A List by Author: Václav Brožek

e-mail:
xbrozek(a)fi.muni.cz
home page:
http://www.fi.muni.cz/~xbrozek/

Two Views on Multiple Mean-Payoff Objectives in Markov Decision Processes

by Tomáš Brázdil, Václav Brožek, Krishnendu Chatterjee, Vojtěch Forejt, Antonín Kučera, A full version of the paper presented at conference LICS 2011. April 2011, 32 pages.

FIMU-RS-2011-02. Available as Postscript, PDF.

Abstract:

We study Markov decision processes (MDPs) with multiple limit-average (or mean-payoff) functions. We consider two different objectives, namely, expectation and satisfaction objectives. Given an MDP with k reward functions, in the expectation objective the goal is to maximize the expected limit-average value, and in the satisfaction objective the goal is to maximize the probability of runs such that the limit-average value stays above a given vector. We show that under the expectation objective, in contrast to the single-objective case, both randomization and memory are necessary for strategies, and that finite-memory randomized strategies are sufficient. Under the satisfaction objective, in contrast to the single-objective case, infinite memory is necessary for strategies, and that randomized memoryless strategies are sufficient for epsilon-approximation, for all epsilon. We further prove that the decision problems for both expectation and satisfaction objectives can be solved in polynomial time and the trade-off curve (Pareto curve) can be epsilon-approximated in time polynomial in the size of the MDP and 1/epsilon, and exponential in the number of reward functions, for all epsilon>0. Our results also reveal flaws in previous work for MDPs with multiple mean-payoff functions under the expectation objective, correct the flaws and obtain improved results.

Qualitative Reachability in Stochastic BPA Games

by Václav Brožek, Tomáš Brázdil, Antonín Kučera, Jan Obdržálek, A full version of the paper presented at STACS 2009. May 2009, 37 pages.

FIMU-RS-2009-01. Available as Postscript, PDF.

Abstract:

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint ‘>0’ or ‘=1’. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in NP intersection co-NP. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.

Discounted Properties of Probabilistic Pushdown Automata

by Tomáš Brázdil, Václav Brožek, Jan Holeček, Antonín Kučera, A full version of the paper presented at LPAR 2008 September 2008, 31 pages.

FIMU-RS-2008-08. Available as Postscript, PDF.

Abstract:

We show that several basic discounted properties of probabilistic pushdown automata related both to terminating and non-terminating runs can be efficiently approximated up to an arbitrarily small given precision.

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.

Abstract:

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.