On the expressivity of the modal mu-calculus

  • J. C. Bradfield
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


We analyse the complexity of the sets of states, in certain classes of infinite systems, that satisfy formulae of the modal mu-calculus. Improving on some of our earlier results, we establish a strong upper bound (namely Δ 2 1 ). We also establish various lower bounds and restricted upper bounds, incidentally providing another proof that the mu-calculus alternation hierarchy does not collapse at level 2.


descriptive complexity logic in computer science verification temporal logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ArN90]
    A. Arnold and D. Niwinski, Fixed point characterization of Büchi automata on infinite trees. J. Inf. Process. Cybern., EIK 26, 451–459 (1990).Google Scholar
  2. [Bra91]
    J. C. Bradfield, Verifying Temporal Properties of Systems. Birkhäuser, Boston, Mass. ISBN 0-8176-3625-0 (1991).Google Scholar
  3. [Bra92]
    J. C. Bradfield, A proof assistant for symbolic model-checking. Proc. CAV '92. LNCS 663 316–329 (1993).Google Scholar
  4. [Bra95]
    J. C. Bradfield, On the expressivity of the modal mu-calculus, Technical Report ECS-LFCS-95-338, LFCS, University of Edinburgh (1995). On-line via http://www.dcs.ed.ac.uk/home/jcb/.Google Scholar
  5. [BrS92]
    J. C. Bradfield and C. Stirling, Local model checking for infinite state spaces. Theoret. Comput. Sci. 96 157–174 (1992).CrossRefGoogle Scholar
  6. [EmL86]
    E. A. Emerson and C.-L. Lei, Efficient model checking in fragments of the propositional mu-calculus. Proc. First IEEE Symp. on Logic in Computer Science 267–278 (1986).Google Scholar
  7. [Esp94]
    J. Esparza, On the decidability of model-checking for several μ-calculi and Petri nets, Proc. CAAP '94, LNCS 787 115–129 (1994).Google Scholar
  8. [EsN94]
    J. Esparza and M. Nielsen, Decidability issues for Petri nets — a survey, J. Inform. Process. Cybernet. 30 143–160 (1994).Google Scholar
  9. [Kai95]
    R. Kaivola, On modal mu-calculus and Büchi tree automata. Inf. Proc. Letters 54 17–22 (1995).CrossRefGoogle Scholar
  10. [Koz83]
    D. Kozen, Results on the propositional mu-calculus. Theoret. Comput. Sci. 27 333–354 (1983).CrossRefGoogle Scholar
  11. [Lar90]
    K. Larsen, Proof systems for satisfiability in Hennessy-Milner logic with recursion. Theoret. Comput. Sci. 72 265–288 (1990).CrossRefGoogle Scholar
  12. [Mos80]
    Y. N. Moschovakis, Descriptive set theory. North-Holland, Amsterdam & New York (1980).Google Scholar
  13. [Niw86]
    D. Niwiński, On fixed point clones. Proc. 13th ICALP, LNCS 226 464–473 (1986).Google Scholar
  14. [Rab70]
    M. O. Rabin, Weakly definable relations and special automata, in Y. Bar-Hillel (ed.) Mathematical Logic and Foundations of Set Theory, North-Holland, Amsterdam (1970), 1–23.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • J. C. Bradfield
    • 1
  1. 1.Laboratory for Foundations of Computer ScienceUniversity of EdinburghEdinburghUK

Personalised recommendations