verlag, Berlin/Heidelberg.Google Scholar
Baral, C. 2003. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press.Google Scholar
Baral, C., Gelfond, M. and Rushton, N. 2007. Probabilistic reasoning with answer sets. In Proceedings of the Ninth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '07), Lifschitz, V. and Niemelä, I., Eds. Lecture Notes in Computer Science, vol. 2923. Springer, Berlin/Heidelberg, 21–33.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 '10), Grünwald, P. and Spirtes, P., Eds. AUAI Press.Google Scholar
Bobillo, F. and Straccia, U. 2007. A fuzzy description logic with product t-norm. In Proceedings of the 16th IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '07). IEEE Computer Society, 652–657.Google Scholar
Cao, T. H. 2000. Annotated fuzzy logic programs. Fuzzy Sets and Systems 113, 2, 277–298.Google Scholar
Damásio, C. V., Medina, J. and Ojeda-Aciego, M. 2004. Sorted multiadjoint logic programs: Termination results and applications. In Proceedings of the Ninth European Conference on Logics in Artificial Intelligence (JELIA '04), Alferes, J. J. and Leite, J., Eds. Lecture Notes in Computer Science, vol. 3229. Springer, Berlin/Heidelberg, 252–265.Google Scholar
Damásio, C. V., Medina, J. and Ojeda-Aciego, M. 2007. Termination of logic programs with imperfect information: Applications and query procedure. Journal of Applied Logic 5, 3, 435–458.CrossRefGoogle Scholar
Damásio, C. V. and Pereira, L. M. 2000. Hybrid probabilistic logic programs as residuated logic programs. In Proceedings of the Seventh European Workshop on Logics in Artificial Intelligence (JELIA '00), Ojeda-Aciego, M., de Guzmán, I., Brewka, G. and Pereira, L. Moniz, Eds. Lecture Notes in Computer Science, vol. 1919. Springer, Berlin/Heidelberg, 57–72.CrossRefGoogle Scholar
Damásio, C. V. and Pereira, L. M. 2001a. Antitonic logic programs. In Proceedings of the Sixth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '01), Eiter, T., Faber, W. and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 2173. Springer, Berlin/Heidelberg, 379–393.Google Scholar
Damásio, C. V. and Pereira, L. M. 2001b. Monotonic and residuated logic programs. In Proceedings of the Sixth European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU '01), Benferhat, S. and Besnard, P., Eds. Lecture Notes in Computer Science, vol. 2143. Springer, Berlin/Heidelberg, 748–759.CrossRefGoogle Scholar
Damásio, C. V. and Pereira, L. M. 2004. Sorted monotonic logic programs and their embeddings. In Proceedings of the Information Processing and Management of Uncertainty (IPMU '04), 807–814.Google Scholar
Davis, M. and Putnam, H. 1960. A computing procedure for quantification theory. Journal of the ACM 7, 3, 201–215.Google Scholar
Fages, F. 1994. Consistency of Clark's completion and existence of stable models. Methods of Logic in Computer Science 1, 51–60.Google Scholar
Fitting, M. 1991. Bilattices and the semantics of logic programming. Journal of Logic Programming 11, 2, 91–116.Google Scholar
Fuhr, N. 2000. Probabilistic datalog: Implementing logical information retrieval for advanced applications. Journal of the American Society for Information Science 51, 2, 95–110.Google Scholar
Gebser, M., Kaufmann, B. and Schaub, T. 2009. The conflict-driven answer set solver clasp: Progress report. In Proceedings of the 10th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '09), Erdem, E., Lin, F. and Schaub, T., Eds. Lecture Notes in Computer Science, vol. 5753. Springer, Berlin/Heidelberg, 509–514.Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the Fifth International Conference and Symposium on Logic Programming (ICLP/SLP '88). MIT Press, 1081–1086.Google Scholar
Giunchiglia, E., Lierler, Y. and Maratea, M. 2004. SAT-based answer set programming. In Proceedings of the 19th national conference on Artifical intelligence (AAAI '04). AAAI Press/The MIT Press, 61–66.Google Scholar
Hähnle, R. 1994. Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence 12, 3–4, 231–263.CrossRefGoogle Scholar
Hájek, P. 2001. Metamathematics of Fuzzy Logic (Trends in Logic). Springer.Google Scholar
Ishizuka, M. and Kanai, N. 1985. Prolog-ELF incorporating fuzzy logic. In Proceedings of the Ninth International Joint Conference on Artificial Intelligence (IJCAI '85), 701–703.Google Scholar
Janssen, J., Heymans, S., Vermeir, D. and DeCock, M. Cock, M. 2008. Compiling fuzzy answer set programs to fuzzy propositional theories. In Proceedings of the 24th International Conference on Logic Programming (ICLP '08), de la Banda, M. Garcia and Pontelli, E., Eds. Lecture Notes in Computer Science, vol. 5366. Springer, Berlin/Heidelberg, 362–376.Google Scholar
