Skip to main content
SpringerLink
Log in

A KLM Perspective on Defeasible Reasoning for Description Logics

  • Chapter
  • First Online:
Description Logic, Theory Combination, and All That

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11560))

Abstract

In this paper we present an approach to defeasible reasoning for the description logic \(\mathcal {ALC}\). The results discussed here are based on work done by Kraus, Lehmann and Magidor (KLM) on defeasible conditionals in the propositional case. We consider versions of a preferential semantics for two forms of defeasible subsumption, and link these semantic constructions formally to KLM-style syntactic properties via representation results. In addition to showing that the semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in description logics. With the semantics of the defeasible version of \(\mathcal {ALC}\) in place, we turn to the investigation of an appropriate form of defeasible entailment for this enriched version of \(\mathcal {ALC}\). This investigation includes an algorithm for the computation of a form of defeasible entailment known as rational closure in the propositional case. Importantly, the algorithm relies completely on classical entailment checks and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of the underlying classical \(\mathcal {ALC}\). Before concluding, we take a brief tour of some existing work on defeasible extensions of \(\mathcal {ALC}\) that go beyond defeasible subsumption.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Given \(X\subseteq \varDelta ^{\mathcal {P}}\), with \(\min _{\prec ^{\mathcal {P}}}X\) we denote the set \(\{x\in X\mid \) for every \(y\in X, y\not \prec ^{\mathcal {P}}x\}\).

References

  1. Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory Implementation and Applications, 2nd edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  2. Baader, F., Hollunder, B.: How to prefer more specific defaults in terminological default logic. In: Bajcsy, R. (ed.) Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI), pp. 669–675. Morgan Kaufmann Publishers (1993)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Baltag, A., Smets, S.: Dynamic belief revision over multi-agent plausibility models. In: van der Hoek, W., Wooldridge, M. (eds.) Proceedings of LOFT, pp. 11–24. University of Liverpool (2006)

    Google Scholar 

  5. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) Logic and the Foundations of Game and Decision Theory (LOFT7). Texts in Logic and Games, no. 3, pp. 13–60. Amsterdam University Press (2008)

    Google Scholar 

  6. Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic semantics of conditional knowledge bases. J. Log. Comput. 9(6), 873–895 (1999)

    Article  MathSciNet  Google Scholar 

  7. Bonatti, P., Faella, M., Petrova, I., Sauro, L.: A new semantics for overriding in description logics. Artif. Intell. 222, 1–48 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bonatti, P., Faella, M., Sauro, L.: Defeasible inclusions in low-complexity DLs. J. Artif. Intell. Res. 42, 719–764 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Bonatti, P., Lutz, C., Wolter, F.: The complexity of circumscription in description logic. J. Artif. Intell. Res. 35, 717–773 (2009)

    Article  MathSciNet  Google Scholar 

  10. Booth, R., Casini, G., Meyer, T., Varzinczak, I.: On the entailment problem for a logic of typicality. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI), pp. 2805–2811 (2015)

    Google Scholar 

  11. Booth, R., Meyer, T., Varzinczak, I.: PTL: a propositional typicality logic. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS (LNAI), vol. 7519, pp. 107–119. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33353-8_9

    Chapter  Google Scholar 

  12. Booth, R., Meyer, T., Varzinczak, I.: A propositional typicality logic for extending rational consequence. In: Fermé, E., Gabbay, D., Simari, G. (eds.) Trends in Belief Revision and Argumentation Dynamics, Studies in Logic - Logic and Cognitive Systems, vol. 48, pp. 123–154. King’s College Publications (2013)

    Google Scholar 

  13. Booth, R., Paris, J.: A note on the rational closure of knowledge bases with both positive and negative knowledge. J. Log. Lang. Inf. 7(2), 165–190 (1998)

    Article  MathSciNet  Google Scholar 

  14. Boutilier, C.: Conditional logics of normality: a modal approach. Artif. Intell. 68(1), 87–154 (1994)

    Article  MathSciNet  Google Scholar 

  15. Britz, K., Casini, G., Meyer, T., Moodley, K., Sattler, U., Varzinczak, I.: Theoretical foundations of defeasible description logics. Technical report. arXiv:1904.07559 [cs.AI], ArXiV (April 2019). http://arxiv.org/abs/1904.07559

