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A Lightweight Defeasible Description Logic in Depth

Quantification in Rational Reasoning and Beyond

Abstract

In this thesis we study KLM-style rational reasoning in defeasible Description Logics. We illustrate that many recent approaches to derive consequences under Rational Closure (and its stronger variants, lexicographic and relevant closure) suffer the fatal drawback of neglecting defeasible information in quantified concepts. We propose novel model-theoretic semantics that are able to derive the missing entailments in two differently strong flavours. Our solution introduces a preference relation to distinguish sets of models in terms of their typicality (amount of defeasible information derivable for quantified concepts). The semantics defined through the most typical (most preferred) sets of models are proven superior to previous approaches in that their entailments properly extend previously derivable consequences, in particular, allowing to derive defeasible consequences for quantified concepts. The dissertation concludes with an algorithmic characterisation of this uniform maximisation of typicality, which accompanies our investigation of the computational complexity for deriving consequences under these new semantics.

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Fig. 1

Notes

  1. 1.

    For a detailed introduction to DLs we refer to [2].

  2. 2.

    These advanced closure operators are alleviating the drawback of Rational Closure known as inheritance blocking.

  3. 3.

    Formally, an inclusion \(BrokenWing\sqcap Fly\sqsubseteq \bot\) is required to imply inconsistency of tweety with Bird  Fly.

  4. 4.

    Formally, the set of all typicality models is finite, hence there are no infinite chains through \(<_t\).

References

  1. 1.

    Baader F, Brandt S, Lutz C (2005) Pushing the \({\cal{EL}}\) envelope. In: Proceedings of the 19th int. joint conference on artificial intelligence, IJCAI, pp 364–369

  2. 2.

    Baader F, Horrocks I, Lutz C, Sattler U (2017) An introduction to description logic. Cambridge University Press, Cambridge

    Book  Google Scholar 

  3. 3.

    Bonatti PA, Faella M, Petrova IM, Sauro L (2015) A new semantics for overriding in description logics. Artif Intell 222:1–48. https://doi.org/10.1016/j.artint.2014.12.010

    MathSciNet  Article  Google Scholar 

  4. 4.

    Casini G, Meyer T, Moodley K, Nortje R (2014) Relevant closure: a new form of defeasible reasoning for description logics. In: Logics in artificial intelligence—14th European conference, JELIA, pp 92–106. https://doi.org/10.1007/978-3-319-11558-0_7

  5. 5.

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

    Article  Google Scholar 

  6. 6.

    Casini G, Straccia U (2010) Rational closure for defeasible description logics. In: Logics in artificial intelligence—12th European conference, JELIA, pp 77–90. https://doi.org/10.1007/978-3-642-15675-5_9

  7. 7.

    Casini G, Straccia U (2012) Lexicographic closure for defeasible description logics. In: Proceedings of the 8th Australasian ontology workshop, pp 28–39

  8. 8.

    Casini G, Straccia U (2013) Defeasible inheritance-based description logics. J Artif Intell Res 48:415–473. https://doi.org/10.1613/jair.4062

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Giordano L, Gliozzi V, Olivetti N, Pozzato GL (2015) Semantic characterization of rational closure: from propositional logic to description logics. Artif Intell 226:1–33. https://doi.org/10.1016/j.artint.2015.05.001

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Kraus S, Lehmann DJ, Magidor M (1990) Nonmonotonic reasoning, preferential models and cumulative logics. Artif Intell 44(1–2):167–207. https://doi.org/10.1016/0004-3702(90)90101-5

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Lehmann DJ, Magidor M (1992) What does a conditional knowledge base entail? Artif Intell 55(1):1–60. https://doi.org/10.1016/0004-3702(92)90041-U

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Peñaloza R, Sertkaya B (2017) Understanding the complexity of axiom pinpointing in lightweight description logics. Artif Intell 250:80–104. https://doi.org/10.1016/j.artint.2017.06.002

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Pensel M (2019) A lightweight defeasible description logic in depth—quantification in rational reasoning and beyond. Ph.D. thesis, Dresden University of Technology, Germany

  14. 14.

    Pensel M, Turhan AY (2017) Including quantification in defeasible reasoning for the description logic \({\cal{EL}}_{\bot }\). In: Logic programming and nonmonotonic reasoning, 14th int. conference, LPNMR. Springer, pp 78–84. https://doi.org/10.1007/978-3-319-61660-5_9

  15. 15.

    Pensel M, Turhan AY (2017) Making quantification relevant again—the case of defeasible \({\cal{EL}}_{\bot }\). In: Proceedings of the 4th int. workshop on defeasible and ampliative reasoning—DARe. CEUR-WS.org

  16. 16.

    Pensel M, Turhan AY (2018) Reasoning in the defeasible description logic \({\cal{EL}}_{\bot }\)—computing standard inferences under rational and relevant semantics. Int J Approx Reason 103:28–70. https://doi.org/10.1016/j.ijar.2018.08.005

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Rector AL, Bechhofer S, Goble CA, Horrocks I, Nowlan A, Solomon D (1997) The grail concept modelling language for medical terminology. Artif Intell Med 9(2):139–171

    Article  Google Scholar 

  18. 18.

    Spackman K (2000) Managing clinical terminology hierarchies using algorithmic calculation of subsumption: experience with snomed-rt. J Am Med Inform Assoc, Fall Symposium Special Issue

  19. 19.

    Varzinczak IJ (2018) A note on a description logic of concept and role typicality for defeasible reasoning over ontologies. Log Univers 12(3–4):297–325. https://doi.org/10.1007/s11787-018-0211-x

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the German Research Foundation (DFG) in GRK 1763—Quantitative Logics and Automata.

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Correspondence to Maximilian Pensel.

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Pensel, M. A Lightweight Defeasible Description Logic in Depth. Künstl Intell 34, 527–531 (2020). https://doi.org/10.1007/s13218-020-00644-z

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