Advertisement

Lean Kernels in Description Logics

  • Rafael PeñalozaEmail author
  • Carlos Mencía
  • Alexey Ignatiev
  • Joao Marques-Silva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10249)

Abstract

Lean kernels (LKs) are an effective optimization for deriving the causes of unsatisfiability of a propositional formula. Interestingly, no analogous notion exists for explaining consequences of description logic (DL) ontologies. We introduce LKs for DLs using a general notion of consequence-based methods, and provide an algorithm for computing them which incurs in only a linear time overhead. As an example, we instantiate our framework to the DL \({\mathcal {ALC}}\). We prove formally and empirically that LKs provide a tighter approximation of the set of relevant axioms for a consequence than syntactic locality-based modules.

Notes

Acknowledgements

We would like to thank Beatriz Peñaloza for her help on statistical methods. Carlos Mencía is supported by grant TIN2016-79190-R.

References

  1. 1.
    Arif, M.F., Mencía, C., Ignatiev, A., Manthey, N., Peñaloza, R., Marques-Silva, J.: BEACON: An efficient SAT-based tool for debugging EL+ ontologies. In: SAT, pp. 521–530 (2016)Google Scholar
  2. 2.
    Romero, A.A.: Ontology module extraction and applications to ontology classification. Ph.D. thesis, University of Oxford, UK (2015)Google Scholar
  3. 3.
    Romero, A.A., Kaminski, M., Grau, B.C., Horrocks, I.: Ontology module extraction via datalog reasoning. In: AAAI, pp. 1410–1416 (2015)Google Scholar
  4. 4.
    Romero, A.A., Kaminski, M., Cuenca Grau, B., Horrocks, I.: Module extraction in expressive ontology languages via datalog reasoning. JAIR 55, 499–564 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baader, F., Brandt, S., Lutz, C.: Pushing the \(\cal{E\!L}\) envelope. In: IJCAI, pp. 364–369 (2005)Google Scholar
  6. 6.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  7. 7.
    Baader, F., Knechtel, M., Peñaloza, R.: Context-dependent views to axioms and consequences of semantic web ontologies. J. Web Semant. 12–13, 22–40 (2012)CrossRefGoogle Scholar
  8. 8.
    Baader, F., Suntisrivaraporn, B.: Debugging SNOMED CT using axiom pinpointing in the description logic \(\cal{EL}^{+}\). In: KR-MED (2008)Google Scholar
  9. 9.
    Bate, A., Motik, B., Grau, B.C., Simancik, F., Horrocks, I.: Extending consequence-based reasoning to SRIQ. In: KR, pp. 187–196 (2016)Google Scholar
  10. 10.
    Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ceylan, İİ., Peñaloza, R.: The bayesian description logic \({\cal{BEL}}\). In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 480–494. Springer, Cham (2014). doi: 10.1007/978-3-319-08587-6_37CrossRefzbMATHGoogle Scholar
  12. 12.
    Grau, B.C., Horrocks, I., Kazakov, Y., Sattler, U.: Just the right amount: Extracting modules from ontologies. In: WWW, pp. 717–726 (2007)Google Scholar
  13. 13.
    Cuenca Grau, B., Horrocks, I., Kazakov, Y., Sattler, U.: Modular reuse of ontologies: Theory and practice. J. Artif. Intell. Res. (JAIR) 31, 273–318 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cuenca Grau, B., Horrocks, I., Kazakov, Y., Sattler, U.: Extracting modules from ontologies: A logic-based approach. In: Stuckenschmidt, H., Parent, C., Spaccapietra, S. (eds.) Modular Ontologies. LNCS, vol. 5445, pp. 159–186. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-01907-4_8CrossRefzbMATHGoogle Scholar
  15. 15.
    Vescovo, C., Klinov, P., Parsia, B., Sattler, U., Schneider, T., Tsarkov, D.: Empirical study of logic-based modules: Cheap is cheerful. In: Alani, H., Kagal, L., Fokoue, A., Groth, P., Biemann, C., Parreira, J.X., Aroyo, L., Noy, N., Welty, C., Janowicz, K. (eds.) ISWC 2013. LNCS, vol. 8218, pp. 84–100. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-41335-3_6CrossRefGoogle Scholar
  16. 16.
    Kalyanpur, A., Parsia, B., Horridge, M., Sirin, E.: Finding all justifications of OWL DL entailments. In: Aberer, K., Choi, K.-S., Noy, N., Allemang, D., Lee, K.-I., Nixon, L., Golbeck, J., Mika, P., Maynard, D., Mizoguchi, R., Schreiber, G., Cudré-Mauroux, P. (eds.) ASWC/ISWC -2007. LNCS, vol. 4825, pp. 267–280. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-76298-0_20CrossRefGoogle Scholar
  17. 17.
    Kaminski, M., Nenov, Y., Grau, B.C.: Datalog rewritability of disjunctive datalog programs and its applications to ontology reasoning. In: AAAI, pp. 1077–1083 (2014)Google Scholar
