Journal of Automated Reasoning

, Volume 39, Issue 3, pp 351–384 | Cite as

Reasoning in Description Logics by a Reduction to Disjunctive Datalog

Article

Abstract

As applications of description logics proliferate, efficient reasoning with knowledge bases containing many assertions becomes ever more important. For such cases, we developed a novel reasoning algorithm that reduces a \(\mathcal{SHIQ}\) knowledge base to a disjunctive datalog program while preserving the set of ground consequences. Queries can then be answered in the resulting program while reusing existing and practically proven optimization techniques of deductive databases, such as join-order optimizations or magic sets. Moreover, we use our algorithm to derive precise data complexity bounds: we show that \(\mathcal{SHIQ}\) is data complete for NP, and we identify an expressive fragment of \(\mathcal{SHIQ}\) with polynomial data complexity.

Keywords

Description logics Disjunctive datalog Data complexity 

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References

  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison Wesley, Reading, MA (1995)MATHGoogle Scholar
  2. 2.
    Alsaç, G., Baral, C.: Reasoning in description logics using declarative logic programming. Technical report, Arizona State University, Arizona. http://www.public.asu.edu/~cbaral/papers/descr-logic-aaai2.pdf (2002)
  3. 3.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, Cambridge, MA (2003)MATHGoogle Scholar
  4. 4.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, MA (1998)Google Scholar
  5. 5.
    Baaz, M., Egly, U., Leitsch, A.: Normal form transformations. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I. Chapt. 5, pp. 273–333. Elsevier Science, Amsterdam (2001)Google Scholar
  6. 6.
    Bachmair, L., Ganzinger, H., Lynch, C., Snyder, W.: Basic paramodulation. Inf. Comput. 121(2), 172–192 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Beeri, C., Ramakrishnan, R.: On the power of magic. In: Proceedings of the 6th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS ’87), pp. 269–283. San Diego, CA, USA (1987)Google Scholar
  8. 8.
    Borgida, A.: On the relative expressiveness of description logics and predicate logics. Artif. Intell. 82(1–2), 353–367 (1996)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Calvanese, D., Giacomo, G.D., Lembo, D., Lenzerini, M., Rosati, R.: Data complexity of query answering in description logics. In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings of the 10th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR 2006), pp. 260–270. Lake District, UK (2006)Google Scholar
  10. 10.
    Calvanese, D., Lenzerini, M., Nardi, D.: Description logics for conceptual data modeling. In: Chomicki, J., Saake, G. (eds.) Logics for Databases and Information Systems, Chapt. 8, pp. 229–263. Kluwer, Boston, MA (1998)Google Scholar
  11. 11.
    Chen, P.P.: The entity-relationship model – toward a unified view of data. ACM Trans. Database Syst. 1(1), 9–36 (1976)CrossRefGoogle Scholar
  12. 12.
    Cumbo, C., Faber, W., Greco, G., Leone, N.: Enhancing the magic-set method for disjunctive datalog programs. In: Demoen, B., Lifschitz, V. (eds.) Proceedings of the 20th Int. Conf. on Logic Programming (ICLP 2004), LNCS, vol. 3132, pp. 371–385. Saint-Malo, France (2004)Google Scholar
  13. 13.
    Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and expressive power of logic programming. ACM Comput. Surv. 33(3), 374–425 (2001)CrossRefGoogle Scholar
  14. 14.
    Eiter, T., Gottlob, G., Mannila, H.: Disjunctive datalog. ACM Trans. Database Syst. 22(3), 364–418 (1997)CrossRefGoogle Scholar
  15. 15.
    Fitting, M.: First-order logic and theorem proving. In: Texts in Computer Science, 2nd edn. Springer, Berlin Heidelberg New York (1996)Google Scholar
  16. 16.
    Grosof, B.N., Horrocks, I., Volz, R., Decker, S.: Description logic programs: combining logic programs with description logic. In: Proceedings of the 12th Int. World Wide Web Conference (WWW 2003), pp. 48–57. Budapest, Hungary (2003)Google Scholar
  17. 17.
    Haarslev, V., Möller, R.,: RACER system description. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) Proceedings of the 1st Int. Joint Conf. on Automated Reasoning (IJCAR 2001), LNAI, vol. 2083, pp. 701–706. Siena, Italy (2001)Google Scholar
  18. 18.
    Haarslev, V., Möller, R., Turhan, A.-Y.: Exploiting pseudo models for TBox and ABox reasoning in expressive description logics. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) Proceedings of the 1st Int. Joint Conf. on Automated Reasoning (IJCAR 2001), LNAI, vol. 2083, pp. 61–75. Siena, Italy (2001)Google Scholar
  19. 19.
