Abstract
We introduce a general method for proving measurability of topologically complex sets by establishing a correspondence between the notion of game tree languages from automata theory and the σ-algebra of \(\mathcal{R}\)-sets, introduced by A. Kolmogorov as a foundation for measure theory. We apply the method to answer positively to an open problem regarding the game interpretation of the probabilistic μ-calculus.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Arnold, A.: The μ-calculus alternation-depth hierarchy is strict on binary trees. ITA 33(4/5), 329–340 (1999)
Arnold, A., Niwiński, D.: Continuous separation of game languages. Fundamenta Informaticae 81(1-3), 19–28 (2007)
Baier, C., Katoen, J.-P.: Principles of Model Checking. The MIT Press (2008)
Blackwell, D.: Borel–programmable functions. Ann. of Probability 6, 321–324 (1978)
Bradfield, J.: The modal mu-calculus alternation hierarchy is strict. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 233–246. Springer, Heidelberg (1996)
Bradfield, J.C.: Fixpoints, games and the difference hierarchy. ITA 37(1), 1–15 (2003)
Bradfield, J.C., Duparc, J., Quickert, S.: Transfinite extension of the mu-calculus. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 384–396. Springer, Heidelberg (2005)
Burgess, J.P.: Classical hierarchies from a modern standpoint. II. R-sets. Fund. Math. 115(2), 97–105 (1983)
Varacca, D., Völzer, H., Winskel, G.: Probabilistic event structures and domains. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 481–496. Springer, Heidelberg (2004)
Jech, T.: Set Theory. Springer Monographs in Mathematics. Springer (2002)
Kanoveǐ, V.G.: A. N. Kolmogorov’s ideas in the theory of operations on sets. Uspekhi Mat. Nauk. 43(6(264)), 93–128 (1988)
Kechris, A.: Classical descriptive set theory. Springer, New York (1995)
Kolmogorov, A.: Operations sur des ensembles. Mat. Sb. 35, 415–422 (1928); (in Russian, summary in French)
Kozen, D.: Results on the propositional mu-calculus. In: Theoretical Computer Science, pp. 333–354 (1983)
Kudlek, M.: Probability in Petri Nets. Fundamenta Informaticae, 67(1) (2005)
Lyapunov, A.A.: \(\mathcal R\)-sets. Trudy Mat. Inst. Steklov. 40, 3–67 (1953)
Michalewski, H., Niwiński, D.: On topological completeness of regular tree languages. In: Constable, R.L., Silva, A. (eds.) Kozen Festschrift. LNCS, vol. 7230, pp. 165–179. Springer, Heidelberg (2012)
Mio, M.: Game Semantics for Probabilistic μ-Calculi. PhD thesis, School of Informatics, University of Edinburgh (2012)
Mio, M.: Probabilistic Modal μ-Calculus with Independent product. Logical Methods in Computer Science 8(4) (2012)
Selivanowski, E.: Sur une classe d’ensembles effectifs (ensembles C). Mat. Sb. 35, 379–413 (1928) (in Russian, summary in French)
Winskel, G.: Distributed probabilistic strategies. In: Proceedings of MFPS (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
Gogacz, T., Michalewski, H., Mio, M., Skrzypczak, M. (2014). Measure Properties of Game Tree Languages. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44522-8_26
Download citation
DOI: https://doi.org/10.1007/978-3-662-44522-8_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44521-1
Online ISBN: 978-3-662-44522-8
eBook Packages: Computer ScienceComputer Science (R0)