The Complexity of the Graded μ-Calculus

  • Orna Kupferman
  • Ulrike Sattler
  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2392)


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.


Modal Logic Description Logic Atomic Proposition Acceptance Condition Kripke Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BB87]
    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. [BC96]
    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. [BFH+94]
    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. [BS99]
    F. Baader and U. Sattler. Expressive number restrictions in description logics. Journal of Logic and Computation, 9(3):319–350, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [CDL99]
    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. [De95]
    G. De Giacomo. Decidability of Class-Based Knowledge Representation Formalisms. PhD thesis, Università degli Studi di Roma “La Sapienza”, 1995.Google Scholar
  7. [DL94a]
    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. [DL94b]
    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. [DLNdN91]
    F. Donini, M. Lenzerini, D. Nardi, and W. Nutt. The complexity of concept languages. In Proc. of KR-91, 1991.Google Scholar
  10. [EJS93]
    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. [Eme97]
    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. [Fin72]
    K. Fine. In so many possible worlds. Notre Dame Journal of Formal Logics, 13:516–520, 1972.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [FL79]
    M.J. Fischer and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and Systems Sciences, 18:194–211, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [GKV97]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [GOR97]
    E. Grädel, M. Otto, and E. Rosen. Two-variable logic with counting is decidable. In Proc. of LICS-97, 1997.Google Scholar
  16. [Grä99]
    E. Grädel. On the restraining power of guards. Journal of Symbolic Logic, 64, 1999.Google Scholar
  17. [HB91]
    B. Hollunder and F. Baader. Qualifying number restrictions in concept languages. In Proc. of KR-91, pages 335–346, 1991.Google Scholar
  18. [HM01]
    V Haarslev and R. Möller. RACER system description. In Proc. of IJCAR-01, volume 2083 of LNAI. Springer-Verlag, 2001.Google Scholar
  19. [Hor98]
    I. Horrocks. Using an Expressive Description Logic: FaCT or Fiction? In Proc. of KR-98, 1998.Google Scholar
  20. [HST00]
    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. [JW95]
    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. [Koz83]
    D. Kozen. Results on the propositional μ-calculus. Theoretical Computer Science, 27:333–354, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [KV98]
    O. Kupferman and M.Y. Vardi. Weak alternating automata and tree automata emptiness. In Proc. STOC-98, pages 224–233, 1998.Google Scholar
  24. [KVW00]
    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. [Lad77]
    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.zbMATHMathSciNetGoogle Scholar
  26. [Lyn77]
    N. Lynch. Log space recognition and translation of parenthesis languages. Journal of the ACM, 24:583–590, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [MS87]
    D.E. Muller and P.E. Schupp. Alternating automata on infinite trees. Theoretical Computer Science, 54:267–276, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [MS95]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [PSMB+91]
    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.CrossRefGoogle Scholar
  30. [PST00]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [Saf89]
    S. Safra. Complexity of automata on infinite objects. PhD thesis, Weizmann Institute of Science, Rehovot, Israel, 1989.Google Scholar
  32. [Sch94]
    K. Schild. Terminological cycles and the propositional μ-calculus. In Proc. of KR-94, pages 509–520. Morgan Kaufmann, 1994.Google Scholar
  33. [SE89]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Tho90]
    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. [Tho97]
    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. [Tob00]
    S. Tobies. The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. Journal of Artificial Intelligence Research, 12:199–217, 2000.zbMATHMathSciNetGoogle Scholar
  37. [Tob01]
    S. Tobies. PSPACE reasoning for graded modal logics. Journal of Logic and Computation, 11(1):85–106, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Var97]
    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. [vD95]
    W van der Hoek and M. De Rijke. Counting objects. Journal of Logic and Computation, 5(3):325–345, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  40. [VW86]
    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.CrossRefMathSciNetGoogle Scholar
  41. [Wal96]
    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. [Wil99]
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Orna Kupferman
    • 1
  • Ulrike Sattler
    • 2
  • Moshe Y. Vardi
    • 3
  1. 1.School of Computer Science and EngineeringHebrew UniversityJerusalemIsrael
  2. 2.Institut für Theoretische InformatikTU DresdenGermany
  3. 3.Department of Computer ScienceRice UniversityHoustonUSA

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