The Hybrid μ-Calculus

  • Ulrike Sattler
  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2083)

Abstract

We present an ExpTime decision procedure for the full μ- Calculus (including converse programs) extended with nominals and a universal program, thus devising a new, highly expressive ExpTime logic. The decision procedure is based on tree automata, and makes explicit the problems caused by nominals and how to overcome them. Roughly speaking, we show how to reason in a logic lacking the tree model property using techniques for logics with the tree model property. The contribution of the paper is two-fold: we extend the family of ExpTime logics, and we present a technique to reason in the presence of nominals.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Areces, P. Blackburn, and M. Marx. The computational complexity of hybrid temporal logics. Logic Journal of the IGPL, 8(5), 2000.Google Scholar
  2. 2.
    F. Baader and B. Hollunder. A terminological knowledge representation system with complete inference algorithm. In Proc. of PDK-91, vol. 567 of LNAI. Springer-Verlag, 1991.Google Scholar
  3. 3.
    G. Bhat and R. Cleaveland. Efficient local model-checking for fragments of the modal μ-calculus. In Proc. of TACAS, vol. 1055 of LNCS. Springer-Verlag, 1996.Google Scholar
  4. 4.
    P. Blackburn. Nominal tense logic. Notre Dame Journal of Formal Logic, 34, 1993.Google Scholar
  5. 5.
    D. Calvanese, G. De Giacomo, and M. Lenzerini. Reasoning in expressive description logics with fixpoints based on automata on infinite trees. In Proc. of IJCAI’99, 1999.Google Scholar
  6. 6.
    D. Calvanese, G. De Giacomo, M. Lenzerini, D. Nardi, and R. Rosati. Description logic framework for information integration. In Proc. of KR-98, 1998.Google Scholar
  7. 7.
    D. Calvanese, M. Lenzerini, and D. Nardi. Description logics for conceptual data modeling. In Logics for Databases and Information Systems. Kluwer Academic Publisher, 1998.Google Scholar
  8. 8.
    E.M. Clarke, O. Grumberg, and K. Hamaguchi. Another look at LTL model checking. In Proc. of CAV’94, vol. 818 of LNCS, pages 415–427. Springer-Verlag, 1994.Google Scholar
  9. 9.
    G. De Giacomo and M. Lenzerini. Boosting the correspondence between description logics and propositional dynamic logics. In Proc. of AAAI-94, 1994.Google Scholar
  10. 10.
    G. De Giacomo and M. Lenzerini. Concept language with number restrictions and fixpoints, and its relationship with μ-calculus. In Proc. of ECAI-94, 1994.Google Scholar
  11. 11.
    G. De Giacomo and M. Lenzerini. Tbox and Abox reasoning in expressive description logics. In Proc. of KR-96. Morgan Kaufmann, 1996.Google Scholar
  12. 12.
    F. Donini, M. Lenzerini, D. Nardi, and W. Nutt. The complexity of concept languages. In Proc. of KR-91. Morgan Kaufmann, 1991.Google Scholar
  13. 13.
    F. M. Donini, M. Lenzerini, D. Nardi, and W. Nutt. The complexity of concept languages. Information and Computation, 134, 1997.Google Scholar
  14. 14.
    E. A. Emerson and C. S. Jutla. Tree automata, μ-calculus, and determinacy. In Proc. of FOCS-91. IEEE, 1991.Google Scholar
  15. 15.
    E. A. Emerson, C. S. Jutla, and A. P. Sistla. On model checking for fragments of the μ-calculus. In Proc. of CAV’93, vol. 697 of LNCS. Springer-Verlag, 1993.Google Scholar
  16. 16.
    D. Fensel, I. Horrocks, F. van Harmelen, S. Decker, M. Erdmann, and M. Klein. OIL in a nutshell. In Proc. EKAW-2000, vol. 1937 of LNAI, 2000. Springer-Verlag.Google Scholar
  17. 17.
    K. Fine. In so many possible worlds. Notre Dame J. of Formal Logics, 13, 1972.Google Scholar
  18. 18.