  16. Britz, K., Casini, G., Meyer, T., Moodley, K., Varzinczak, I.: Ordered interpretations and entailment for defeasible description logics. Technical report, CAIR, CSIR Meraka and UKZN, South Africa (2013). http://tinyurl.com/cydd6yy

  17. Britz, K., Casini, G., Meyer, T., Varzinczak, I.: Preferential role restrictions. In: Proceedings of the 26th International Workshop on Description Logics, pp. 93–106 (2013)

    Google Scholar 

  18. Britz, K., Heidema, J., Labuschagne, W.: Semantics for dual preferential entailment. J. Philos. Log. 38, 433–446 (2009)

    Article  MathSciNet  Google Scholar 

  19. Britz, K., Heidema, J., Meyer, T.: Semantic preferential subsumption. In: Lang, J., Brewka, G. (eds.) Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR), pp. 476–484. AAAI Press/MIT Press (2008)

    Google Scholar 

  20. Britz, K., Heidema, J., Meyer, T.: Modelling object typicality in description logics. In: Nicholson, A., Li, X. (eds.) AI 2009. LNCS (LNAI), vol. 5866, pp. 506–516. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10439-8_51

    Chapter  Google Scholar 

  21. Britz, K., Meyer, T., Varzinczak, I.: Semantic foundation for preferential description logics. In: Wang, D., Reynolds, M. (eds.) AI 2011. LNCS (LNAI), vol. 7106, pp. 491–500. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25832-9_50

    Chapter  Google Scholar 

  22. Britz, K., Varzinczak, I.: Defeasible modalities. In: Proceedings of the 14th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), pp. 49–60 (2013)

    Google Scholar 

  23. Britz, K., Varzinczak, I.: Introducing role defeasibility in description logics. In: Michael, L., Kakas, A. (eds.) JELIA 2016. LNCS (LNAI), vol. 10021, pp. 174–189. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48758-8_12

    Chapter  MATH  Google Scholar 

  24. Britz, K., Varzinczak, I.: Toward defeasible \(\cal{SROIQ}\). In: Proceedings of the 30th International Workshop on Description Logics (2017)

    Google Scholar 

  25. Britz, K., Varzinczak, I.: From KLM-style conditionals to defeasible modalities, and back. J. Appl. Non-Class. Log. (JANCL) 28(1), 92–121 (2018)

    Article  MathSciNet  Google Scholar 

  26. Britz, K., Varzinczak, I.: Preferential accessibility and preferred worlds. J. Log. Lang. Inf. (JoLLI) 27(2), 133–155 (2018)

    Article  MathSciNet  Google Scholar 

  27. Britz, K., Varzinczak, I.: Rationality and context in defeasible subsumption. In: Ferrarotti, F., Woltran, S. (eds.) FoIKS 2018. LNCS, vol. 10833, pp. 114–132. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-90050-6_7

    Chapter  MATH  Google Scholar 

  28. Casini, G., Meyer, T., Moodley, K., Sattler, U., Varzinczak, I.: Introducing defeasibility into OWL ontologies. In: Arenas, M., et al. (eds.) ISWC 2015. LNCS, vol. 9367, pp. 409–426. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25010-6_27

    Chapter  Google Scholar 

  29. Casini, G., Meyer, T., Moodley, K., Varzinczak, I.: Nonmonotonic reasoning in description logics: rational closure for the ABox. In: Proceedings of the 26th International Workshop on Description Logics, pp. 600–615 (2013)

    Google Scholar 

  30. Casini, G., Straccia, U.: Rational closure for defeasible description logics. In: Janhunen, T., Niemelä, I. (eds.) JELIA 2010. LNCS (LNAI), vol. 6341, pp. 77–90. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15675-5_9

    Chapter  MATH  Google Scholar 

  31. Casini, G., Straccia, U.: Lexicographic closure for defeasible description logics. In: Proceedings of the 8th Australasian Ontology Workshop (AOW), vol. 969, pp. 4–15. CEUR Workshop Proceedings (2012)

