18 research outputs found
Church Synthesis Problem for Noisy Input
Abstract. We study two variants of infinite games with imperfect in-formation. In the first variant, in each round player-1 may decide to hide his move from player-2. This captures situations where the input signal is subject to fluctuations (noises), and every error in the input signal can be detected by the controller. In the second variant, all of player-1 moves are visible to player-2; however, after the game ends, player-1 may change some of his moves. This captures situations where the input signal is subject to fluctuations; however, the controller cannot detect errors in the input signal. We consider several cases, according to the amount of errors allowed in the input signal: a fixed number of errors, finitely many errors and the case where the rate of errors is bounded by a threshold. For each of these cases we consider games with regular and mean-payoff winning conditions. We investigate the decidability of these games. There is a natural reduction for some of these games to (perfect infor-mation) multidimensional mean-payoff games recently considered in [6]. However, the decidability of the winner of multidimensional mean-payoff games was stated as an open question. We prove its decidability and provide tight complexity bounds.
The Complexity of Nash Equilibria in Limit-Average Games
We study the computational complexity of Nash equilibria in concurrent games
with limit-average objectives. In particular, we prove that the existence of a
Nash equilibrium in randomised strategies is undecidable, while the existence
of a Nash equilibrium in pure strategies is decidable, even if we put a
constraint on the payoff of the equilibrium. Our undecidability result holds
even for a restricted class of concurrent games, where nonzero rewards occur
only on terminal states. Moreover, we show that the constrained existence
problem is undecidable not only for concurrent games but for turn-based games
with the same restriction on rewards. Finally, we prove that the constrained
existence problem for Nash equilibria in (pure or randomised) stationary
strategies is decidable and analyse its complexity.Comment: 34 page
On the Complexity of Heterogeneous Multidimensional Games
We study two-player zero-sum turn-based games played on multidimensional weighted graphs with heterogeneous quantitative objectives. Our objectives are defined starting from the measures Inf, Sup, LimInf, and LimSup of the weights seen along the play, as well as on the window mean-payoff (WMP) measure recently introduced in [Krishnendu,Doyen,Randour,Raskin, Inf. Comput., 2015]. Whereas multidimensional games with Boolean combinations of classical mean-payoff objectives are undecidable [Velner, FOSSACS, 2015], we show that CNF/DNF Boolean combinations for heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} lead to EXPTIME-completeness with exponential memory strategies for both players. We also identify several interesting fragments with better complexities and memory requirements, and show that some of them are solvable in PTIME
Robust Multidimensional Mean-Payoff Games are Undecidable
Mean-payoff games play a central role in quantitative synthesis and
verification. In a single-dimensional game a weight is assigned to every
transition and the objective of the protagonist is to assure a non-negative
limit-average weight. In the multidimensional setting, a weight vector is
assigned to every transition and the objective of the protagonist is to satisfy
a boolean condition over the limit-average weight of each dimension, e.g.,
\LimAvg(x_1) \leq 0 \vee \LimAvg(x_2)\geq 0 \wedge \LimAvg(x_3) \geq 0. We
recently proved that when one of the players is restricted to finite-memory
strategies then the decidability of determining the winner is inter-reducible
with Hilbert's Tenth problem over rationals (a fundamental long-standing open
problem). In this work we allow arbitrary (infinite-memory) strategies for both
players and we show that the problem is undecidable
Hyperplane Separation Technique for Multidimensional Mean-Payoff Games
We consider both finite-state game graphs and recursive game graphs (or
pushdown game graphs), that can model the control flow of sequential programs
with recursion, with multi-dimensional mean-payoff objectives. In pushdown
games two types of strategies are relevant: global strategies, that depend on
the entire global history; and modular strategies, that have only local memory
and thus do not depend on the context of invocation. We present solutions to
several fundamental algorithmic questions and our main contributions are as
follows: (1) We show that finite-state multi-dimensional mean-payoff games can
be solved in polynomial time if the number of dimensions and the maximal
absolute value of the weight is fixed; whereas if the number of dimensions is
arbitrary, then problem is already known to be coNP-complete. (2) We show that
pushdown graphs with multi-dimensional mean-payoff objectives can be solved in
polynomial time. (3) For pushdown games under global strategies both single and
multi-dimensional mean-payoff objectives problems are known to be undecidable,
and we show that under modular strategies the multi-dimensional problem is also
undecidable (whereas under modular strategies the single dimensional problem is
NP-complete). We show that if the number of modules, the number of exits, and
the maximal absolute value of the weight is fixed, then pushdown games under
modular strategies with single dimensional mean-payoff objectives can be solved
in polynomial time, and if either of the number of exits or the number of
modules is not bounded, then the problem is NP-hard. (4) Finally we show that a
fixed parameter tractable algorithm for finite-state multi-dimensional
mean-payoff games or pushdown games under modular strategies with
single-dimensional mean-payoff objectives would imply the solution of the
long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282
On the complexity of heterogeneous multidimensional quantitative games
In this paper, we study two-player zero-sum turn-based games played on a
finite multidimensional weighted graph. In recent papers all dimensions use the
same measure, whereas here we allow to combine different measures. Such
heterogeneous multidimensional quantitative games provide a general and natural
model for the study of reactive system synthesis. We focus on classical
measures like the Inf, Sup, LimInf, and LimSup of the weights seen along the
play, as well as on the window mean-payoff (WMP) measure. This new measure is a
natural strengthening of the mean-payoff measure. We allow objectives defined
as Boolean combinations of heterogeneous constraints. While multidimensional
games with Boolean combinations of mean-payoff constraints are undecidable, we
show that the problem becomes EXPTIME-complete for DNF/CNF Boolean combinations
of heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} and that
exponential memory strategies are sufficient for both players to win. We
provide a detailed study of the complexity and the memory requirements when the
Boolean combination of the measures is replaced by an intersection.
EXPTIME-completeness and exponential memory strategies still hold for the
intersection of measures in {WMP, Inf, Sup, LimInf, LimSup}, and we get
PSPACE-completeness when WMP measure is no longer considered. To avoid
EXPTIME-or PSPACE-hardness, we impose at most one occurrence of WMP measure and
fix the number of Sup measures, and we propose several refinements (on the
number of occurrences of the other measures) for which we get polynomial
algorithms and lower memory requirements. For all the considered classes of
games, we also study parameterized complexity