Skip to main content

Union and Intersection of All Justifications

  • 480 Accesses

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13261)

Abstract

We present new algorithms for computing the union and intersection of all justifications for a given ontological consequence without first computing the set of all justifications. Our goal is to use these sets to explain the consequences and, if needed, repair them. Through an empirical evaluation, we show that our approach behaves well in practice for expressive description logics. In particular, the union of all justifications can be computed much faster than with existing justification-enumeration approaches.

Keywords

  • Justifications
  • Axiom pinpointing
  • Ontology Repairs

This work is partially funded by the Norwegian Research Council via the SIRIUS centre (Grant Nr.237898) and the BPI-France (PSPC AIDA: 2019-PSPC-09).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-031-06981-9_4
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-031-06981-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.

Notes

  1. 1.

    We restrict to \(\mathcal {ALCH}\) in this section as condor only accepts \(\mathcal {ALCH}\)-TBoxes.

  2. 2.

    The implementation is vailable at https://github.com/JieyingChenChen/IntersectionAndUnionOfAllJust.

References

  1. Arif, M.F., Mencía, C., Ignatiev, A., Manthey, N., Peñaloza, R., Marques-Silva, J.: BEACON: an efficient SAT-based tool for debugging \({\cal{EL}}{^+}\) ontologies. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 521–530. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_32

    CrossRef  MATH  Google Scholar 

  2. Arif, M.F., Mencía, C., Marques-Silva, J.: Efficient axiom pinpointing with EL2MCS. In: Hölldobler, S., Krötzsch, M., Peñaloza, R., Rudolph, S. (eds.) KI 2015. LNCS (LNAI), vol. 9324, pp. 225–233. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24489-1_17

    CrossRef  Google Scholar 

  3. Arif, M.F., Mencía, C., Marques-Silva, J.: Efficient MUS enumeration of horn formulae with applications to axiom pinpointing. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 324–342. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24318-4_24

    CrossRef  MATH  Google Scholar 

  4. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope further. In: Proceedings of the OWLED 2008 DC Workshop on OWL: Experiences and Directions (2008)

    Google Scholar 

  5. Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  6. Baader, F., Hollunder, B.: Embedding defaults into terminological knowledge representation formalisms. J. Autom. Reason. 14(1), 149–180 (1995)

    MathSciNet  CrossRef  Google Scholar 

  7. Baader, F., Peñaloza, R.: Automata-based axiom pinpointing. J. Autom. Reason. 45(2), 91–129 (2010)

    MathSciNet  CrossRef  Google Scholar 

  8. Baader, F., Peñaloza, R.: Axiom pinpointing in general tableaux. J. Log. Comput. 20(1), 5–34 (2010)

    MathSciNet  CrossRef  Google Scholar 

  9. Baader, F., Peñaloza, R., Suntisrivaraporn, B.: Pinpointing in the description logic \(\cal{EL}^+\). In: Hertzberg, J., Beetz, M., Englert, R. (eds.) KI 2007. LNCS (LNAI), vol. 4667, pp. 52–67. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74565-5_7

    CrossRef  Google Scholar 

  10. Bail, S.P.: The justificatory structure of OWL ontologies. Ph.D. thesis, University of Manchester, UK (2013)

    Google Scholar 

  11. Chang, C.-L., Lee, R.C.-T.: Symbolic Logic and Mechanical Theorem Proving. Academic Press (2014)

    Google Scholar 

  12. Chen, J., Ludwig, M., Ma, Y., Walther, D.: Zooming in on ontologies: minimal modules and best excerpts. In: d’Amato, C., et al. (eds.) ISWC 2017. LNCS, vol. 10587, pp. 173–189. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68288-4_11

    CrossRef  Google Scholar 

  13. Chen, J., Ludwig, M., Ma, Y., Walther, D.: Computing minimal projection modules for \(\cal{ELH}^{r}\)-terminologies. In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 355–370. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19570-0_23

