Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T13:19:50.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Fuzzy answer sets approximations

Published online by Cambridge University Press:  25 September 2013

MARIO ALVIANO  and
RAFAEL PEÑALOZA
MARIO ALVIANO
Affiliation:
Department of Mathematics and Computer Science, University of Calabria, 87036 Rende (CS), Italy (e-mail: [email protected])
RAFAEL PEÑALOZA
Affiliation:
Dresden University of Technology, 01062 Dresden, GermanyCenter for Advancing Electronics Dresden (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Fuzzy answer set programming (FASP) is a recent formalism for knowledge representation that enriches the declarativity of answer set programming by allowing propositions to be graded. To now, no implementations of FASP solvers are available and all current proposals are based on compilations of logic programs into different paradigms, like mixed integer programs or bilevel programs. These approaches introduce many auxiliary variables which might affect the performance of a solver negatively. To limit this downside, operators for approximating fuzzy answer sets can be introduced: Given a FASP program, these operators compute lower and upper bounds for all atoms in the program such that all answer sets are between these bounds. This paper analyzes several operators of this kind which are based on linear programming, fuzzy unfounded sets and source pointers. Furthermore, the paper reports on a prototypical implementation, also describing strategies for avoiding computations of these operators when they are guaranteed to not improve current bounds. The operators and their implementation can be used to obtain more constrained mixed integer or bilevel programs, or even for providing a basis for implementing a native FASP solver. Interestingly, the semantics of relevant classes of programs with unique answer sets, like positive programs and programs with stratified negation, can be already computed by the prototype without the need for an external tool.

Keywords

fuzzy logicanswer set programmingsearch-space pruning operators
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 13 , Special Issue 4-5: 29th International Conference on Logic Programming , July 2013 , pp. 753 - 767
Copyright
Copyright © 2013 [MARIO ALVIANO and RAFAEL PEÑALOZA] 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Partially supported by Regione Calabria within the PIA project KnowRex POR FESR 2007–2013.

†

Partially supported by DFG under grant BA 1122/17-1 and within the Cluster of Excellence ‘cfAED’.

