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 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MATHGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  8. [Jut95]
    Jutla, C.S.: Determinization and memoryless winning strategies. Information and Computation 133(2), 117–134 (1997)MATHCrossRefMathSciNetGoogle Scholar
  9. [Koz83]
    Kozen, D.: Results on the propositional μ-calculus. Theoretical Computer Science 27, 333–354 (1983)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  12. [MS87]
    Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theoretical Computer Science 54, 267–276 (1987)MATHCrossRefMathSciNetGoogle 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

Personalised recommendations