Technical Reports

The report FIMU-RS-2006-02

Stochastic Games with Branching-Time Winning Objectives

by Tomáš Brázdil, Václav Brožek, Vojtěch Forejt, Antonín Kučera, A full version of the paper presented at LICS 2006. September 2006, 37 pages.

FIMU-RS-2006-02. Available as Postscript, PDF.


We consider stochastic turn-based games where the winning objectives are given by formulae of the branching-time logic PCTL. These games are generally not determined and winning strategies may require memory and/or randomization. Our main results concern history-dependent strategies. In particular, we show that the problem whether there exists a history-dependent winning strategy in 1.5-player games is highly undecidable, even for objectives formulated in the L(F^{=5/8},F^{=1},F^{>0},G^{=1}) fragment of PCTL. On the other hand, we show that the problem becomes decidable (and in fact EXPTIME-complete) for the L(F^{=1},F^{>0},G^{=1}) fragment of PCTL, where winning strategies require only finite memory. This result is tight in the sense that winning strategies for L(F^{=1},F^{>0},G^{=1},G^{>0}) objectives may already require infinite memory.

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