Abstract
We investigate quantitative extensions of modal logic and the modal μ-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative μ-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative μ-calculus.
Similar content being viewed by others
References
de Alfaro, L.: Quantitative verification and control via the mu-calculus. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR. LNCS, vol. 2761, pp. 102–126. Springer, Berlin (2003)
de Alfaro, L., Majumdar, R.: Quantitative solution of omega-regular games. J. Comput. Syst. Sci. 68(2), 374–397 (2004)
de Alfaro, L., Henzinger, T.A., Majumdar, R.: Discounting the future in systems theory. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP. Lecture Notes in Computer Science, vol. 2719, pp. 1022–1037. Springer, Berlin (2003)
de Alfaro, L., Faella, M., Henzinger, T.A., Majumdar, R., Stoelinga, M.: Model checking discounted temporal properties. Theor. Comput. Sci. 345(1), 139–170 (2005)
de Alfaro, L., Faella, M., Stoelinga, M.: Linear and branching system metrics. Technical Report ucsc-crl-05-01, School of Engineering, University of California, Santa Cruz (2005)
Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model-checking for fragments of μ-calculus. In: Proceedings of the 5th International Conference on Computer Aided Verification, CAV ’93, vol. 697, pp. 385–396. Springer, Berlin (1993)
Gawlitza, T., Seidl, H.: Computing game values for crash games. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA. Lecture Notes in Computer Science, vol. 4762, pp. 177–191. Springer, Berlin (2007)
Gimbert, H., Zielonka, W.: Perfect information stochastic priority games. In: Arge, L., Cachin, C., Jurdzinski, T., Tarlecki, A. (eds.) ICALP. Lecture Notes in Computer Science, vol. 4596, pp. 850–861. Springer, Berlin (2007)
Grädel, E.: Finite model theory and descriptive complexity. In: Finite Model Theory and Its Applications, pp. 125–230. Springer, Berlin (2007)
Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS. Lecture Notes in Computer Science, vol. 1770, pp. 290–301. Springer, Berlin (2000)
Martin, D.A.: Borel determinacy. Ann. Math. 102, 363–371 (1975)
McIver, A., Morgan, C.: Results on the quantitative μ-calculus qMμ. ACM Trans. Comput. Log. 8(1) (2007)
Stirling, C.: Games and modal mu-calculus. In: Margaria, T., Steffen, B. (eds.) TACAS. Lecture Notes in Computer Science, vol. 1055, pp. 298–312. Springer, Berlin (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fischer, D., Grädel, E. & Kaiser, Ł. Model Checking Games for the Quantitative μ-Calculus. Theory Comput Syst 47, 696–719 (2010). https://doi.org/10.1007/s00224-009-9201-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-009-9201-y