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The Complexity of the Graded μ-Calculus

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Automated Deduction—CADE-18 (CADE 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2392))

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Abstract

In classical logic, existential and universal quantifiers express that there exists at least one individual satisfying a formula, or that all individuals satisfy a formula. In many logics, these quantifiers have been generalized to express that, for a non-negative integer n, at least n individuals or all but n individuals satisfy a formula. In modal logics, graded modalities generalize standard existential and universal modalities in that they express, e.g., that there exist at least n accessible worlds satisfying a certain formula. Graded modalities are useful expressive means in knowledge representation; they are present in a variety of other knowledge representation formalisms closely related to modal logic.

A natural question that arises is how the generalization of the existential and universal modalities affects the satisfiability problem for the logic and its computational complexity, especially when the numbers in the graded modalities are coded in binary. In this paper we study the graded μ -calculus, which extends graded modal logic with fixed-point operators, or, equivalently, extends classical μ-calculus with graded modalities. We prove that the satisfiability problem for graded μ-calculus is EXPTIME-complete - not harder than the satisfiability problem for μ-calculus, even when the numbers in the graded modalities are coded in binary.

Supported in Part by NSF grants CCR-9988322, IIS-9908435, IIS-9978135, and EIA-0086264, and by BSF grant 9800096.

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References

  1. B. Banieqbal and H. Barringer. Temporal logic with fixed points. In Temporal Logic in Specification, volume 398 of LNCS, pages 62–74. Springer-Verlag, 1987.

    Google Scholar 

  2. G. Bhat and R. Cleaveland. Efficient local model-checking for fragments of the modal μ-calculus. In Proc. ofTACAS-96, LNCS 1055. Springer-Verlag, 1996.

    Google Scholar 

  3. F. Baader, E. Franconi, B. Hollunder, B. Nebel, and H.J. Profitlich. An empirical analysis of optimization techniques for terminological representation systems, or: Making KRIS get a move on. Applied Artificial Intelligence, 4:109–132, 1994.

    Google Scholar 

  4. F. Baader and U. Sattler. Expressive number restrictions in description logics. Journal of Logic and Computation, 9(3):319–350, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Calvanese, G. De Giacomo, and M. Lenzerini. Reasoning in expressive description logics with fixpoints based on automata on infinite trees. In IJCAI’99, 1999.

    Google Scholar 

  6. G. De Giacomo. Decidability of Class-Based Knowledge Representation Formalisms. PhD thesis, Università degli Studi di Roma “La Sapienza”, 1995.

    Google Scholar 

  7. G. De Giacomo and M. Lenzerini. Boosting the correspondence between description logics and propositional dynamic logics. In Proc. of AAAI-94, 1994.

    Google Scholar 

  8. G. De Giacomo and M. Lenzerini. Concept language with number restrictions and fixpoints, and its relationship with mu-calculus. In Proc. of ECAI-94, 1994.

    Google Scholar 

  9. F. Donini, M. Lenzerini, D. Nardi, and W. Nutt. The complexity of concept languages. In Proc. of KR-91, 1991.

    Google Scholar 

  10. E.A. Emerson, C. Jutla, and A.P. Sistla. On model-checking for fragments of μ-calculus. In Proc. 4th CAV, LNCS 697, pages 385–396. Springer-Verlag, 1993.

    Google Scholar 

  11. E.A. Emerson. Model checking and the μ-calculus. In Descriptive Complexity and Finite Models, pages 185–214. American Mathematical Society, 1997.

    Google Scholar 

  12. K. Fine. In so many possible worlds. Notre Dame Journal of Formal Logics, 13:516–520, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  13. M.J. Fischer and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and Systems Sciences, 18:194–211, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Grädel, Ph. G. Kolaitis, and M. Y. Vardi. The decision problem for 2-variable first-order logic. Bulletin of Symbolic Logic, 3:53–69, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Grädel, M. Otto, and E. Rosen. Two-variable logic with counting is decidable. In Proc. of LICS-97, 1997.

    Google Scholar 

  16. E. Grädel. On the restraining power of guards. Journal of Symbolic Logic, 64, 1999.

    Google Scholar 

  17. B. Hollunder and F. Baader. Qualifying number restrictions in concept languages. In Proc. of KR-91, pages 335–346, 1991.

    Google Scholar 

  18. V Haarslev and R. Möller. RACER system description. In Proc. of IJCAR-01, volume 2083 of LNAI. Springer-Verlag, 2001.

    Google Scholar 

  19. I. Horrocks. Using an Expressive Description Logic: FaCT or Fiction? In Proc. of KR-98, 1998.

    Google Scholar 

  20. I. Horrocks, U. Sattler, and S. Tobies. Reasoning with individuals for the description logic shiq. In Proc. of CADE-17, LNCS 1831, Germany, 2000. Springer-Verlag.