Kifer, M. and Li, A. 1988. On the semantics of rule-based expert systems with uncertainty. In Proceedings of the Second International Conference on Database Theory (ICDT '88), Gyssens, M., Paredaens, J. and Van Gucht, D., Eds. Lecture Notes in Computer Science, vol. 326. Springer, Berlin/Heidelberg, 102–117.Google Scholar
Kifer, M. and Subrahmanian, V. S. 1992. Theory of generalized annotated logic programming and its applications. Journal of Logic Programming 12, 3–4, 335–367.Google Scholar
Lakshmanan, L. V. S. 1994. An epistemic foundation for logic programming with uncertainty. In Proceedings of the 14th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS '94), Thiagarajan, P., Ed. Lecture Notes in Computer Science, vol. 880. Springer, Berlin/Heidelberg, 89–100.Google Scholar
Lakshmanan, L. V. S. 1997. Towards a Generalized Theory of Deductive Databases with Uncertainty. PhD Thesis, Concordia University.Google Scholar
Lakshmanan, L. V. S. and Sadri, F. 1994. Modeling uncertainty in deductive databases. In Proceedings of the Fifth International Conference on Database and Expert Systems Applications (DEXA '94), Karagiannis, D., Ed. Lecture Notes in Computer Science, vol. 856. Springer, Berlin/Heidelberg, 724–733.CrossRefGoogle Scholar
Lakshmanan, L. V. S. and Sadri, F. 1997. Uncertain deductive databases: A hybrid approach. Information Systems 22, 9, 483–508.Google Scholar
Lakshmanan, L. V. S. and Shiri, N. 2001. A parametric approach to deductive databases with uncertainty. IEEE Transactions on Knowledge and Data Engineering 13, 4, 554–570.CrossRefGoogle Scholar
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S. and Scarcello, F. 2006. The DLV system for knowledge representation and reasoning. ACM Transactions on Computational Logic 7, 3, 499–562.CrossRefGoogle Scholar
Lin, F. and Zhao, Y. 2004. ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence 157, 1–2, 115–137.Google Scholar
Liu, L. and Truszczyński, M. 2005. Pbmodels—software to compute stable models by pseudoboolean solvers. In Proceedings of the Eighth international conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '05), Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. Lecture Notes in Computer Science, vol. 3662. Springer, Berlin/Heidelberg, 410–415.CrossRefGoogle Scholar
Loyer, Y. and Straccia, U. 2002. The well-founded semantics in normal logic programs with uncertainty. In Proceedings of the Sixth International Symposium on Functional and Logic Programming (FLOPS '02), Hu, Z. and Rodríguez-Artalejo, M., Eds. Lecture Notes in Computer Science, vol. 2441. Springer, 152–166.Google Scholar
Loyer, Y. and Straccia, U. 2003. The approximate well-founded semantics for logic programs with uncertainty. In Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS '03), Rovan, B. and Vojtáš, P., Eds. Lecture Notes in Computer Science, vol. 2747. Springer, Berlin/Heidelberg, 541–550.Google Scholar
Loyer, Y. and Straccia, U. 2006. Epistemic foundation of stable model semantics. Journal of Theory and Practice of Logic Programming 6, 355–393.CrossRefGoogle Scholar
Lukasiewicz, T. 1998. Probabilistic logic programming. In Proceedings of the 13th European Conference on Artificial Intelligence (ECAI '98). J. Wiley & Sons, 388–392.Google Scholar
Lukasiewicz, T. 1999. Many-valued disjunctive logic programs with probabilistic semantics. In Proceedings of the Fifth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '99), Gelfond, M., Leone, N. and Pfeifer, G., Eds. Lecture Notes in Computer Science, vol. 1730. Springer, Berlin/Heidelberg, 277–289.Google Scholar
Lukasiewicz, T. 2006. Fuzzy description logic programs under the answer set semantics for the semantic web. In Proceedings of the Second International Conference on Rules and Rule Markup Languages for the Semantic Web (RuleML '06), 89–96.Google Scholar
Lukasiewicz, T. and Straccia, U. 2007a. Tightly integrated fuzzy description logic programs under the answer set semantics for the semantic web. In Proceedings of the First International Conference on Web Reasoning and Rule Systems (RR '07), Marchiori, M., Pan, J. and Marie, C., Eds. Lecture Notes in Computer Science, vol. 4524. Springer, Berlin/Heidelberg, 289–298.Google Scholar
Lukasiewicz, T. and Straccia, U. 2007b. Top-k retrieval in description logic programs under vagueness for the semantic web. In Proceedings of the First International Conference on Scalable Uncertainty Management (SUM '07), Prade, H. and Subrahmanian, V., Eds. Lecture Notes in Computer Science, vol. 4772. Springer, Berlin/Heidelberg, 16–30.Google Scholar
Madrid, N. and Ojeda-Aciego, M. 2008. Towards a fuzzy answer set semantics for residuated logic programs. In Proceedings of the 2008 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology (WI-IAT '08), 260–264.Google Scholar
