Studia Logica

, Volume 89, Issue 3, pp 343–363 | Cite as

On the Proof Theory of the Modal mu-Calculus

  • Thomas StuderEmail author


We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary.


Infinitary proof theory μ-calculus 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldwinckle, John, and Robin Cockett, ‘The proof theory of modal μ logics’, in Proc. Fixed Points in Computer Science, 2001.Google Scholar
  2. 2.
    Bradfield, Julian, and Colin Stirling, ‘Modal mu-calculi’, in Patrick Blackburn, Johan van Benthem, and Frank Wolter, (eds.), Handbook of Modal Logic, Elsevier, 2007, pp. 721–756.Google Scholar
  3. 3.
    Brotherston, James, Sequent Calculus Proof Systems for Inductive Definitions, Ph.D. thesis, University of Edinburgh, 2006.Google Scholar
  4. 4.
    Brotherston, James, and Alex Simpson, ‘Complete sequent calculi for induction and infinite descent’, in Proceedings of LICS-22, 2007, pp. 51–60.Google Scholar
  5. 5.
    Brünnler, Kai, and Martin Lange, ‘Cut-free sequent systems for temporal logics’, Journal of Logic and Algebraic Programming, (to appear).Google Scholar
  6. 6.
    Brünnler, Kai, and Thomas Studer, ‘Syntactic cut-elimination for common knowledge’, in Methods for Modalities 5, 2007.Google Scholar
  7. 7.
    Dax, Christian, Martin Hofmann, and Martin Lange, ‘A proof system for the linear time μ-calculus’, in Proc. 26th Conf. on Foundations of Software Technology and Theoretical Computer Science, FSTTCS’06, vol. 4337 of LNCS, Springer, 2006, pp. 274–285.Google Scholar
  8. 8.
    Emerson, E. Allen, and Charanjit S. Jutla, ‘The complexity of tree automata and logics of programs’, in 29th Annual Symposium on Foundations of Computer Science FOCS, IEEE, 1988, pp. 328–337.Google Scholar
  9. 9.
    Fischer Michael J., Richard E. Ladner (1979) ‘Propositional dynamic logic of regular programs’. Journal of Computing and System Science 18(2): 194–211CrossRefGoogle Scholar
  10. 10.
    Gaintzarain, Joxe, Montserrat Hermo, Paqui Lucio, Marisa Navarro, and Fernando Orejas, ‘A cut-free and invariant-free sequent calculus for PLTL’, in Jacques Duparc, and Thomas A. Henzinger, (eds.), Computer Science Logic CSL 2007, Springer, 2007, pp. 481–495.Google Scholar
  11. 11.
    Jäger Gerhard, Mathis Kretz, Thomas Studer (2007) ‘Cut-free common knowledge’. Journal of Applied Logic 5: 681–689CrossRefGoogle Scholar
  12. 12.
    Jäger, Gerhard, Mathis Kretz, and Thomas Studer, ‘Canonical completeness of infinitary mu’, Journal of Logic and Algebraic Programming, (to appear).Google Scholar
  13. 13.
    Kozen Dexter (1983) ‘Results on the propositional modal μ–calculus’. Theoretical Computer Science 27: 333–354CrossRefGoogle Scholar
  14. 14.
    Kozen Dexter (1988) ‘A finite model theorem for the propositional μ–calculus’. Studia Logica 47(3): 233–241CrossRefGoogle Scholar
  15. 15.
    Leivant, Daniel, ‘A proof theoretic methodology for propositional dynamic logic’, in Proceedings of the International Colloquium on Formalization of Programming Concepts, Springer LNCS, 1981, pp. 356–373.Google Scholar
  16. 16.
    Niwinski Damian, Igor Walukiewicz (1996) ‘Games for the mu-calculus’. Theoretical Computer Science 163(1&2): 99–116CrossRefGoogle Scholar
  17. 17.
    Santocanale, Luigi, ‘A calculus of circular proofs and its categorical semantics’, in FoSSaCS ’02: Proceedings of the 5th International Conference on Foundations of Software Science and Computation Structures, Springer, 2002, pp. 357–371.Google Scholar
  18. 18.
    Sprenger, Christoph, and Mads Dam, ‘On the structure of inductive reasoning: Circular and tree-shaped proofs in the mu-calculus’, in Proc. FOSSACS’03, Springer LNCS, 2003, pp. 425–440.Google Scholar
  19. 19.
    Walukiewicz, Igor, ‘A complete deductive system for the μ–calculus’, in Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Science Press, 1993, pp. 136–147.Google Scholar
  20. 20.
    Walukiewicz Igor (2000) ‘Completeness of Kozen’s axiomatization of the propositional μ–calculus’. Information and Computation 157: 142–182CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institut für Informatik und angewandte MathematikUniversität BernBernSwitzerland

Personalised recommendations