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Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an Approximator of AFT

Published online by Cambridge University Press:  01 September 2021

FANGFANG LIU Open the ORCID record for FANGFANG LIU [Opens in a new window]  and
JIA-HUAI YOU Open the ORCID record for JIA-HUAI YOU [Opens in a new window]
FANGFANG LIU
Affiliation:
School of Computer Engineering and Science, Shanghai University, Shanghai, China (e-mail: ffliu@shu.edu.cn)
JIA-HUAI YOU
Affiliation:
Department of Computing Science, University of Alberta, Edmonton, Canada (e-mail: jyou@ualberta.ca)
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Abstract

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Approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid minimal knowledge and negation as failure (MKNF) knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the abstraction power of AFT, characterizes not only the well-founded semantics but also two-valued as well as three-valued semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved approximator for these knowledge bases, of which the least stable fixpoint is information richer than the one formulated from Knorr et al.’s construction. This leads to an improved computation for the well-founded semantics. This work is built on an extension of AFT that supports consistent as well as inconsistent pairs in the induced product bilattice, to deal with inconsistencies that arise in the context of hybrid MKNF knowledge bases. This part of the work can be considered generalizing the original AFT from symmetric approximators to arbitrary approximators.

Keywords

approximation fixpoint theoryhybrid MKNF knowledge baseslogic programsanswer set semanticsdescription logicsinconsistencies
Type
Original Article
Information
Theory and Practice of Logic Programming , Volume 22 , Issue 2: Special Issue on the International Joint Conference on Rules and Reasoning, RuleML+RR 2019 , March 2022 , pp. 305 - 334
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Amendola, G., Eiter, T., Fink, M., Leone, N. and Moura, J. 2016. Semi-equilibrium models for paracoherent answer set programs. Artificial Intellegence 234, 219–271.Google Scholar
Antic, C., Eiter, T. and Fink, M. 2013. Hex semantics via approximation fixpoint theory. In Proceedings of the 12th International Conference on Logic Programming and Nonmonotonic Reasoning, LPNMR-13. Corunna, Spain, 102–115.Google Scholar
Bi, Y., You, J. and Feng, Z. 2014. A generalization of approximation fixpoint theory and application. In Proceedings of the 8th International Conference on Web Reasoning and Rule Systems, RR-14. Athens, Greece, 45–59.Google Scholar
Bogaerts, B. 2019. Weighted abstract dialectical frameworks through the lens of approximation fixpoint theory. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence, AAAI-19. AAAI Press, Hawaii, USA, 2686–2693.Google Scholar
Bogaerts, B. and Cruz-Filipe, L. 2018. Fixpoint semantics for active integrity constraints. Artificial Intellegence 255, 43–70.CrossRefGoogle Scholar
Bogaerts, B. and den Broeck, G. V. 2015. Knowledge compilation of logic programs using approximation fixpoint theory. Theory and Practice of Logic Programming 15, 4–5, 464–480.CrossRefGoogle Scholar
Bogaerts, B., Jansen, J., Cat, B. D., Janssens, G., Bruynooghe, M., and Denecker, M. 2016. Bootstrapping inference in the IDP knowledge base system. New Generation Computing 34, 3, 193–220.CrossRefGoogle Scholar
Bogaerts, B., Vennekens, J. and Denecker, M. 2015. Grounded fixpoints and their applications in knowledge representation. Artificial Intelligence 224, 51–71.CrossRefGoogle Scholar
Bogaerts, B., Vennekens, J. and Denecker, M. 2018. Safe inductions and their applications in knowledge representation. Artificial Intelligence 259, 167–185.CrossRefGoogle Scholar
Bona, G. D. and Hunter, A. 2017. Localising iceberg inconsistencies. Artificial Intelligence 246, 118–151.CrossRefGoogle Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2000. Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In Logic-Based Artificial Intelligence. Springer, 127–144.Google Scholar