References

Achs, Á. 1997. Evaluation strategies of fuzzy datalog. Acta Cybernetica 13, 1, 85–102.Google Scholar
Achs, Á. and Kiss, A. 1995. Fuzzy extension of datalog. Acta Cybernetica 12, 2, 153–166.Google Scholar
Alviano, M., Faber, W., Leone, N., Perri, S., Pfeifer, G. and Terracina, G. 2011. The disjunctive datalog system DLV. In Datalog 2.0, Gottlob, G., Ed. Vol. 6702, Springer Berlin/Heidelberg, 282–301.Google Scholar
Baral, C. 2003. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press.CrossRefGoogle Scholar
Blondeel, M., Schockaert, S., De Cock, M. and Vermeir, D. 2012. NP-completeness of fuzzy answer set programming under Łukasiewicz semantics. In Working Papers of the ECAI-2012 Workshop in Weighted Logics for Artificial Intelligence WL4AI, Godo, L. and Prade, H., Eds. 43–50.Google Scholar
Calimeri, F., Ianni, G., Ricca, F., Alviano, M., Bria, A., Catalano, G., Cozza, S., Faber, W., Febbraro, O., Leone, N., Manna, M., Martello, A., Panetta, C., Perri, S., Reale, K., Santoro, M. C., Sirianni, M., Terracina, G. and Veltri, P. 2011. The third answer set programming competition: Preliminary report of the system competition track. In 11th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2011), Delgrande, J. and Faber, W., Eds. Lecture Notes in Computer Science, vol. 6645, Springer Berlin/Heidelberg, 388–403.10.1007/978-3-642-20895-9_46CrossRefGoogle Scholar
Damásio, C. V. and Pereira, L. M. 2001. Antitonic logic programs. In Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'01). Springer-Verlag, London, UK, 379–392.Google Scholar
Delgrande, J. P., Schaub, T., Tompits, H. and Woltran, S. 2008. Belief revision of logic programs under answer set semantics. In Principles of Knowledge Representation and Reasoning: Proceedings of the Eleventh International Conference, KR 2008 Sydney, Australia, September 16-19, 2008, Brewka, G. and Lang, J., Eds. 411–421.Google Scholar
Dubois, D., Lang, J. and Prade, H. 1991. Fuzzy sets in approximate reasoning, part 2: Logical approaches. Fuzzy Sets and Systems 40, 1, 203–244.CrossRefGoogle Scholar
Gebser, M., Kaufmann, B., Neumann, A. and Schaub, T. 2007. Conflict-driven answer set solving. In Twentieth International Joint Conference on Artificial Intelligence (IJCAI-07). Morgan Kaufmann Publishers, 386–392.Google Scholar
Gebser, M., Schaub, T. and Thiele, S. 2007. Gringo: A new grounder for answer set programming. In Logic Programming and Nonmonotonic Reasoning — 9th International Conference, LPNMR'07, Baral, C., Brewka, G., and Schlipf, J., Eds. Lecture Notes in Computer Science, vol. 4483, Springer Verlag, Tempe, Arizona, 266–271.10.1007/978-3-540-72200-7_24CrossRefGoogle Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–385.10.1007/BF03037169CrossRefGoogle Scholar
Janssen, J. 2011. Foundations of Fuzzy Answer Set Programming. PhD thesis, Ghent University.10.2991/978-94-91216-59-6_3CrossRefGoogle Scholar
Janssen, J., Vermeir, D., Schockaert, S. and Cock, M. D. 2012. Reducing fuzzy answer set programming to model finding in fuzzy logics. Theory and Practice of Logic Programming 12, 6, 811–842.CrossRefGoogle Scholar
Klement, E. P., Mesiar, R. and Pap, E. 2000. Triangular Norms. Trends in Logic, Studia Logica Library. Springer-Verlag.10.1007/978-94-015-9540-7CrossRefGoogle Scholar
Lierler, Y. and Maratea, M. 2004. Cmodels-2: SAT-based answer set solver enhanced to non-tight programs. In Proceedings of the 7th International Conference on Logic Programming and Non-Monotonic Reasoning (LPNMR-7), Lifschitz, V. and Niemelä, I., Eds. LNAI, vol. 2923, Springer, 346–350.Google Scholar
Lifschitz, V. 2002. Answer set programming and plan generation. Artificial Intelligence 138, 39–54.CrossRefGoogle Scholar
Lin, F. and You, J.-H. 2002. Abduction in logic programming: A new definition and an abductive procedure based on rewriting. Artificial Intelligence 140, 1/2, 175–205.10.1016/S0004-3702(02)00227-8CrossRefGoogle Scholar
Loyer, Y. and Straccia, U. 2009. Approximate well-founded semantics, query answering and generalized normal logic programs over lattices. Annals Mathematics and Artificial Intelligence 55, 3–4, 389–417.10.1007/s10472-008-9099-0CrossRefGoogle Scholar
Lukasiewicz, T. 2006. Fuzzy description logic programs under the answer set semantics for the semantic web. In Proc. 2nd International Conference on Rules and Rule Markup Languages for the Semantic Web (RuleML 2006), Eiter, T., Franconi, E., Hodgson, R. and Stephens, S., Eds. IEEE Computer Society, 89–96.10.1109/RULEML.2006.12CrossRefGoogle Scholar
Marek, V. W. and Remmel, J. B. 2004. Answer set programming with default logic. In , Proceedings of the 10th International Workshop on Non-Monotonic Reasoning (NMR 2004), Whistler, Canada, June 6-8, 2004, Delgrande, J. P. and Schaub, T., Eds. 276–284.Google Scholar
Marek, V. W. and Truszczyński, M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm – A 25-Year Perspective, Apt, K. R., Marek, V. W., Truszczyński, M. and Warren, D. S., Eds. Springer Verlag, 375–398.10.1007/978-3-642-60085-2_17CrossRefGoogle Scholar
Niemelä, I. 1999. Logic programming with stable model semantics as constraint programming paradigm. Annals of Mathematics and Artificial Intelligence 25, 3–4, 241–273.CrossRefGoogle Scholar
Nieuwenborgh, D. V., Cock, M. D. and Vermeir, D. 2007a. Computing fuzzy answer sets using dlvhex. In Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), Porto, Portugal, September 8-13, 2007, Lecture Notes in Computer Science, vol. 4670, 449–450.Google Scholar
Nieuwenborgh, D. V., Cock, M. D. and Vermeir, D. 2007b. An introduction to fuzzy answer set programming. Annals Mathematics and Artificial Intelligence 50, 3–4, 363–388.10.1007/s10472-007-9080-3CrossRefGoogle Scholar
Simons, P., Niemelä, I. and Soininen, T. 2002. Extending and implementing the stable model semantics. Artificial Intelligence 138, 181–234.CrossRefGoogle Scholar
Van Gelder, A., Ross, K. A. and Schlipf, J. S. 1991. The Well-founded semantics for general logic programs. Journal of the ACM 38, 3, 620–650.10.1145/116825.116838CrossRefGoogle Scholar
Supplementary material: PDF

Alviano et al. supplementary material

Appendix

Download Alviano et al. supplementary material(PDF)
PDF 260.1 KB