Abstract
The approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices, and has been useful in dealing with semantics issues for various types of logic programs. The theory in the current form, however, only deals with consistent pairs on a bilattice, and it thus does not apply to situations where inconsistency may be part of a fixpoint construction. This is the case for FOL-programs, where a rule set and a first-order theory are tightly integrated. In this paper, we develop an extended theory of AFT that treats consistent as well as inconsistent pairs on a bilattice. We then apply the extended theory to FOL-programs and explore various possibilities on semantics. This leads to novel formulations of approximating operators, and new well-founded semantics and characterizations of answer sets for FOL-programs. The work reported here shows how consistent approximations may be extended to capture wider classes of logic programs whose semantics can be treated uniformly.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alviano, M., Calimeri, F., Faber, W., Leone, N., Perri, S.: Unfounded sets and well-founded semantics of answer set programs with aggregates. J. Artif. Intell. Res. 42 (2011)
Antic, C., Eiter, T., Fink, M.: Hex semantics via approximation fixpoint theory. In: Cabalar, P., Son, T.C. (eds.) LPNMR 2013. LNCS, vol. 8148, pp. 102–115. Springer, Heidelberg (2013)
Bi, Y., You, J.H., Feng, Z.: A generalization of approximation fixpoint theory and its application to FOL-programs. Tech. rep., College of Computer Science and Technology, Tianjin University, Tianjin, China (June 2014), http://xinwang.tju.edu.cn/drupal/sites/default/files/basic_page/full-proof.pdf
de Bruijn, J., Eiter, T., Tompits, H.: Embedding approaches to combining rules and ontologies into autoepistemic logic. In: Proc. KR 2008, pp. 485–495 (2008)
de Bruijn, J., Pearce, D., Polleres, A., Valverde, A.: Quantified equilibrium logic and hybrid rules. In: Marchiori, M., Pan, J.Z., Marie, C.d.S. (eds.) RR 2007. LNCS, vol. 4524, pp. 58–72. Springer, Heidelberg (2007)
Denecker, M., Marek, V., Truszczynski, M.: Ultimate approximation and its application in nonmonotonic knowledge representation systems. Information and Computation 192(1), 84–121 (2004)
Eiter, T., Ianni, G., Lukasiewicz, T., Schindlauer, R., Tompits, H.: Combining answer set programming with description logics for the semantic web. Artifical Intelligence 172(12-13), 1495–1539 (2008)
Eiter, T., Ianni, G., Schindlauer, R., Tompits, H.: A uniform integration of higher-order reasoning and external evaluations in answer-set programming. In: Proc. IJCAI 2005, pp. 90–96 (2005)
Eiter, T., Lukasiewicz, T., Ianni, G., Schindlauer, R.: Well-founded semantics for description logic programs in the semantic web. ACM Transactions on Computational Logic 12(2) (2011)
Lukasiewicz, T.: A novel combination of answer set programming with description logics for the semantic web. IEEE TKDE 22(11), 1577–1592 (2010)
McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artifical Intelligence 13(27-39), 171–172 (1980)
Motik, B., Rosati, R.: Reconciling description logics and rules. Journal of the ACM 57(5), 1–62 (2010)
Pelov, N., Denecker, M., Bruynooghe, M.: Well-founded and stable semantics of logic programs with aggregates. Theory and Practice of Logic Programming 7, 301–353 (2007)
Rosati, R.: On the decidability and complexity of integrating ontologies and rules. Journal of Web Semantics 3(1), 61–73 (2005)
Rosati, R.: DL+log: Tight integration of description logics and disjunctive datalog. In: Proc. KR 2006, pp. 68–78 (2006)
Shen, Y.D.: Well-supported semantics for description logic programs. In: Proc. IJCAI 2011, pp. 1081–1086 (2011)
Shen, Y.-D., Wang, K.: Extending logic programs with description logic expressions for the semantic web. In: Aroyo, L., Welty, C., Alani, H., Taylor, J., Bernstein, A., Kagal, L., Noy, N., Blomqvist, E. (eds.) ISWC 2011, Part I. LNCS, vol. 7031, pp. 633–648. Springer, Heidelberg (2011)
Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5(2), 285–309 (1955)
Vennekens, J., Denecker, M., Bruynooghe, M.: FO(ID) as an extension of DL with rules. Ann. Math. Artif. Intell. 58(1-2), 85–115 (2010)
You, J.H., Shen, Y.D., Wang, K.: Well-supported semantics for logic programs with generalized rules. In: Erdem, E., Lee, J., Lierler, Y., Pearce, D. (eds.) Correct Reasoning. LNCS, vol. 7265, pp. 576–591. Springer, Heidelberg (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Bi, Y., You, JH., Feng, Z. (2014). A Generalization of Approximation Fixpoint Theory and Application. In: Kontchakov, R., Mugnier, ML. (eds) Web Reasoning and Rule Systems. RR 2014. Lecture Notes in Computer Science, vol 8741. Springer, Cham. https://doi.org/10.1007/978-3-319-11113-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-11113-1_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11112-4
Online ISBN: 978-3-319-11113-1
eBook Packages: Computer ScienceComputer Science (R0)