    E. Franconi and U. Sattler. A data warehouse conceptual data model for multidimensional aggregation: a preliminary report. AI✻IA Notizie, 1, 1999.Google Scholar
  19. 19.
    V. Haarslev and R. Möller. Expressive abox reasoning with number restrictions, role hierarchies, and transitively closed roles. In Proc. of KR-00, 2000.Google Scholar
  20. 20.
    Gerard J. Holzmann. The spin model checker. IEEE Trans. on Software Engineering, 23(5), 1997.Google Scholar
  21. 21.
    I. Horrocks. Using an Expressive Description Logic: FaCT or Fiction? In Proc. Of KR-98, 1998.Google Scholar
  22. 22.
    I. Horrocks, U. Sattler, and S. Tobies. Practical reasoning for very expressive description logics. Logic Journal of the IGPL, 8(3), May 2000.Google Scholar
  23. 23.
    D. Kozen. Results on the propositional μ-calculus. In Proc. of ICALP’82, vol. 140 of LNCS. Springer-Verlag, 1982.Google Scholar
  24. 24.
    O. Kupferman and M. Y. Vardi. μ-calculus synthesis. In Proc. MFCS’00, LNCS. Springer-Verlag, 2000.Google Scholar
  25. 25.
    O. Kupferman and M.Y. Vardi. Weak alternating automata and tree automata emptiness. In Proc. of STOC-98, 1998.Google Scholar
  26. 26.
    H. Levesque and R. J. Brachman. Expressiveness and tractability in knowledge representation and reasoning. Computational Intelligence, 3, 1987.Google Scholar
  27. 27.
    D. E. Muller and P. E. Schupp. Alternating automata on infinite trees. Theoretical Computer Science, 54(1-2), 1987.Google Scholar
  28. 28.
    B. Nebel. Reasoning and Revision in Hybrid Representation Systems. LNAI. Springer-Verlag, 1990.Google Scholar
  29. 29.
    P. F. Patel-Schneider and I. Horrocks. DLP and FaCT. In Proc. TABLEAUX-99, vol. 1397 of LNAI. Springer-Verlag, 1999.Google Scholar
  30. 30.
    A. Prior. Past, Present and Future. Oxford University Press, 1967.Google Scholar
  31. 31.
    A. Rector and I. Horrocks. Experience building a large, re-usable medical ontology using a description logic with transitivity and concept inclusions. In Proc. of the AAAI Spring Symposium on Ontological Engineering. AAAI Press, 1997.Google Scholar
  32. 32.
    K. Schild. A correspondence theory for terminological logics: Preliminary report. In Proc. of IJCAI-91, 1991.Google Scholar
  33. 33.
    K. Schild. Terminological cycles and the propositional μ-calculus. In Proc. Of KR-94, 1994. Morgan Kaufmann.Google Scholar
  34. 34.
    M. Schmidt-Schauβ and G. Smolka. Attributive concept descriptions with complements. Artificial Intelligence, 48(1), 1991.Google Scholar
  35. 35.
    R. S. Streett and E. A. Emerson. An automata theoretic decision procedure for the propositional μ-calculus. Information and Computation, 81(3), 1989.Google Scholar
  36. 36.
    W. Thomas. Languages, automata, and logic. In Handbook of Formal Language Theory, vol 1. Springer-Verlag, 1997.Google Scholar
  37. 37.
    S. Tobies. The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. J. of Artificial Intelligence Research, 12, 2000.Google Scholar
  38. 38.
    S. Tobies. PSPACE reasoning for graded modal logics. J. of Logic and Computation, 2001. To appear.Google Scholar
  39. 39.
    M. Y. Vardi. What makes modal logic so robustly decidable? In Descriptive Complexity and Finite Models, American Mathematical Society, 1997.Google Scholar
  40. 40.
    M. Y. Vardi. Reasoning about the past with two-way automata. In Proc. Of ICALP’98, vol. 1443 of LNCS, 1998. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Heidelberg 2001

Authors and Affiliations

  • Ulrike Sattler
    • 1
  • Moshe Y. Vardi
    • 2
  1. 1.LuFG Theor. InformatikRWTH AachenGermany
  2. 2.Department of Computer ScienceRice UniversityHoustonUSA

Personalised recommendations