Denecker, M., Marek, V. W. and Truszczynski, M. 2003. Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence 143, 1, 79–122.CrossRefGoogle Scholar
Denecker, M., Marek, V. W. and Truszczynski, M. 2004. Ultimate approximation and its application in nonmonotonic knowledge representation systems. Information and Computation 192, 1, 84–121.CrossRefGoogle Scholar
Denecker, M. and Vennekens, J. 2007. Well-founded semantics and the algebraic theory of non-monotone inductive definitions. In Proceedings of the 9th International Conference on Logic Programming and Nonmonotonic Reasoning, LPNMR-07. Tempe, USA. 84–96.Google Scholar
Dung, P. M. 1995. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificail Intelligence 77, 2, 321–358.CrossRefGoogle Scholar
Fitting, M. 2002. Fixpoint semantics for logic programming a survey. Theoretical Computer Science 278, 1–2, 25–51.CrossRefGoogle 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. MIT Press, Seattle, Washington, 1070–1080.Google Scholar
Ji, J., Liu, F. and You, J. 2017. Well-founded operators for normal hybrid MKNF knowledge bases. Theory and Practice of Logic Programming 17, 5–6, 889–905.Google Scholar
Kaminski, T., Knorr, M. and Leite, J. 2015. Efficient paraconsistent reasoning with ontologies and rules. In Proceedings of the 24th International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, 25–31 July 2015, Yang, Q. and Wooldridge, M. J., Eds. AAAI Press, 3098–3105.Google Scholar
Killen, S. and You, J. 2021. Unfounded sets for disjunctive hybrid MKNF knowledge bases. In Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning, KR-21 (to appear). IJCAI Inc.CrossRefGoogle Scholar
Knorr, M., Alferes, J. J. and Hitzler, P. 2011. Local closed world reasoning with description logics under the well-founded semantics. Artificial Intelligence 175, 9–10, 1528–1554.CrossRefGoogle Scholar
Lifschitz, V. 1991. Nonmonotonic databases and epistemic queries. In Proceedings of the 12th Joint Conference on Artificial Intelligence, IJCAI-91. Sydney, Australia, 381–386.Google Scholar
Liu, F. and You, J. 2017. Three-valued semantics for hybrid MKNF knowledge bases revisited. Artificial Intelligence 252, 123–138.Google Scholar
Liu, F. and You, J. 2019. Alternating fixpoint operator for hybrid MKNF knowledge bases as an approximator of AFT. In Proceedings of the Third International Joint Conference on Rules and Reasoning, RuleML+RR 2019, Fodor, P., Montali, M., Calvanese, D., and Roman, D., Eds. LNCS, vol. 11784. Springer, Bolzano, Italy, 113–127.Google Scholar
Markowsky, G. 1976. Chain-complete posets and directed sets with applications. Algebra Universalis 6, 1, 53–68.Google Scholar
Motik, B. and Rosati, R. 2007. A faithful integration of description logics with logic programming. In Proceedings of the 19th International Joint Conference on Artificial Intelligence, IJCAI-07. Hyderabad, India, 477–482.Google Scholar
Motik, B. and Rosati, R. 2010. Reconciling description logics and rules. Journal of the ACM 57, 5, 1–62.Google Scholar
Pearce, D. and Valverde, A. 2008. Quantified equilibrium logic and foundations for answer set programs. In Proceedings of the 24th International Conference on Logic Programming, ICLP-08, Udine, Italy, de la Banda, M. G. and Pontelli, E., Eds. Lecture Notes in Computer Science, vol. 5366. Springer, 546–560.Google Scholar
Pelov, N., Denecker, M. and Bruynooghe, M. 2007. Well-founded and stable semantics of logic programs with aggregates. Theory and Practice of Logic Programming 7, 3, 301–353.Google Scholar
Sakama, C. and Inoue, K. 1995. Paraconsistent stable semantics for extended disjunctive programs. Journal of Logic and Computation 5, 3, 265–285.CrossRefGoogle Scholar
Strass, H. 2013. Approximating operators and semantics for abstract dialectical frameworks. Artificial Intelligence 205, 39–70.CrossRefGoogle Scholar
Tarski, A. 1955. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5, 2, 285–309.CrossRefGoogle Scholar
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. The Journal of the ACM 23, 4, 733–742.CrossRefGoogle Scholar
Vennekens, J., Gilis, D. and Denecker, M. 2006. Splitting an operator: Algebraic modularity results for logics with fixpoint semantics. ACM Transactions on Computational Logic 7, 4, 765–797.CrossRefGoogle Scholar