    CrossRef  Google Scholar 

  14. Chen, J., Ludwig, M., Walther, D.: On computing minimal EL-subsumption modules. In: Proceedings of the Joint Ontology Workshops 2016 Episode 2: The French Summer of Ontology co-located with the 9th International Conference on Formal Ontology in Information Systems (FOIS 2016). CEUR Workshop Proceedings, Annecy, France, 6–9 July 2016, vol. 1660. CEUR-WS.org (2016)

    Google Scholar 

  15. Chen, J., Ludwig, M., Walther, D.: Computing minimal subsumption modules of ontologies. In: Proceedings of GCAI 2018, pp. 41–53 (2018)

    Google Scholar 

  16. Chen, J., Ma, Y., Peñaloza, R., Yang, H.: Union and intersection of all justifications (extended abstract). In: Proceedings of the 34th International Workshop on Description Logics, DL 2021. CEUR Workshop Proceedings, vol. 2954. CEUR-WS.org (2021)

    Google Scholar 

  17. Cucala, D.T., Grau, B.C., Horrocks, I.: Consequence-based reasoning for description logics with disjunction, inverse roles, number restrictions, and nominals. In: Lang, J. (ed.) Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, pp. 1970–1976. ijcai.org (2018)

    Google Scholar 

  18. Cucala, D.T., Grau, B.C., Horrocks, I.: Sequoia: a consequence based reasoner for SROIQ. In: Simkus, M., Weddell, G.E. (eds.) Proceedings of the 32nd International Workshop on Description Logics. CEUR Workshop Proceedings, vol. 2373. CEUR-WS.org (2019)

    Google Scholar 

  19. Glimm, B., Horrocks, I., Motik, B., Stoilos, G., Wang, Z.: HermiT: an OWL 2 reasoner. J. Autom. Reason. 53(3), 245–269 (2014)

    CrossRef  Google Scholar 

  20. Horridge, M., Bechhofer, S.: The OWL API: a Java API for OWL ontologies. Semant. Web 2(1), 11–21 (2011)

    CrossRef  Google Scholar 

  21. Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible SROIQ. In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings of the Tenth International Conference on Principles of Knowledge Representation and Reasoning, pp. 57–67. AAAI Press (2006)

    Google Scholar 

  22. Janota, M., Marques-Silva, J.: cmMUS: a tool for circumscription-based MUS membership testing. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS (LNAI), vol. 6645, pp. 266–271. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20895-9_30

    CrossRef  Google Scholar 

  23. Kalyanpur, A., Parsia, B., Horridge, M., Sirin, E.: Finding all justifications of OWL DL entailments. In: Aberer, K., et al. (eds.) ASWC/ISWC -2007. LNCS, vol. 4825, pp. 267–280. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-76298-0_20

    CrossRef  Google Scholar 

  24. Kalyanpur, A., Parsia, B., Sirin, E., Hendler, J.: Debugging unsatisfiable classes in OWL ontologies. J. Web Semant. 3(4), 268–293 (2005)

    CrossRef  Google Scholar 

  25. Kalyanpur, A.A.: Debugging and repair of OWL ontologies. Ph.D. thesis (2006)

    Google Scholar 

  26. Kazakov, Y., Skočovský, P.: Enumerating justifications using resolution. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 609–626. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_40

    CrossRef  Google Scholar 

  27. Koopmann, P., Chen, J.: Deductive module extraction for expressive description logics. In: Bessiere, C. (ed.) Proceedings of IJCAI 2020, pp. 1636–1643. ijcai.org (2020)

    Google Scholar 

  28. Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)

    MathSciNet  CrossRef  Google Scholar 

  29. Liffiton, M.H., Sakallah, K.A.: On finding all minimally unsatisfiable subformulas. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 173–186. Springer, Heidelberg (2005). https://doi.org/10.1007/11499107_13

    CrossRef  Google Scholar 

  30. Manthey, N., Peñaloza, R., Rudolph, S.: Efficient axiom pinpointing in EL using sat technology. In: Description Logics (2016)

