Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Towards Constructive DL for Abstraction and Refinement

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

This work explores some aspects of a new and natural semantical dimension that can be accommodated within the syntax of description logics which opens up when passing from the classical truth-value interpretation to a constructive interpretation. We argue that such a strengthened interpretation is essential to represent applications with partial information adequately and to achieve consistency under abstraction as well as robustness under refinement. We introduce a constructive version of \(\mathcal{ALC}\), called \({c\mathcal{ALC}}\), for which we give a sound and complete Hilbert axiomatisation and a Gentzen tableau calculus showing finite model property and decidability.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Alechina, N., Mendler, M., de Paiva, V., Ritter, E.: Categorical and Kripke semantics for constructive S4 modal logic. In: Fribourg, L. (ed.) Proc. of Computer Science Logic 2001 (CSL 2001). Lecture Notes in Computer Science, vol. 2142, pp. 292–307. Springer, New York (2001)

    Chapter  Google Scholar 

  2. Artale, A., Franconi, E.: A survey of temporal extensions of description logics. Ann. Math. Artif. Intell. 30(1–4) (2001)

  3. Artale, A., Lutz, C., Toman, D.: A description logic of change. In: Int’l Workshop on Description Logics (DL 2006), pp. 97–108 (2006)

  4. Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  5. Bellin, G., de Paiva, V., Ritter, E.: Extended Curry-Howard correspondence for a basic constructive modal logic. In: Methods for Modalities II (2001)

  6. Benford, F.: The law of anomalous numbers. In: Proc. Amer. Phil. Soc., pp. 551–572 (1938)

  7. Borgida, A.: Diachronic description logics. In: Int’l Workshop on Description Logics (DL 2001), pp. 106–112 (2001)

  8. Botazzo, L., Ferrari, M., Fiorentini, C., Fiorino, G.: A constructive semantics for ALC. In: Int’l Workshop on Description Logics (DL 2007), pp. 219–226 (2007)

  9. Brachman, R.J., McGuinness, D.L., Patel-Schneider, P.F., Resnick, L.A., Borgida, A.: Living with CLASSIC: When and How to Use a KL-ONE-Like Language. In: Principles of Semantic Networks, 401–456, Morgan Kaufmann, (1991)

  10. Braüner, T., de Paiva, V.: Intuitionistic hybrid logic. J. Appl. Logic 4(3), 231–255 (2006) (Methods for Modalities 3 (M4M-3))

    Article  MATH  Google Scholar 

  11. Brunet, O.: A logic for partial system description. J. Log. Comput. 14(4), 507–528 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Calvanese, D., De Giacomo, G., Lenzerini, M.: Semi-structured data with constraints and incomplete information. In: Int’l Workshop on Description Logics (DL 1998) (1998)

  13. de Paiva, V.: Constructive description logics: what, why and how. In: Context Representation and Reasoning, Riva del Garda (2006)

  14. Dürig, M., Studer, Th.: Probabilistic ABox reasoning: preliminary results. In: Int’l Workshop on Description Logics (DL 2005) (2005)

  15. Ewald, W.B.: Intuitionistic tense and modal logic. J. Symb. Log. 51 (1986)

  16. Fischer-Servi, G.: Semantics for a class of intuitionistic modal calculi. In: Dalla Chiara, M.L. (ed.) Italian Studies in the Philosophy of Science, pp. 59–72. Reidel, Dordrecht (1980)

    Google Scholar 

  17. Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional Modal Logics. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  18. Hölldobler, S., Nga, N.H., Khang, T.D.: The fuzzy description logic ALCFLH. In: Int’l Workshop on Description Logics (DL 2005) (2005)

  19. Ma, Y., Hitzler, P., Lin, Z.: Paraconsistent resolution for four-valued description logics. In: Int’l Workshop on Description Logics (DL 2007) (2007)

  20. Mendler, M., de Paiva, V.: Constructive CK for contexts. In: Serafini, L., Bouquet, P. (eds.) Context Representation and Reasoning (CRR-2005). CEUR Proceedings, vol. 13 (2005) (Also presented at the Association for Symbolic Logic Annual Meeting, Stanford University, USA, 22 March 2005)

  21. Mendler, M., Scheele, S.: Towards constructive description logics for abstraction and refinement. Technical Report 77(2008), University of Bamberg (2008)

  22. Mendler, M., Scheele, S.: Exponential speedup in \(\mathcal{UL}\) subsumption checking relative to general tboxes for the constructive semantics. In: Grau, B.C., Horrocks, I., Motik, B., Sattler, U. (eds.) 22nd International Workshop on Description Logics (DL 2009). CEUR Workshop Proceedings, vol. 477. CEUR, 27–30 July 2009

  23. Odintsov, S.P., Wansing, H.: Inconsistency-tolerant description logic. Part II: a tableau algorithm for \(\mathcal{CALC}^\textsf{C}\). J. Appl. Logic 6(3), 343–360 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Paschke, A.: Typed hybrid description logic programs with order-sorted semantic web type systems on OWL and RDFS. Technical report, TU Munich (2005)

  25. Patel-Schneider, P.F.: A four-valued semantics for terminological logics. Artif. Intell. 38, 319–351 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Plotkin, G., Stirling, C.: A framework for intuitionistic modal logics. In: Theoretical Aspects of Reasoning About Knowledge. Monterey (1986)

  27. Sattler, U.: A concept language extended with different kinds of transitive roles. In: Görz, G., Hölldobler, S. (eds.) 20. Deutsche Jahrestagung für Künstliche Intelligenz, number 1137. Springer, New York (1996)

    Google Scholar 

  28. Simpson, A.K.: The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh (1994)

  29. Straccia, U.: Fuzzy ALC with fuzzy concrete domains. In: Int’l Workshop on Description Logics (DL 2005) (2005)

  30. Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, vol. 2. North-Holland, Amsterdam (1988)

    Google Scholar 

  31. Troelstra, A.S.: Realizability. In: Buss, S.R. (ed.) Handbook of Proof Theory, chapter VI, pp. 407–474. Elsevier, Amsterdam (1998)

    Chapter  Google Scholar 

  32. van Dalen, D.: Intuitionistic logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 3, chapter 4, pp. 225–339. Reidel, Dordrecht (1986)

    Google Scholar 

  33. Wijesekera, D.: Constructive modal logic I. Ann. Pure Appl. Logic 50, 271–301 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stephan Scheele.

Additional information

An extended abstract of this work has been presented at the 21th International Workshop on Description Logics (DL2008).

This work is funded by the German Research Council (DFG) as part of the project SPACMODL grant No. ME 1427/4-1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mendler, M., Scheele, S. Towards Constructive DL for Abstraction and Refinement. J Autom Reasoning 44, 207–243 (2010). https://doi.org/10.1007/s10817-009-9151-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-009-9151-8

Keywords

Navigation