Madrid, N. and Ojeda-Aciego, M. 2009. On coherence and consistence in fuzzy answer set semantics for residuated logic programs. In Proceedings of the Eighth International Workshop on Fuzzy Logic and Applications (WILF '09), Gesù, V. Di, Pal, S. and Petrosino, A., Eds. Lecture Notes in Computer Science, vol. 5571. Springer, Berlin/Heidelberg, 60–67.Google Scholar
Madrid, N. and Ojeda-Aciego, M. 2011. On the existence and unicity of stable models in normal residuated logic programs. International Journal on Computer Mathematics, to appear.CrossRefGoogle Scholar
Nerode, A., Remmel, J. B. and Subrahmanian, V. S. 1997. Annotated nonmonotonic rule systems. Theoretical Computer Science 171, 1–2, 111–146.Google Scholar
Ng, R. and Subrahmanian, V. S. 1993. A semantical framework for supporting subjective and conditional probabilities in deductive databases. Journal of Automated Reasoning 10, 2, 191–235.CrossRefGoogle Scholar
Ng, R. and Subrahmanian, V. S. 1994. Stable semantics for probabilistic deductive databases. Information and Computation 110, 1, 42–83.CrossRefGoogle Scholar
Nicolas, P., Garcia, L. and Stéphan, I. 2005. Possibilistic stable models. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI'05). Morgan Kaufmann Publishers Inc. San Francisco, CA, USA.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.CrossRefGoogle Scholar
Novák, V., Perfilieva, I. and Mockor, J. 1999. Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers.CrossRefGoogle Scholar
Saad, E. 2009a. Extended fuzzy logic programs with fuzzy answer set semantics. In Proceedings of the Third International Conference on Scalable Uncertainty Management (SUM '09), Godo, L. and Pugliese, A., Eds. Lecture Notes in Computer Science, vol. 5785. Springer, Berlin/Heidelberg, 223–239.Google Scholar
Saad, E. 2009b. Probabilistic reasoning by SAT solvers. In Proceedings of the 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU '09), Sossai, C. and Chemello, G., Eds. Lecture Notes in Computer Science, vol. 5590. Springer, Berlin/Heidelberg, 663–675.CrossRefGoogle Scholar
Shapiro, E. Y. 1983. Logic programs with uncertainties: A tool for implementing rule-based systems. In Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI '83), Bundy, A., Ed. William Kaufmann, 529–532.Google Scholar
Simons, P. 2000. Extending and Implementing the Stable Model Semantics. PhD Thesis, Helsinki University of Technology.Google Scholar
Straccia, U. 2005. Query answering in normal logic programs under uncertainty. In Proceedings of the Eighth European Conferences on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU '05), Godo, L., Ed. Lecture Notes in Computer Science, vol. 3571. Springer, Berlin/Heidelberg, 470–470.Google Scholar
Straccia, U. 2006. Annotated answer set programming. In Proceedings of the 11th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU '06).Google Scholar
Straccia, U. 2008. Managing uncertainty and vagueness in description logics, logic programs and description logic programs. In Proceedings of the 4th International Summer School 2008 on Reasoning Web, Baroglio, C., Bonatti, P. A., uszynski, J. M., Marchiori, M., Polleres, A. and Schaffert, S., Eds. Lecture Notes in Computer Science, vol. 5224. Springer, Berlin/Heidelberg, 54–103.Google Scholar
Straccia, U., Ojeda-Aciego, M. and Damásio, C. V. 2009. On fixed-points of multivalued functions on complete lattices and their application to generalized logic programs. SIAM Journal on Computing 38, 5, 1881–1911.CrossRefGoogle Scholar
Subrahmanian, V. S. 1994. Amalgamating knowledge bases. ACM Transactions on Database Systems 19, 2, 291–331.CrossRefGoogle Scholar
Tarski, A. 1955. A lattice theoretical fixpoint theorem and its application. Pacific Journal of Mathematics 5, 285–309.Google Scholar
van Emden, M. H. 1986. Quantitative deduction and its fixpoint theory. Journal of Logic Programming 30, 1, 37–53.Google Scholar
Van Nieuwenborgh, D., De Cock, M. and Vermeir, D. 2007a. Computing fuzzy answer sets using DLVHEX. In Proceedings of the 23rd International Conference on Logic Programming (ICLP '07), Dahl, V. and Niemelä, I., Eds. Lecture Notes in Computer Science, vol. 4670. Springer, Berlin/Heidelberg, 449–450.Google Scholar
Van Nieuwenborgh, D., De Cock, M. and Vermeir, D. 2007b. An introduction to fuzzy answer set programming. Annals of Mathematics and Artificial Intelligence 50, 3–4, 363–388.Google Scholar
Vojtás, P. 2001. Fuzzy logic programming. Fuzzy Sets and Systems 124, 3, 361–370.CrossRefGoogle Scholar
Wagner, G. 1998. Negation in fuzzy and possibilistic logic programs. Uncertainty Theory in Artificial Intelligence Series 3, 113–128.Google Scholar