  18. 18.
    Kazakov, Y.: Consequence-driven reasoning for Horn SHIQ ontologies. In: Boutilier, C. (ed.) IJCAI 2009, pp. 2040–2045 (2009)Google Scholar
  19. 19.
    Kazakov, Y., Krötzsch, M., Simancik, F.: The incredible ELK - from polynomial procedures to efficient reasoning with \(\cal{E\!L}\) ontologies. JAR 53(1), 1–61 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Büning, H.K., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 339–401. IOS Press (2009)Google Scholar
  21. 21.
    Kullmann, O.: Investigations on autark assignments. Discrete Appl. Math. 107(1–3), 99–137 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kullmann, O., Lynce, I., Marques-Silva, J.: Categorisation of clauses in conjunctive normal forms: Minimally unsatisfiable sub-clause-sets and the lean kernel. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 22–35. Springer, Heidelberg (2006). doi: 10.1007/11814948_4CrossRefzbMATHGoogle Scholar
  23. 23.
    Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liffiton, M.H., Previti, A., Malik, A., Marques-Silva, J.: Fast, flexible MUS enumeration. Constraints 21(2), 223–250 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liffiton, M., Sakallah, K.: Searching for autarkies to trim unsatisfiable clause sets. In: Kleine Büning, H., Zhao, X. (eds.) SAT 2008. LNCS, vol. 4996, pp. 182–195. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-79719-7_18CrossRefGoogle Scholar
  26. 26.
    Ludwig, M., Peñaloza, R.: Error-tolerant reasoning in the description logic \(\cal{E\!L}\). In: JELIA, pp. 107–121 (2014)Google Scholar
  27. 27.
    Marques-Silva, J., Ignatiev, A., Mencía, C., Peñaloza, R.: Efficient reasoning for inconsistent horn formulae. In: Michael, L., Kakas, A. (eds.) JELIA 2016. LNCS (LNAI), vol. 10021, pp. 336–352. Springer, Cham (2016). doi: 10.1007/978-3-319-48758-8_22CrossRefGoogle Scholar
  28. 28.
    Marques-Silva, J., Ignatiev, A., Morgado, A., Manquinho, V.M., Lynce, I.: Efficient autarkies. In: ECAI, pp. 603–608 (2014)Google Scholar
  29. 29.
    Minoux, M.: LTUR: A simplified linear-time unit resolution algorithm for horn formulae and computer implementation. Inf. Process. Lett. 29(1), 1–12 (1988)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Peñaloza, R., Sertkaya, B.: On the complexity of axiom pinpointing in the \(\cal{E\!L}\) family of description logics. In: KR (2010)Google Scholar
  31. 31.
    Riguzzi, F., Bellodi, E., Lamma, E., Zese, R.: Probabilistic description logics under the distribution semantics. Semant. Web 6(5), 477–501 (2015)CrossRefGoogle Scholar
  32. 32.
    Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with complements. Artif. Intell. 48(1), 1–26 (1991)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sebastiani, R., Vescovi, M.: Axiom pinpointing in lightweight description logics via horn-SAT encoding and conflict analysis. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 84–99. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02959-2_6CrossRefGoogle Scholar
  34. 34.
    Sebastiani, R., Vescovi, M.: Axiom pinpointing in large \(\cal{EL}^+\) ontologies via SAT and SMT techniques. Technical Report DISI-15-010, DISI, University of Trento, Italy, April 2015. http://disi.unitn.it/rseba/elsat/elsat_techrep.pdf
  35. 35.
    Simancik, F., Kazakov, Y., Horrocks, I.: Consequence-based reasoning beyond horn ontologies. In: IJCAI 2011, pp. 1093–1098 (2011). IJCAI/AAAIGoogle Scholar
  36. 36.
    Suntisrivaraporn, B.: Module extraction and incremental classification: A pragmatic approach for \(\cal{EL}^+\) ontologies. In: ESWC, pp. 230–244 (2008)Google Scholar
  37. 37.
    Suntisrivaraporn, B.: Polynomial-Time Reasoning Support for Design and Maintenance of Large-Scale Biomedical Ontologies. Ph.D. thesis, TU Dresden (2009)Google Scholar
  38. 38.
    Suntisrivaraporn, B., Qi, G., Ji, Q., Haase, P.: A modularization-based approach to finding all justifications for OWL DL entailments. In: Domingue, J., Anutariya, C. (eds.) ASWC 2008. LNCS, vol. 5367, pp. 1–15. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-89704-0_1CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Rafael Peñaloza
    • 1
    Email author
  • Carlos Mencía
    • 2
  • Alexey Ignatiev
    • 3
  • Joao Marques-Silva
    • 3
  1. 1.Free University of Bozen-BolzanoBolzanoItaly
  2. 2.University of OviedoGijónSpain
  3. 3.University of LisbonLisbonPortugal

Personalised recommendations