    Google Scholar 

  32. Casini, G., Straccia, U.: Defeasible inheritance-based description logics. J. Artif. Intell. Res. (JAIR) 48, 415–473 (2013)

    Article  MathSciNet  Google Scholar 

  33. Casini, G., Meyer, T., Moodley, K., Nortjé, R.: Relevant closure: a new form of defeasible reasoning for description logics. In: Fermé, E., Leite, J. (eds.) JELIA 2014. LNCS (LNAI), vol. 8761, pp. 92–106. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11558-0_7

    Chapter  MATH  Google Scholar 

  34. Casini, G., Straccia, U.: Defeasible inheritance-based description logics. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), pp. 813–818 (2011)

    Google Scholar 

  35. Casini, G., Straccia, U., Meyer, T.: A polynomial time subsumption algorithm for nominal safe \(\cal{ELO}_\bot \) under rational closure. Inf. Sci. (2018). https://doi.org/10.1016/j.ins.2018.09.037

    Article  Google Scholar 

  36. Donini, F., Nardi, D., Rosati, R.: Description logics of minimal knowledge and negation as failure. ACM Trans. Comput. Log. 3(2), 177–225 (2002)

    Article  MathSciNet  Google Scholar 

  37. Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In: Gabbay, D., Hogger, C., Robinson, J. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp. 439–513. Oxford University Press (1994)

    Google Scholar 

  38. Gärdenfors, P., Makinson, D.: Nonmonotonic inference based on expectations. Artif. Intell. 65(2), 197–245 (1994)

    Article  MathSciNet  Google Scholar 

  39. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Preferential description logics. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 257–272. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75560-9_20

    Chapter  Google Scholar 

  40. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Reasoning about typicality in preferential description logics. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 192–205. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87803-2_17

    Chapter  MATH  Google Scholar 

  41. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.: Analytic tableaux calculi for KLM logics of nonmonotonic reasoning. ACM Trans. Comput. Log. 10(3), 18:1–18:47 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.: \(\cal{ALC}+{T}\): a preferential extension of description logics. Fundam. Inform. 96(3), 341–372 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: A minimal model semantics for nonmonotonic reasoning. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS (LNAI), vol. 7519, pp. 228–241. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33353-8_18

    Chapter  Google Scholar 

  44. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.: A non-monotonic description logic for reasoning about typicality. Artif. Intell. 195, 165–202 (2013)

    Article  MathSciNet  Google Scholar 

  45. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.: Semantic characterization of rational closure: from propositional logic to description logics. Artif. Intell. 226, 1–33 (2015)

    Article  MathSciNet  Google Scholar 

  46. Governatori, G.: Defeasible description logics. In: Antoniou, G., Boley, H. (eds.) RuleML 2004. LNCS, vol. 3323, pp. 98–112. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30504-0_8

    Chapter  Google Scholar 

  47. Grosof, B., Horrocks, I., Volz, R., Decker, S.: Description logic programs: combining logic programs with description logic. In: Proceedings of the 12th International Conference on World Wide Web (WWW), pp. 48–57. ACM (2003)

    Google Scholar 

  48. Heymans, S., Vermeir, D.: A defeasible ontology language. In: Meersman, R., Tari, Z. (eds.) OTM 2002. LNCS, vol. 2519, pp. 1033–1046. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36124-3_66

    Chapter  Google Scholar 

  49. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44, 167–207 (1990)

    Article  MathSciNet  Google Scholar 

  50. Lehmann, D.: What does a conditional knowledge base entail? In: Brachman, R., Levesque, H. (eds.) Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR), pp. 212–222 (1989)

    Google Scholar 

  51. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell. 15(1), 61–82 (1995)

    Article  MathSciNet  Google Scholar 

  52. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artif. Intell. 55, 1–60 (1992)

    Article  MathSciNet  Google Scholar 

  53. Padgham, L., Zhang, T.: A terminological logic with defaults: a definition and an application. In: Bajcsy, R. (ed.) Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI), pp. 662–668. Morgan Kaufmann Publishers (1993)

