Workshop on Logic Programming

WLP 2013: Declarative Programming and Knowledge Management pp 65-82 | Cite as

On Axiomatic Rejection for the Description Logic \(\mathcal {ALC}\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8439)

Abstract

Traditional proof calculi are mainly studied for formalising the notion of valid inference, i.e., they axiomatise the valid sentences of a logic. In contrast, the notion of invalid inference received less attention. Logical calculi which axiomatise invalid sentences are commonly referred to as complementary calculi or rejection systems. Such calculi provide a proof-theoretic account for deriving non-theorems from other non-theorems and are applied, in particular, for specifying proof systems for nonmonotonic logics. In this paper, we present a sound and complete sequent-type rejection system which axiomatises concept non-subsumption for the description logic \(\mathcal {ALC}\). Description logics are well-known knowledge-representation languages formalising ontological reasoning and provide the logical underpinning for semantic-web reasoning. We also discuss the relation of our calculus to a well-known tableau procedure for \(\mathcal {ALC}\). Although usually tableau calculi are syntactic variants of standard sequent-type systems, for \(\mathcal {ALC}\) it turns out that tableaux are rather syntactic counterparts of complementary sequent-type systems. As a consequence, counter models for witnessing concept non-subsumption can easily be obtained from a rejection proof. Finally, by the well-known relationship between \(\mathcal {ALC}\) and multi-modal logic \(\mathbf {K}\), we also obtain a complementary sequent-type system for the latter logic, generalising a similar calculus for standard \(\mathbf {K}\) as introduced by Goranko.

References

  1. 1.
    Łukasiewicz, J.: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. Clarendon Press, Oxford (1957)Google Scholar
  2. 2.
    Bonatti, P.A.: A Gentzen system for non-theorems. Technical report CD-TR 93/52, Technische Universität Wien, Institut für Informationssysteme (1993)Google Scholar
  3. 3.
    Tiomkin, M.L.: Proving unprovability. In: Proceedings of the LICS ’88, pp. 22–26. IEEE Computer Society (1988)Google Scholar
  4. 4.
    Dyckhoff, R.: Contraction-free sequent calculi for intuitionistic logic. J. Symbolic Logic 57(3), 795–807 (1992)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Kreisel, G., Putnam, H.: Eine Unableitbarkeitsbeweismethode für den Intuitionistischen Aussagenkalkül. Archiv für Mathematische Logik und Grundlagenforschung 3(1–2), 74–78 (1957)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Goranko, V.: Refutation systems in modal logic. Studia Logica 53, 299–324 (1994)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Skura, T.: Refutations and proofs in S4. In: Wansing, H. (ed.) Proof Theory of Modal Logic, pp. 45–51. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  8. 8.
    Oetsch, J., Tompits, H.: Gentzen-type refutation systems for three-valued logics with an application to disproving strong equivalence. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS, vol. 6645, pp. 254–259. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  9. 9.
    Wybraniec-Skardowska, U.: On the notion and function of the rejection of propositions. Acta Universitatis Wratislaviensis Logika 23(2754), 179–202 (2005)Google Scholar
  10. 10.
    Caferra, R., Peltier, N.: Accepting/rejecting propositions from accepted/rejected propositions: a unifying overview. Int. J. Intell. Syst. 23(10), 999–1020 (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Bonatti, P.A., Olivetti, N.: Sequent calculi for propositional nonmonotonic logics. ACM Trans. Comput. Logic 3(2), 226–278 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Reiter, R.: A logic for default reasoning. Artif. Intell. 13(1–2), 81–132 (1980)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Moore, R.C.: Semantical considerations on nonmonotonic logic. In: Proceedings of the IJCAI ’83, pp. 272–279. William Kaufmann (1983)Google Scholar
  14. 14.
    McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artif. Intell. 13(1–2), 27–39 (1980)CrossRefMATHGoogle Scholar
  15. 15.
    Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, New York (2003)Google Scholar
  16. 16.
    Rademaker, A.: A Proof Theory for Description Logics. Springer, New York (2012)CrossRefMATHGoogle Scholar
  17. 17.
    Borgida, A., Franconi, E., Horrocks, I., McGuinness, D.L., Patel-Schneider, P.F.: Explaining \(\cal ALC\) subsumption. In: Proceedings of the DL ’99. CEUR Workshop Proceedings, vol. 22 (1999)Google Scholar
  18. 18.
    Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Studia Logica 69, 5–40 (2001)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Takeuti, G.: Proof Theory. Studies in Logic and the Foundations of Mathematics Series. North-Holland, Amsterdam (1975)Google Scholar
  20. 20.
    Schild, K.: A Correspondence theory for terminological logics: preliminary report. In: Proceedings of the IJCAI ’91, pp. 466–471. Morgan Kaufmann Publishers Inc. (1991)Google Scholar
  21. 21.
    Horrocks, I.: The FaCT system. In: de Swart, H. (ed.) TABLEAUX 1998. LNCS (LNAI), vol. 1397, pp. 307–312. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  22. 22.
    Goranko, V., Otto, M.: Model theory of modal logic. In Blackburn, P., Wolter, F., van Benthem, J. (eds.) Handbook of Modal Logic, pp. 255–325. Elsevier (2006)Google Scholar
  23. 23.
    Fermüller, C., Leitsch, A., Hustadt, U., Tammet, T.: Resolution decision procedures. In: Robinson, A., Vorkonov, A. (eds.) Handbook of Automated Reasoning, pp. 1791–1849. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut Für Informationssysteme 184/3Technische Universität WienViennaAustria

Personalised recommendations