TY - CONF AB - Graph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first player is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a meanpayoff condition). There are two variants of the problem, namely, the threshold problem where the quantitative goal is to ensure that the mean-payoff value is above a threshold, and the value problem where the quantitative goal is to ensure the optimal mean-payoff value; in both cases ensuring the qualitative parity objective. The previous best-known algorithms for game graphs with n vertices, m edges, parity objectives with d priorities, and maximal absolute reward value W for mean-payoff objectives, are as follows: O(nd+1 . m . w) for the threshold problem, and O(nd+2 · m · W) for the value problem. Our main contributions are faster algorithms, and the running times of our algorithms are as follows: O(nd-1 · m ·W) for the threshold problem, and O(nd · m · W · log(n · W)) for the value problem. For mean-payoff parity objectives with two priorities, our algorithms match the best-known bounds of the algorithms for mean-payoff games (without conjunction with parity objectives). Our results are relevant in synthesis of reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective). AU - Chatterjee, Krishnendu AU - Henzinger, Monika H AU - Svozil, Alexander ID - 552 SN - 978-395977046-0 T2 - Leibniz International Proceedings in Informatics TI - Faster algorithms for mean-payoff parity games VL - 83 ER - TY - CONF AB - We consider two player, zero-sum, finite-state concurrent reachability games, played for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. Player 1 wins iff a designated goal state is eventually visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed. Our main results are as follows: We show that: (i) the optimal bound on the patience of optimal and -optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary. AU - Chatterjee, Krishnendu AU - Hansen, Kristofer AU - Ibsen-Jensen, Rasmus ID - 553 SN - 978-395977046-0 T2 - Leibniz International Proceedings in Informatics TI - Strategy complexity of concurrent safety games VL - 83 ER - TY - JOUR AB - In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14, 1477–1500 (doi:10.4310/CMS.2016.v14. n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ? {4, 5, . . .}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d = 2) and three (d = 3) space dimensions. AU - Gerencser, Mate AU - Jentzen, Arnulf AU - Salimova, Diyora ID - 560 IS - 2207 JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences SN - 13645021 TI - On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions VL - 473 ER - TY - BOOK AB - This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality. AU - Erdös, László AU - Yau, Horng ID - 567 SN - 9-781-4704-3648-3 TI - A Dynamical Approach to Random Matrix Theory VL - 28 ER - TY - JOUR AB - We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C). AU - Franek, Peter AU - Krcál, Marek ID - 568 IS - 2 JF - Homology, Homotopy and Applications SN - 15320073 TI - Persistence of zero sets VL - 19 ER -