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.


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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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