The Complexity of Enriched μ-Calculi

  • Piero A. Bonatti
  • Carsten Lutz
  • Aniello Murano
  • Moshe Y. Vardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4052)


The fully enriched μ -calculus is the extension of the propositional μ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched μ-calculus is known to be decidable and ExpTime-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched μ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and ExpTime-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are ExpTime-complete, and then reducing satisfiability in the relevant logics to this problem. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPT) and fully enriched automata (FEA) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched μ-calculus extends the standard μ-calculus.


Description Logic Atomic Proposition Forest Model Kripke Structure Input Tree 
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  1. [BHS02]
    Baader, F., Horrocks, I., Sattler, U.: Description logics for the semantic web. KI – Künstliche Intelligenz, 3,(2002)Google Scholar
  2. [BM+03]
    Baader, F., McGuiness, D.L., Nardi, D., Patel-Schneider, P.: The Description Logic Handbook: Theory, implementation and applications. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  3. [BC96]
    Bhat, G., Cleaveland, R.: Efficient local model-checking for fragments of the modal mu-calculus. In: Margaria, T., Steffen, B. (eds.) TACAS 1996. LNCS, vol. 1055, pp. 107–126. Springer, Heidelberg (1996)Google Scholar
  4. [BL+06]
    P.A. Bonatti, C. Lutz, A. Murano and M.Y. Vardi. The Complexity of Enriched μ-calculi. Chair for Automata Theory, Institute for Theoretical Computer Science, Dresden University of Technology, LTCS-Report, LTCS-06-02, Germany,(2006) see
  5. [BP04]
    Bonatti, P.A., Peron, A.: On the undecidability of logics with converse, nominals, recursion and counting. Artificial Intelligence 158(1), 75–96 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CGL01]
    Calvanese, D., De Giacomo, G., Lenzerini, M.: Reasoning in expressive description logics with fixpoints based on automata on infinite trees. In: Proc. of the 16th Int. Joint Conf. on Artificial Intelligence (IJCAI 1999), pp. 84–89 (1999)Google Scholar
  7. [FL79]
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and Systems Sciences 18, 194–211 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Jut95]
    Jutla, C.S.: Determinization and memoryless winning strategies. Information and Computation 133(2), 117–134 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Koz83]
    Kozen, D.: Results on the propositional μ-calculus. Theoretical Computer Science 27, 333–354 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [KSV02]
    Vardi, M.Y., Kupferman, O., Sattler, U.: The Complexity of the Graded mgr-Calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002)Google Scholar
  11. [KVW00]
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. Journal of the ACM 47(2), 312–360 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [MS87]
    Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theoretical Computer Science 54, 267–276 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Saf89]
    Safra, S.: Complexity of automata on infinite objects. PhD thesis, Weizmann Institute of Science, Rehovot, Israel (1989)Google Scholar
  14. [SV01]
    Vardi, M.Y., Sattler, U.: The Hybrid mgr-Calculus. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 76–91. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. [Tho90]
    Thomas, W.: Automata on Infinite Objects. In: Handbook of Theoretical Computer Science, pp. 133–191 (1990)Google Scholar
  16. [Tho97]
    Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Language Theory, vol. III, pp. 389–455 (1997)Google Scholar
  17. [Var98]
    Vardi, M.Y.: Reasoning about the Past with Two-Way Automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Piero A. Bonatti
    • 1
  • Carsten Lutz
    • 2
  • Aniello Murano
    • 1
  • Moshe Y. Vardi
    • 3
  1. 1.Dipartimento di Scienze FisicheUniversità di Napoli “Federico II”NapoliItaly
  2. 2.TU Dresden, Institute for Theoretical Computer ScienceDresdenGermany
  3. 3.Dept. of Computer ScienceMicrosoft Research and Rice UniversityUSA

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