370 research outputs found
Playing Stochastically in Weighted Timed Games to Emulate Memory
Weighted timed games are two-player zero-sum games played in a timed automaton equipped with integer weights. We consider optimal reachability objectives, in which one of the players, that we call Min, wants to reach a target location while minimising the cumulated weight. While knowing if Min has a strategy to guarantee a value lower than a given threshold is known to be undecidable (with two or more clocks), several conditions, one of them being the divergence, have been given to recover decidability. In such weighted timed games (like in untimed weighted games in the presence of negative weights), Min may need finite memory to play (close to) optimally. This is thus tempting to try to emulate this finite memory with other strategic capabilities. In this work, we allow the players to use stochastic decisions, both in the choice of transitions and of timing delays. We give for the first time a definition of the expected value in weighted timed games, overcoming several theoretical challenges. We then show that, in divergent weighted timed games, the stochastic value is indeed equal to the classical (deterministic) value, thus proving that Min can guarantee the same value while only using stochastic choices, and no memory
Solving Stochastic B\"uchi Games on Infinite Arenas with a Finite Attractor
We consider games played on an infinite probabilistic arena where the first
player aims at satisfying generalized B\"uchi objectives almost surely, i.e.,
with probability one. We provide a fixpoint characterization of the winning
sets and associated winning strategies in the case where the arena satisfies
the finite-attractor property. From this we directly deduce the decidability of
these games on probabilistic lossy channel systems.Comment: In Proceedings QAPL 2013, arXiv:1306.241
Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs
The winning condition of a parity game with costs requires an arbitrary, but fixed bound on the distance between occurrences of odd colors and the next occurrence of a larger even one. Such games quantitatively extend parity games while retaining most of their attractive properties, i.e, determining the winner is in NP and co-NP and one player has positional winning strategies.
We show that the characteristics of parity games with costs are vastly different when asking for strategies realizing the minimal such bound: the solution problem becomes PSPACE-complete and exponential memory is both necessary in general and always sufficient. Thus, playing parity games with costs optimally is harder than just winning them. Moreover, we show that the tradeoff between the memory size and the realized bound is gradual in general
Playing Stochastically in Weighted Timed Games to Emulate Memory
Weighted timed games are two-player zero-sum games played in a timed
automaton equipped with integer weights. We consider optimal reachability
objectives, in which one of the players, that we call Min, wants to reach a
target location while minimising the cumulated weight. While knowing if Min has
a strategy to guarantee a value lower than a given threshold is known to be
undecidable (with two or more clocks), several conditions, one of them being
the divergence, have been given to recover decidability. In such weighted timed
games (like in untimed weighted games in the presence of negative weights), Min
may need finite memory to play (close to) optimally. This is thus tempting to
try to emulate this finite memory with other strategic capabilities. In this
work, we allow the players to use stochastic decisions, both in the choice of
transitions and of timing delays. We give for the first time a definition of
the expected value in weighted timed games, overcoming several theoretical
challenges. We then show that, in divergent weighted timed games, the
stochastic value is indeed equal to the classical (deterministic) value, thus
proving that Min can guarantee the same value while only using stochastic
choices, and no memory
Reaching Your Goal Optimally by Playing at Random with No Memory
Shortest-path games are two-player zero-sum games played on a graph equipped with integer weights. One player, that we call Min, wants to reach a target set of states while minimising the total weight, and the other one has an antagonistic objective. This combination of a qualitative reachability objective and a quantitative total-payoff objective is one of the simplest settings where Min needs memory (pseudo-polynomial in the weights) to play optimally. In this article, we aim at studying a tradeoff allowing Min to play at random, but using no memory. We show that Min can achieve the same optimal value in both cases. In particular, we compute a randomised memoryless ?-optimal strategy when it exists, where probabilities are parametrised by ?. We also show that for some games, no optimal randomised strategies exist. We then characterise, and decide in polynomial time, the class of games admitting an optimal randomised memoryless strategy
Strategy Synthesis for Multi-dimensional Quantitative Objectives
Multi-dimensional mean-payoff and energy games provide the mathematical
foundation for the quantitative study of reactive systems, and play a central
role in the emerging quantitative theory of verification and synthesis. In this
work, we study the strategy synthesis problem for games with such
multi-dimensional objectives along with a parity condition, a canonical way to
express -regular conditions. While in general, the winning strategies
in such games may require infinite memory, for synthesis the most relevant
problem is the construction of a finite-memory winning strategy (if one
exists). Our main contributions are as follows. First, we show a tight
exponential bound (matching upper and lower bounds) on the memory required for
finite-memory winning strategies in both multi-dimensional mean-payoff and
energy games along with parity objectives. This significantly improves the
triple exponential upper bound for multi energy games (without parity) that
could be derived from results in literature for games on VASS (vector addition
systems with states). Second, we present an optimal symbolic and incremental
algorithm to compute a finite-memory winning strategy (if one exists) in such
games. Finally, we give a complete characterization of when finite memory of
strategies can be traded off for randomness. In particular, we show that for
one-dimension mean-payoff parity games, randomized memoryless strategies are as
powerful as their pure finite-memory counterparts.Comment: Conference version published in CONCUR 2012, LNCS 7454. Journal
version published in Acta Informatica, volume 51, issue 3-4, Springer, 201
Life Is Random, Time Is Not: Markov Decision Processes with Window Objectives
The window mechanism was introduced by Chatterjee et al. [Krishnendu Chatterjee et al., 2015] to strengthen classical game objectives with time bounds. It permits to synthesize system controllers that exhibit acceptable behaviors within a configurable time frame, all along their infinite execution, in contrast to the traditional objectives that only require correctness of behaviors in the limit. The window concept has proved its interest in a variety of two-player zero-sum games, thanks to the ability to reason about such time bounds in system specifications, but also the increased tractability that it usually yields. In this work, we extend the window framework to stochastic environments by considering the fundamental threshold probability problem in Markov decision processes for window objectives. That is, given such an objective, we want to synthesize strategies that guarantee satisfying runs with a given probability. We solve this problem for the usual variants of window objectives, where either the time frame is set as a parameter, or we ask if such a time frame exists. We develop a generic approach for window-based objectives and instantiate it for the classical mean-payoff and parity objectives, already considered in games. Our work paves the way to a wide use of the window mechanism in stochastic models
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