    Google Scholar 

  31. Manthey, N., Peñaloza, R., Rudolph, S.: SATPin: axiom pinpointing for lightweight description logics through incremental SAT. Künstliche Intell. 34(3), 389–394 (2020)

    CrossRef  Google Scholar 

  32. Ozaki, A., Peñaloza, R.: Consequence-based axiom pinpointing. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds.) SUM 2018. LNCS (LNAI), vol. 11142, pp. 181–195. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00461-3_13

    CrossRef  Google Scholar 

  33. Parsia, B., Matentzoglu, N., Gonçalves, R.S., Glimm, B., Steigmiller, A.: The OWL reasoner evaluation (ORE) 2015 competition report. J. Autom. Reason. 59, 455–482 (2015)

    MathSciNet  CrossRef  Google Scholar 

  34. Parsia, B., Sirin, E., Kalyanpur, A.: Debugging owl ontologies. In: Proceedings of the 14th International Conference on World Wide Web, pp. 633–640 (2005)

    Google Scholar 

  35. Peñaloza, R.: Axiom pinpointing. In: Cota, G., Daquino, M., Pozzato, G.L. (eds.) Applications and Practices in Ontology Design, Extraction, and Reasoning. Studies on the Semantic Web, vol. 49, pp. 162–177. IOS Press (2020)

    Google Scholar 

  36. Peñaloza, R.: Error-tolerance and error management in lightweight description logics. Künstliche Intell. 34(4), 491–500 (2020)

    CrossRef  Google Scholar 

  37. Peñaloza, R., Mencía, C., Ignatiev, A., Marques-Silva, J.: Lean kernels in description logics. In: Blomqvist, E., Maynard, D., Gangemi, A., Hoekstra, R., Hitzler, P., Hartig, O. (eds.) ESWC 2017. LNCS, vol. 10249, pp. 518–533. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58068-5_32

    CrossRef  Google Scholar 

  38. Penaloza, R., Sertkaya, B.: Understanding the complexity of axiom pinpointing in lightweight description logics. Artif. Intell. 250, 80–104 (2017)

    MathSciNet  CrossRef  Google Scholar 

  39. Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)

    MathSciNet  CrossRef  Google Scholar 

  40. Sattler, U., Schneider, T., Zakharyaschev, M.: Which kind of module should I extract? In: Proceedings of DL 2009. CEUR Workshop Proceedings, vol. 477. CEUR-WS.org (2009)

    Google Scholar 

  41. Schlobach, S., Cornet, R.: Non-standard reasoning services for the debugging of description logic terminologies. In: Gottlob, G., Walsh, T. (eds.) Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, pp. 355–362. Morgan Kaufmann (2003)

    Google Scholar 

  42. 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). https://doi.org/10.1007/978-3-642-02959-2_6

    CrossRef  Google Scholar 

  43. Simancik, F., Kazakov, Y., Horrocks, I.: Consequence-based reasoning beyond horn ontologies. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, Barcelona, Catalonia, Spain, 16–22 July 2011, pp. 1093–1098. IJCAI/AAAI (2011)

    Google Scholar 

  44. Suntisrivaraporn, B.: Polynomial time reasoning support for design and maintenance of large-scale biomedical ontologies. Ph.D. thesis, Dresden University of Technology, Germany (2009)

    Google Scholar 

  45. 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). https://doi.org/10.1007/978-3-540-89704-0_1

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jieying Chen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Chen, J., Ma, Y., Peñaloza, R., Yang, H. (2022). Union and Intersection of All Justifications. In: , et al. The Semantic Web. ESWC 2022. Lecture Notes in Computer Science, vol 13261. Springer, Cham. https://doi.org/10.1007/978-3-031-06981-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-06981-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-06980-2

  • Online ISBN: 978-3-031-06981-9

  • eBook Packages: Computer ScienceComputer Science (R0)