    Heymans, S., Vermeir, D.: Integrating semantic web reasoning and answer set programming. In: Vos, M.D., Provetti, A. (eds.) Proceedings of the 2nd Int. Workshop on Answer Set Programming, Advances in Theory and Implementation (ASP’03), CEUR Workshop Proceedings, vol. 78, pp. 194–208. Messina, Italy (2003)Google Scholar
  20. 20.
    Horrocks, I., Patel-Schneider, P.F., van Harmelen, F.: From SHIQ and RDF to OWL: the making of a web ontology language. J. Web Sem. 1(1), 7–26 (2003)Google Scholar
  21. 21.
    Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for very expressive description logics. Log. J. IGPL 8(3), 239–263 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hustadt, U., Motik, B., Sattler, U.: Reducing \(\mathcal{SHIQ}^-\) description logic to disjunctive datalog programs. In: Dubois, D., Welty, C.A., Williams, M.-A. (eds.) Proceedings of the 9th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR 2004), pp. 152–162. Whistler, Canada (2004)Google Scholar
  23. 23.
    Hustadt, U., Motik, B., Sattler, U.: A decomposition rule for decision procedures by resolution-based calculi. In: Baader, F., Voronkov, A. (eds.) Proceedings of the 11th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2004), LNAI, vol. 3452, pp. 21–35. Montevideo, Uruguay (2005a)Google Scholar
  24. 24.
    Hustadt, U., Motik, B., Sattler, U.: Data complexity of reasoning in very expressive description logics. In: Proceedings of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), pp. 466–471. Edinburgh, UK (2005b)Google Scholar
  25. 25.
    Hustadt, U., Schmidt, R.A.: On the relation of resolution and tableaux proof systems for description logics. In: Thomas, D. (ed.) Proceedings of the 16th Int. Joint Conf. on Artificial Intelligence (IJCAI ’99), pp. 110–115. Stockholm, Sweden (1999)Google Scholar
  26. 26.
    Kazakov, Y., de Nivelle, H.: A resolution decision procedure for the guarded fragment with transitive guards. In: Basin, D., Rusinowitch, M. (eds.) Proceedings of the 2nd Int. Joint Conf. on Automated Reasoning (IJCAR 2004), LNAI, vol. 3097, pp. 122–136. Cork, Ireland (2004)Google Scholar
  27. 27.
    Kazakov, Y., Motik, B.: A resolution-based decision procedure for \(\mathcal{SHIQ}\) . In: Furbach, U., Harrison, J., Shankar, N. (eds.) Proceedings of the 3rd Int. Joint Conf. on Automated Reasoning (IJCAR 2006), LNAI, vol. 4130, pp. 662–667. Seattle, WA (2006)Google Scholar
  28. 28.
    Motik, B.: Reasoning in description logics using resolution and deductive databases. Ph.D. thesis, Univesität Karlsruhe, Germany (2006)Google Scholar
  29. 29.
    Motik, B., Sattler, U.: A comparison of reasoning techniques for querying large description logic ABoxes. In: Hermann, M., Voronkov, A. (eds.) Proceedings of the 13th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2006), pp. 227–241. Phnom Penh, Cambodia (2006), (accepted for publication)Google Scholar
  30. 30.
    Nebel, B.: Terminological cycles: semantics and computational properties. In: Sowa, J.F. (ed.) Principles of Semantic Networks: Explorations in the Representation of Knowledge, pp. 331–361. Morgan Kaufmann, San Mateo, CA (1991)Google Scholar
  31. 31.
    Nieuwenhuis, R., Rubio, A.: Theorem proving with ordering and equality constrained clauses. J. Symb. Comput. 19(4), 312–351 (1995)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Nonnengart, A., Weidenbach, C.: Computing small clause normal forms. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I. Chapt. 6, pp. 335–367. Elsevier, Amsterdam (2001)Google Scholar
  33. 33.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading, MA (1993)Google Scholar
  34. 34.
    Parsia, B., Sirin, E.: Pellet: an OWL-DL reasoner. Poster, In: Proceedings of the 3rd Int. Semantic Web Conference (ISWC 2004), Hiroshima, Japan, 7–11 November (2004)Google Scholar
  35. 35.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1996)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Schaerf, A.: Query answering in concept-based knowledge representation systems: algorithms, complexity, and semantic issues. Ph.D. thesis, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza,” Italy (1994)Google Scholar
  37. 37.
    Swift, T.: Deduction in ontologies via ASP. In: Lifschitz, V., Niemelä, I. (eds.) Proceedings of the 7th Int. Conf. on Logic Programming and Nonmonotonic Reasoning (LPNMR 2004), LNCS, vol. 2923, pp. 275–288. Fort Lauderdale, FL (2004)Google Scholar
  38. 38.
    Tobies, S.: Complexity results and practical algorithms for logics in knowledge representation. Ph.D. thesis, RWTH Aachen, Germany (2001)Google Scholar
  39. 39.
    Tsarkov, D., Horrocks, I.: FaCT++ description logic reasoner: system description. In: Proceedings of the 3rd Int. Joint Conf. on Automated Reasoning (IJCAR 2006), LNAI, vol. 4130, pp. 292–297. Seattle, WA (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Department of Computer ScienceUniversity of ManchesterManchesterUK

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