    Google Scholar 

  21. D. Janin and I. Walukiewicz. Automata for the modal μ-calculus and related results. In Proc. of MFCS-95, LNCS, pages 552–562. Springer-Verlag, 1995.

    Google Scholar 

  22. D. Kozen. Results on the propositional μ-calculus. Theoretical Computer Science, 27:333–354, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  23. O. Kupferman and M.Y. Vardi. Weak alternating automata and tree automata emptiness. In Proc. STOC-98, pages 224–233, 1998.

    Google Scholar 

  24. O. Kupferman, M.Y. Vardi, and P. Wolper. An automata-theoretic approach to branching-time model checking. Journal of the ACM, 47(2):312–360, March 2000.

    Google Scholar 

  25. R. E. Ladner. The computational complexity of provability in systems of modal propositional logic. SIAM Journal of Control and Optimization, 6(3):467–480, 1977.

    MATH  MathSciNet  Google Scholar 

  26. N. Lynch. Log space recognition and translation of parenthesis languages. Journal of the ACM, 24:583–590, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  27. D.E. Muller and P.E. Schupp. Alternating automata on infinite trees. Theoretical Computer Science, 54:267–276, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  28. D.E. Muller and P.E. Schupp. Simulating alternating tree automata by nondeter-ministic automata: New results and new proofs of theorems of Rabin, McNaughton and Safra. Theoretical Computer Science, 141:69–107, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  29. P. Patel-Schneider, D. McGuinness, R. Brachman, L. Resnick, and A. Borgida. The CLASSIC knowledge representation system: Guiding principles and implementation rationale. SIGART Bulletin, 2(3): 108–113, 1991.

    Article  Google Scholar 

  30. L. Pacholski, W. Szwast, and L. Tendera. Complexity results for first-order two-variable logic with counting. SIAM Journal of Computing, 29(4): 1083–1117, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Safra. Complexity of automata on infinite objects. PhD thesis, Weizmann Institute of Science, Rehovot, Israel, 1989.

    Google Scholar 

  32. K. Schild. Terminological cycles and the propositional μ-calculus. In Proc. of KR-94, pages 509–520. Morgan Kaufmann, 1994.

    Google Scholar 

  33. R.S. Streett and E.A. Emerson. An automata theoretic decision procedure for the propositional μ-calculus. Information and Computation, 81(3):249–264, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. Thomas. Automata on infinite objects. In J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 165–191. North Holland, 1990.

    Google Scholar 

  35. W. Thomas. Languages, automata, and logic. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Language Theory, volume III, pages 389–455, 1997.

    Google Scholar 

  36. S. Tobies. The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. Journal of Artificial Intelligence Research, 12:199–217, 2000.

    MATH  MathSciNet  Google Scholar 

  37. S. Tobies. PSPACE reasoning for graded modal logics. Journal of Logic and Computation, 11(1):85–106, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  38. M.Y. Vardi. What makes modal logic so robustly decidable? In Descriptive Complexity and Finite Models, pages 149–183. American Mathematical Society, 1997.

    Google Scholar 

  39. W van der Hoek and M. De Rijke. Counting objects. Journal of Logic and Computation, 5(3):325–345, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  40. M.Y. Vardi and P. Wolper. Automata-theoretic techniques for modal logics of programs. Journal of Computer and System Science, 32(2): 182–221, 1986.

    Article  MathSciNet  Google Scholar 

  41. I. Walukiewicz. Monadic second order logic on tree-like structures. In Proc. of STACS-96, LNCS, pages 401–413. Springer-Verlag, 1996.

    Google Scholar 

  42. T. Wilke. CTL+ is exponentially more succinct than CTL. In Proc. of FSTTCS-99, volume 1738 of LNCS, pages 110–121. Springer-Verlag, 1999.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel

    Orna Kupferman

  2. Institut für Theoretische Informatik, TU Dresden, Germany

    Ulrike Sattler

  3. Department of Computer Science, Rice University, Houston, TX, 77251-1892, USA

    Moshe Y. Vardi

Authors
  1. Orna Kupferman
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  2. Ulrike Sattler
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  3. Moshe Y. Vardi
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Manchester, Oxford Rd, Manchester, M13 9PL, UK

    Andrei Voronkov

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Kupferman, O., Sattler, U., Vardi, M.Y. (2002). The Complexity of the Graded μ-Calculus. In: Voronkov, A. (eds) Automated Deduction—CADE-18. CADE 2002. Lecture Notes in Computer Science(), vol 2392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45620-1_34

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  • DOI: https://doi.org/10.1007/3-540-45620-1_34

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