    Google Scholar 

  54. Pensel, M., Turhan, A.: Making quantification relevant again - the case of defeasible \(el_{\bot }\). In: Booth, R., Casini, G., Varzinczak, I.J. (eds.) Proceedings of the 4th International Workshop on Defeasible and Ampliative Reasoning (DARe), pp. 44–57 (2017)

    Google Scholar 

  55. Pensel, M., Turhan, A.: Reasoning in the defeasible description logic \(\cal{EL}_\bot \) - computing standard inferences under rational and relevant semantics. Int. J. Approx. Reason. 103, 28–70 (2018)

    Article  MathSciNet  Google Scholar 

  56. Qi, G., Pan, J.Z., Ji, Q.: Extending description logics with uncertainty reasoning in possibilistic logic. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol. 4724, pp. 828–839. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75256-1_72

    Chapter  MATH  Google Scholar 

  57. Quantz, J., Royer, V.: A preference semantics for defaults in terminological logics. In: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR), pp. 294–305 (1992)

    Google Scholar 

  58. Quantz, J., Ryan, M.: Preferential default description logics. Technical report, TU Berlin (1993). www.tu-berlin.de/fileadmin/fg53/KIT-Reports/r110.pdf

  59. Rott, H.: Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  60. Sengupta, K., Krisnadhi, A.A., Hitzler, P.: Local closed world semantics: grounded circumscription for OWL. In: Aroyo, L., et al. (eds.) ISWC 2011. LNCS, vol. 7031, pp. 617–632. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25073-6_39

    Chapter  Google Scholar 

  61. Shoham, Y.: Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence. MIT Press, Cambridge (1988)

    Google Scholar 

  62. Straccia, U.: Default inheritance reasoning in hybrid KL-ONE-style logics. In: Bajcsy, R. (ed.) Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI), pp. 676–681. Morgan Kaufmann Publishers (1993)

    Google Scholar 

  63. Varzinczak, I.: A note on a description logic of concept and role typicality for defeasible reasoning over ontologies. Logica Universalis 12(3–4), 297–325 (2018)

    Article  MathSciNet  Google Scholar 

  64. Xiao, G., et al.: Ontology-based data access: a survey. In: Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, 13–19 July 2018, Stockholm, Sweden, pp. 5511–5519 (2018). https://doi.org/10.24963/ijcai.2018/777

Download references

Acknowledgments

Giovanni Casini and Thomas Meyer have received funding from the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agr. No. 690974 (MIREL). The work of Thomas Meyer has been supported in part by the National Research Foundation of South Africa (grant No. UID 98019).

Author information

Authors and Affiliations

  1. CAIR, Stellenbosch University, Stellenbosch, South Africa

    Katarina Britz & Ivan Varzinczak

  2. CSC, Université du Luxembourg, Esch-sur-Alzette, Luxembourg

    Giovanni Casini

  3. CAIR, University of Cape Town, Cape Town, South Africa

    Thomas Meyer

  4. CRIL, Université d’Artois & CNRS, Lens, France

    Ivan Varzinczak

Authors
  1. Katarina Britz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Casini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ivan Varzinczak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Meyer .

Editor information

Editors and Affiliations

  1. University of Bremen, Bremen, Germany

    Carsten Lutz

  2. University of Manchester, Manchester, UK

    Uli Sattler

  3. University of Iowa, Iowa City, IA, USA

    Cesare Tinelli

  4. TU Dresden, Dresden, Germany

    Anni-Yasmin Turhan

  5. University of Liverpool, Liverpool, UK

    Frank Wolter

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Britz, K., Casini, G., Meyer, T., Varzinczak, I. (2019). A KLM Perspective on Defeasible Reasoning for Description Logics. In: Lutz, C., Sattler, U., Tinelli, C., Turhan, AY., Wolter, F. (eds) Description Logic, Theory Combination, and All That. Lecture Notes in Computer Science(), vol 11560. Springer, Cham. https://doi.org/10.1007/978-3-030-22102-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22102-7_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22101-0

  • Online ISBN: 978-3-030-22102-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation