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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Entropic risk (ERisk) is an established risk measure in finance, quantifying risk by an exponential re-weighting of rewards. We study ERisk for the first time in the context of turn-based stochastic games with the total reward objective. This gives rise to an objective function that demands the control of systems in a risk-averse manner. We show that the resulting games are determined and, in particular, admit optimal memoryless deterministic strategies. This contrasts risk measures that previously have been considered in the special case of Markov decision processes and that require randomization and/or memory. We provide several results on the decidability and the computational complexity of the threshold problem, i.e. whether the optimal value of ERisk exceeds a given threshold. In the most general case, the problem is decidable subject to Shanuel’s conjecture. If all inputs are rational, the resulting threshold problem can be solved using algebraic numbers, leading to decidability via a polynomial-time reduction to the existential theory of the reals. Further restrictions on the encoding of the input allow the solution of the threshold problem in NP∩coNP. Finally, an approximation algorithm for the optimal value of ERisk is provided.

Christel Baier, Krishnendu Chatterjee, Tobias Meggendorfer, and Jakob Piribauer. Entropic Risk for Turn-Based Stochastic Games. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{baier_et_al:LIPIcs.MFCS.2023.15, author = {Baier, Christel and Chatterjee, Krishnendu and Meggendorfer, Tobias and Piribauer, Jakob}, title = {{Entropic Risk for Turn-Based Stochastic Games}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.15}, URN = {urn:nbn:de:0030-drops-185491}, doi = {10.4230/LIPIcs.MFCS.2023.15}, annote = {Keywords: Stochastic games, risk-aware verification} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Spatial games form a widely-studied class of games from biology and physics modeling the evolution of social behavior. Formally, such a game is defined by a square (d by d) payoff matrix M and an undirected graph G. Each vertex of G represents an individual, that initially follows some strategy i ∈ {1,2,…,d}. In each round of the game, every individual plays the matrix game with each of its neighbors: An individual following strategy i meeting a neighbor following strategy j receives a payoff equal to the entry (i,j) of M. Then, each individual updates its strategy to its neighbors' strategy with the highest sum of payoffs, and the next round starts. The basic computational problems consist of reachability between configurations and the average frequency of a strategy. For general spatial games and graphs, these problems are in PSPACE. In this paper, we examine restricted setting: the game is a prisoner’s dilemma; and G is a subgraph of grid. We prove that basic computational problems for spatial games with prisoner’s dilemma on a subgraph of a grid are PSPACE-hard.

Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Complexity of Spatial Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 11:1-11:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2022.11, author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Jecker, Isma\"{e}l and Svoboda, Jakub}, title = {{Complexity of Spatial Games}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.11}, URN = {urn:nbn:de:0030-drops-174038}, doi = {10.4230/LIPIcs.FSTTCS.2022.11}, annote = {Keywords: spatial games, computational complexity, prisoner’s dilemma, dynamical systems} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Given a Markov chain M = (V, v_0, δ), with state space V and a starting state v_0, and a probability threshold ε, an ε-core is a subset C of states that is left with probability at most ε. More formally, C ⊆ V is an ε-core, iff ℙ[reach (V\C)] ≤ ε. Cores have been applied in a wide variety of verification problems over Markov chains, Markov decision processes, and probabilistic programs, as a means of discarding uninteresting and low-probability parts of a probabilistic system and instead being able to focus on the states that are likely to be encountered in a real-world run. In this work, we focus on the problem of computing a minimal ε-core in a Markov chain. Our contributions include both negative and positive results: (i) We show that the decision problem on the existence of an ε-core of a given size is NP-complete. This solves an open problem posed in [Jan Kretínský and Tobias Meggendorfer, 2020]. We additionally show that the problem remains NP-complete even when limited to acyclic Markov chains with bounded maximal vertex degree; (ii) We provide a polynomial time algorithm for computing a minimal ε-core on Markov chains over control-flow graphs of structured programs. A straightforward combination of our algorithm with standard branch prediction techniques allows one to apply the idea of cores to find a subset of program lines that are left with low probability and then focus any desired static analysis on this core subset.

Ali Ahmadi, Krishnendu Chatterjee, Amir Kafshdar Goharshady, Tobias Meggendorfer, Roodabeh Safavi, and Ðorđe Žikelić. Algorithms and Hardness Results for Computing Cores of Markov Chains. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 29:1-29:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ahmadi_et_al:LIPIcs.FSTTCS.2022.29, author = {Ahmadi, Ali and Chatterjee, Krishnendu and Goharshady, Amir Kafshdar and Meggendorfer, Tobias and Safavi, Roodabeh and \v{Z}ikeli\'{c}, Ðor{\d}e}, title = {{Algorithms and Hardness Results for Computing Cores of Markov Chains}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {29:1--29:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.29}, URN = {urn:nbn:de:0030-drops-174216}, doi = {10.4230/LIPIcs.FSTTCS.2022.29}, annote = {Keywords: Markov Chains, Cores, Complexity} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Product graphs arise naturally in formal verification and program analysis. For example, the analysis of two concurrent threads requires the product of two component control-flow graphs, and for language inclusion of deterministic automata the product of two automata is constructed. In many cases, the component graphs have constant treewidth, e.g., when the input contains control-flow graphs of programs. We consider the algorithmic analysis of products of two constant-treewidth graphs with respect to three classic specification languages, namely, (a) algebraic properties, (b) mean-payoff properties, and (c) initial credit for energy properties.
Our main contributions are as follows. Consider a graph G that is the product of two constant-treewidth graphs of size n each. First, given an idempotent semiring, we present an algorithm that computes the semiring transitive closure of G in time Õ(n⁴). Since the output has size Θ(n⁴), our algorithm is optimal (up to polylog factors). Second, given a mean-payoff objective, we present an O(n³)-time algorithm for deciding whether the value of a starting state is non-negative, improving the previously known O(n⁴) bound. Third, given an initial credit for energy objective, we present an O(n⁵)-time algorithm for computing the minimum initial credit for all nodes of G, improving the previously known O(n⁸) bound. At the heart of our approach lies an algorithm for the efficient construction of strongly-balanced tree decompositions of constant-treewidth graphs. Given a constant-treewidth graph G' of n nodes and a positive integer λ, our algorithm constructs a binary tree decomposition of G' of width O(λ) with the property that the size of each subtree decreases geometrically with rate (1/2 + 2^{-λ}).

Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Quantitative Verification on Product Graphs of Small Treewidth. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 42:1-42:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2021.42, author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas}, title = {{Quantitative Verification on Product Graphs of Small Treewidth}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {42:1--42:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.42}, URN = {urn:nbn:de:0030-drops-155533}, doi = {10.4230/LIPIcs.FSTTCS.2021.42}, annote = {Keywords: graph algorithms, algebraic paths, mean-payoff, initial credit for energy} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Graphs and games on graphs are fundamental models for the analysis of reactive systems, in particular, for model-checking and the synthesis of reactive systems. The class of ω-regular languages provides a robust specification formalism for the desired properties of reactive systems. In the classical infinitary formulation of the liveness part of an ω-regular specification, a "good" event must happen eventually without any bound between the good events. A stronger notion of liveness is bounded liveness, which requires that good events happen within d transitions. Given a graph or a game graph with n vertices, m edges, and a bounded liveness objective, the previous best-known algorithmic bounds are as follows: (i) O(dm) for graphs, which in the worst-case is O(n³); and (ii) O(n² d²) for games on graphs. Our main contributions improve these long-standing algorithmic bounds. For graphs we present: (i) a randomized algorithm with one-sided error with running time O(n^{2.5} log n) for the bounded liveness objectives; and (ii) a deterministic linear-time algorithm for the complement of bounded liveness objectives. For games on graphs, we present an O(n² d) time algorithm for the bounded liveness objectives.

Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, and Alexander Svozil. Faster Algorithms for Bounded Liveness in Graphs and Game Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 124:1-124:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chatterjee_et_al:LIPIcs.ICALP.2021.124, author = {Chatterjee, Krishnendu and Henzinger, Monika and Kale, Sagar Sudhir and Svozil, Alexander}, title = {{Faster Algorithms for Bounded Liveness in Graphs and Game Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {124:1--124:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.124}, URN = {urn:nbn:de:0030-drops-141930}, doi = {10.4230/LIPIcs.ICALP.2021.124}, annote = {Keywords: Graphs, Game Graphs, B\"{u}chi} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

A vector addition system with states (VASS) consists of a finite set of states and counters. A transition changes the current state to the next state, and every counter is either incremented, or decremented, or left unchanged. A state and value for each counter is a configuration; and a computation is an infinite sequence of configurations with transitions between successive configurations. A probabilistic VASS consists of a VASS along with a probability distribution over the transitions for each state. Qualitative properties such as state and configuration reachability have been widely studied for VASS. In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS. For a counter, the cost of a configuration is the value of the counter; and the long-run average value of a computation for the counter is the long-run average of the costs of the configurations in the computation. The multi-dimensional long-run average problem given a VASS and a threshold value for each counter, asks whether there is a computation such that for each counter the long-run average value for the counter does not exceed the respective threshold. For probabilistic VASS, instead of the existence of a computation, we consider whether the expected long-run average value for each counter does not exceed the respective threshold. Our main results are as follows: we show that the multi-dimensional long-run average problem (a) is NP-complete for integer-valued VASS; (b) is undecidable for natural-valued VASS (i.e., nonnegative counters); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating.

Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 23:1-23:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2020.23, author = {Chatterjee, Krishnendu and Henzinger, Thomas A. and Otop, Jan}, title = {{Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {23:1--23:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.23}, URN = {urn:nbn:de:0030-drops-128359}, doi = {10.4230/LIPIcs.CONCUR.2020.23}, annote = {Keywords: vector addition systems, mean-payoff, multidimension, probabilistic semantics} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.

Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Simplified Game of Life: Algorithms and Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 22:1-22:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2020.22, author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Jecker, Isma\"{e}l and Svoboda, Jakub}, title = {{Simplified Game of Life: Algorithms and Complexity}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {22:1--22:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.22}, URN = {urn:nbn:de:0030-drops-126903}, doi = {10.4230/LIPIcs.MFCS.2020.22}, annote = {Keywords: game of life, cellular automata, computational complexity, dynamical systems} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

The Price of Anarchy (PoA) is a well-established game-theoretic concept to shed light on coordination issues arising in open distributed systems. Leaving agents to selfishly optimize comes with the risk of ending up in sub-optimal states (in terms of performance and/or costs), compared to a centralized system design. However, the PoA relies on strong assumptions about agents' rationality (e.g., resources and information) and interactions, whereas in many distributed systems agents interact locally with bounded resources. They do so repeatedly over time (in contrast to "one-shot games"), and their strategies may evolve.
Using a more realistic evolutionary game model, this paper introduces a realized evolutionary Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection in dynamic distributed systems with multiple equilibria, based on local interactions of simple memoryless agents.
Considering a fundamental game related to virus propagation on networks, we present analytical bounds on the ePoA in basic network topologies and for different strategy update dynamics. In particular, deriving stationary distributions of the stochastic evolutionary process, we find that the Nash equilibria are not always the most abundant states, and that different processes can feature significant off-equilibrium behavior, leading to a significantly higher ePoA compared to the PoA studied traditionally in the literature.

Laura Schmid, Krishnendu Chatterjee, and Stefan Schmid. The Evolutionary Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 21:1-21:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{schmid_et_al:LIPIcs.OPODIS.2019.21, author = {Schmid, Laura and Chatterjee, Krishnendu and Schmid, Stefan}, title = {{The Evolutionary Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.21}, URN = {urn:nbn:de:0030-drops-118071}, doi = {10.4230/LIPIcs.OPODIS.2019.21}, annote = {Keywords: Evolutionary Games, Virus Propagation, Price of Anarchy, Analysis} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

We study Markov decision processes and turn-based stochastic games with parity conditions. There are three qualitative winning criteria, namely, sure winning, which requires all paths to satisfy the condition, almost-sure winning, which requires the condition to be satisfied with probability 1, and limit-sure winning, which requires the condition to be satisfied with probability arbitrarily close to 1. We study the combination of two of these criteria for parity conditions, e.g., there are two parity conditions one of which must be won surely, and the other almost-surely. The problem has been studied recently by Berthon et al. for MDPs with combination of sure and almost-sure winning, under infinite-memory strategies, and the problem has been established to be in NP cap co-NP. Even in MDPs there is a difference between finite-memory and infinite-memory strategies. Our main results for combination of sure and almost-sure winning are as follows: (a) we show that for MDPs with finite-memory strategies the problem is in NP cap co-NP; (b) we show that for turn-based stochastic games the problem is co-NP-complete, both for finite-memory and infinite-memory strategies; and (c) we present algorithmic results for the finite-memory case, both for MDPs and turn-based stochastic games, by reduction to non-stochastic parity games. In addition we show that all the above complexity results also carry over to combination of sure and limit-sure winning, and results for all other combinations can be derived from existing results in the literature. Thus we present a complete picture for the study of combinations of two qualitative winning criteria for parity conditions in MDPs and turn-based stochastic games.

Krishnendu Chatterjee and Nir Piterman. Combinations of Qualitative Winning for Stochastic Parity Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 6:1-6:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2019.6, author = {Chatterjee, Krishnendu and Piterman, Nir}, title = {{Combinations of Qualitative Winning for Stochastic Parity Games}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.6}, URN = {urn:nbn:de:0030-drops-109089}, doi = {10.4230/LIPIcs.CONCUR.2019.6}, annote = {Keywords: Two Player Games, Stochastic Games, Parity Winning Conditions} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

The fundamental model-checking problem, given as input a model and a specification, asks for the algorithmic verification of whether the model satisfies the specification. Two classical models for reactive systems are graphs and Markov decision processes (MDPs). A basic specification formalism in the verification of reactive systems is the strong fairness (aka Streett) objective, where given different types of requests and corresponding grants, the requirement is that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All omega-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. Consider graphs/MDPs with n vertices, m edges, and a Streett objectives with k pairs, and let b denote the size of the description of the Streett objective for the sets of requests and grants. The current best-known algorithm for the problem requires time O(min(n^2, m sqrt{m log n}) + b log n). In this work we present randomized near-linear time algorithms, with expected running time O~(m + b), where the O~ notation hides poly-log factors. Our randomized algorithms are near-linear in the size of the input, and hence optimal up to poly-log factors.

Krishnendu Chatterjee, Wolfgang Dvořák, Monika Henzinger, and Alexander Svozil. Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2019.7, author = {Chatterjee, Krishnendu and Dvo\v{r}\'{a}k, Wolfgang and Henzinger, Monika and Svozil, Alexander}, title = {{Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.7}, URN = {urn:nbn:de:0030-drops-109093}, doi = {10.4230/LIPIcs.CONCUR.2019.7}, annote = {Keywords: model checking, graph games, Streett games} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

A vector addition system with states (VASS) consists of a finite set of states and counters. A configuration is a state and a value for each counter; a transition changes the state and each counter is incremented, decremented, or left unchanged. While qualitative properties such as state and configuration reachability have been studied for VASS, we consider the long-run average cost of infinite computations of VASS. The cost of a configuration is for each state, a linear combination of the counter values. In the special case of uniform cost functions, the linear combination is the same for all states. The (regular) long-run emptiness problem is, given a VASS, a cost function, and a threshold value, if there is a (lasso-shaped) computation such that the long-run average value of the cost function does not exceed the threshold. For uniform cost functions, we show that the regular long-run emptiness problem is (a) decidable in polynomial time for integer-valued VASS, and (b) decidable but nonelementarily hard for natural-valued VASS (i.e., nonnegative counters). For general cost functions, we show that the problem is (c) NP-complete for integer-valued VASS, and (d) undecidable for natural-valued VASS. Our most interesting result is for (c) integer-valued VASS with general cost functions, where we establish a connection between the regular long-run emptiness problem and quadratic Diophantine inequalities. The general (nonregular) long-run emptiness problem is equally hard as the regular problem in all cases except (c), where it remains open.

Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Long-Run Average Behavior of Vector Addition Systems with States. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 27:1-27:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2019.27, author = {Chatterjee, Krishnendu and Henzinger, Thomas A. and Otop, Jan}, title = {{Long-Run Average Behavior of Vector Addition Systems with States}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.27}, URN = {urn:nbn:de:0030-drops-109293}, doi = {10.4230/LIPIcs.CONCUR.2019.27}, annote = {Keywords: vector addition systems, mean-payoff, Diophantine inequalities} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

Crypto-currencies are digital assets designed to work as a medium of exchange, e.g., Bitcoin, but they are susceptible to attacks (dishonest behavior of participants). A framework for the analysis of attacks in crypto-currencies requires (a) modeling of game-theoretic aspects to analyze incentives for deviation from honest behavior; (b) concurrent interactions between participants; and (c) analysis of long-term monetary gains. Traditional game-theoretic approaches for the analysis of security protocols consider either qualitative temporal properties such as safety and termination, or the very special class of one-shot (stateless) games. However, to analyze general attacks on protocols for crypto-currencies, both stateful analysis and quantitative objectives are necessary. In this work our main contributions are as follows: (a) we show how a class of concurrent mean-payoff games, namely ergodic games, can model various attacks that arise naturally in crypto-currencies; (b) we present the first practical implementation of algorithms for ergodic games that scales to model realistic problems for crypto-currencies; and (c) we present experimental results showing that our framework can handle games with thousands of states and millions of transitions.

Krishnendu Chatterjee, Amir Kafshdar Goharshady, Rasmus Ibsen-Jensen, and Yaron Velner. Ergodic Mean-Payoff Games for the Analysis of Attacks in Crypto-Currencies. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 11:1-11:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2018.11, author = {Chatterjee, Krishnendu and Kafshdar Goharshady, Amir and Ibsen-Jensen, Rasmus and Velner, Yaron}, title = {{Ergodic Mean-Payoff Games for the Analysis of Attacks in Crypto-Currencies}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.11}, URN = {urn:nbn:de:0030-drops-95497}, doi = {10.4230/LIPIcs.CONCUR.2018.11}, annote = {Keywords: Crypto-currency, Quantitative Verification, Mean-payoff Games} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

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 mean-payoff 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(n^(d+1)·m·W) for the threshold problem, and O(n^(d+2)·m·W) for the value problem.
Our main contributions are faster algorithms, and the running times of our algorithms are as follows: O(n^(d-1)·m·W) for the threshold problem, and O(n^d·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).

Krishnendu Chatterjee, Monika Henzinger, and Alexander Svozil. Faster Algorithms for Mean-Payoff Parity Games. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 39:1-39:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2017.39, author = {Chatterjee, Krishnendu and Henzinger, Monika and Svozil, Alexander}, title = {{Faster Algorithms for Mean-Payoff Parity Games}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.39}, URN = {urn:nbn:de:0030-drops-80809}, doi = {10.4230/LIPIcs.MFCS.2017.39}, annote = {Keywords: graph games, mean-payoff parity games} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

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 epsilon-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.

Krishnendu Chatterjee, Kristoffer Arnsfelt Hansen, and Rasmus Ibsen-Jensen. Strategy Complexity of Concurrent Safety Games. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 55:1-55:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2017.55, author = {Chatterjee, Krishnendu and Hansen, Kristoffer Arnsfelt and Ibsen-Jensen, Rasmus}, title = {{Strategy Complexity of Concurrent Safety Games}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {55:1--55:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.55}, URN = {urn:nbn:de:0030-drops-81203}, doi = {10.4230/LIPIcs.MFCS.2017.55}, annote = {Keywords: Concurrent games, Reachability and safety, Patience of strategies} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Evolutionary graph theory studies the evolutionary dynamics in a population structure given as a connected graph. Each node of the graph represents an individual of the population, and edges determine how offspring are placed. We consider the classical birth-death Moran process where there are two types of individuals, namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates the reproductive strength. The evolutionary dynamics happens as follows: in the initial step, in a population of all resident individuals a mutant is introduced, and then at each step, an individual is chosen proportional to the fitness of its type to reproduce, and the offspring replaces a neighbor uniformly at random. The process stops when all individuals are either residents or mutants. The probability that all individuals in the end are mutants is called the fixation probability, which is a key factor in the rate of evolution. We consider the problem of approximating the fixation probability. The class of algorithms that is extremely relevant for approximation of the fixation probabilities is the Monte-Carlo simulation of the process. Previous results present a polynomial-time Monte-Carlo algorithm for undirected graphs when $r$ is given in unary. First, we present a simple modification: instead of simulating each step, we discard ineffective steps, where no node changes type (i.e., either residents replace residents, or mutants replace mutants). Using the above simple modification and our result that the number of effective steps is concentrated around the expected number of effective steps, we present faster polynomial-time Monte-Carlo algorithms for undirected graphs. Our algorithms are always at least a factor O(n^2/log n) faster as compared to the previous algorithms, where n is the number of nodes, and is polynomial even if r is given in binary. We also present lower bounds showing that the upper bound on the expected number of effective steps we present is asymptotically tight for undirected graphs.

Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Martin A. Nowak. Faster Monte-Carlo Algorithms for Fixation Probability of the Moran Process on Undirected Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 61:1-61:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2017.61, author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Nowak, Martin A.}, title = {{Faster Monte-Carlo Algorithms for Fixation Probability of the Moran Process on Undirected Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {61:1--61:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.61}, URN = {urn:nbn:de:0030-drops-81213}, doi = {10.4230/LIPIcs.MFCS.2017.61}, annote = {Keywords: Graph algorithms, Evolutionary biology, Monte-Carlo algorithms} }

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**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

Nested weighted automata (NWA) present a robust and convenient automata-theoretic formalism for quantitative specifications.
Previous works have considered NWA that processed input words only in the forward direction. It is natural to allow the automata to process input words backwards as well, for example, to measure the maximal or average time between a response and the preceding request. We therefore introduce and study bidirectional NWA that can process input
words in both directions. First, we show that bidirectional NWA can express interesting quantitative properties that are not expressible by forward-only NWA. Second, for the fundamental decision problems of emptiness and universality, we establish decidability and complexity results for the new framework which match the best-known results for the special case of forward-only NWA. Thus, for NWA, the increased expressiveness of bidirectionality is achieved at no additional computational complexity. This is in stark contrast to the unweighted case, where bidirectional finite automata are no more expressive but exponentially more succinct than their forward-only counterparts.

Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Bidirectional Nested Weighted Automata. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 5:1-5:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2017.5, author = {Chatterjee, Krishnendu and Henzinger, Thomas A. and Otop, Jan}, title = {{Bidirectional Nested Weighted Automata}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.5}, URN = {urn:nbn:de:0030-drops-77763}, doi = {10.4230/LIPIcs.CONCUR.2017.5}, annote = {Keywords: weighted automata, nested weighted automata, complexity, bidirectional} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Graph games with omega-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify omega-regular winning conditions. Graph games with parity conditions are equivalent to mu-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators.
We consider graph games with n vertices and parity conditions with c priorities (equivalently, a mu-calculus formula with c alternations of least and greatest fixed points). While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires O(n^c) symbolic operations and linear space; and (b) an improved algorithm that requires O(n^{c/2+1}) symbolic operations but also O(n^{c/2+1}) space (i.e., exponential space).
In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires O(n^{c/2+1}) symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires O(n^{c/3+1}) symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations.

Krishnendu Chatterjee, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. Improved Set-Based Symbolic Algorithms for Parity Games. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 18:1-18:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2017.18, author = {Chatterjee, Krishnendu and Dvor\'{a}k, Wolfgang and Henzinger, Monika and Loitzenbauer, Veronika}, title = {{Improved Set-Based Symbolic Algorithms for Parity Games}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.18}, URN = {urn:nbn:de:0030-drops-76830}, doi = {10.4230/LIPIcs.CSL.2017.18}, annote = {Keywords: model checking, graph games, parity games, symbolic computation, progress measure} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We present a logic that extends CTL (Computation Tree Logic) with operators that express synchronization properties. A property is synchronized in a system if it holds in all paths of a certain length. The new logic is obtained by using the same path quantifiers and temporal operators as in CTL, but allowing a different order of the quantifiers. This small syntactic variation induces a logic that can express non-regular properties for which known extensions of MSO with equality of path length are undecidable. We show that our variant of CTL is decidable and that the model-checking problem is in Delta_3^P = P^{NP^{NP}}, and is hard for the class of problems solvable in polynomial time using a parallel access to an NP oracle. We analogously consider quantifier exchange in extensions of CTL, and we present operators defined using basic operators of CTL* that express the occurrence of infinitely many synchronization points. We show that the model-checking problem remains in Delta_3^P. The distinguishing power of CTL and of our new logic coincide if the Next operator is allowed in the logics, thus the classical bisimulation quotient can be used for state-space reduction before model checking.

Krishnendu Chatterjee and Laurent Doyen. Computation Tree Logic for Synchronization Properties. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 98:1-98:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatterjee_et_al:LIPIcs.ICALP.2016.98, author = {Chatterjee, Krishnendu and Doyen, Laurent}, title = {{Computation Tree Logic for Synchronization Properties}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {98:1--98:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.98}, URN = {urn:nbn:de:0030-drops-62334}, doi = {10.4230/LIPIcs.ICALP.2016.98}, annote = {Keywords: Computation Tree Logic, Synchronization, model-checking, complexity} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

While weighted automata provide a natural framework to express quantitative properties, many basic properties like average response time cannot be expressed with weighted automata. Nested weighted automata extend weighted automata and consist of a master automaton and a set of slave automata that are invoked by the master automaton. Nested weighted automata are strictly more expressive than weighted automata (e.g., average response time can be expressed with nested weighted automata), but the basic decision questions have higher complexity (e.g., for deterministic automata, the emptiness question for nested weighted automata is PSPACE-hard, whereas the corresponding complexity for weighted automata is PTIME). We consider a natural subclass of nested weighted automata where at any point at most a bounded number k of slave automata can be active. We focus on automata whose master value function is the limit average. We show that these nested weighted automata with bounded width are strictly more expressive than weighted automata (e.g., average response time with no overlapping requests can be expressed with bound k=1, but not with non-nested weighted automata). We show that the complexity of the basic decision problems (i.e., emptiness and universality) for the subclass with k constant matches the complexity for weighted automata. Moreover, when k is part of the input given in unary we establish PSPACE-completeness.

Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Nested Weighted Limit-Average Automata of Bounded Width. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 24:1-24:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2016.24, author = {Chatterjee, Krishnendu and Henzinger, Thomas A. and Otop, Jan}, title = {{Nested Weighted Limit-Average Automata of Bounded Width}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.24}, URN = {urn:nbn:de:0030-drops-64397}, doi = {10.4230/LIPIcs.MFCS.2016.24}, annote = {Keywords: weighted automata, nested weighted automata, complexity, mean-payoff} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Games on graphs provide the appropriate framework to study several central problems in computer science, such as verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or Büchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized Büchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized Büchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with n vertices, m edges, and generalized Büchi objectives with k conjunctions. First, we present an algorithm with running time O(k*n^2), improving the previously known O(k*n*m) and O(k^2*n^2) worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with k_1 conjunctions in the antecedent and k_2 conjunctions in the consequent, and present an O(k_1 k_2 n^{2.5})-time algorithm, improving the previously known O(k_1*k_2*n*m)-time algorithm for m > n^{1.5}.

Krishnendu Chatterjee, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. Conditionally Optimal Algorithms for Generalized Büchi Games. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 25:1-25:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2016.25, author = {Chatterjee, Krishnendu and Dvor\'{a}k, Wolfgang and Henzinger, Monika and Loitzenbauer, Veronika}, title = {{Conditionally Optimal Algorithms for Generalized B\"{u}chi Games}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.25}, URN = {urn:nbn:de:0030-drops-64403}, doi = {10.4230/LIPIcs.MFCS.2016.25}, annote = {Keywords: generalized B\"{u}chi objective, GR(1) objective, conditional lower bounds, graph games, graph algorithms, computer-aided verification} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We consider data-structures for answering reachability and distance queries on constant-treewidth graphs with n nodes, on the standard RAM computational model with wordsize W=Theta(log n). Our first contribution is a data-structure that after O(n) preprocessing time, allows (1) pair reachability queries in O(1) time; and (2) single-source reachability queries in O(n/log n) time. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. The data-structure uses at all times O(n) space. Our second contribution is a space-time tradeoff data-structure for distance queries. For any epsilon in [1/2,1], we provide a data-structure with polynomial preprocessing time that allows pair queries in O(n^{1-\epsilon} alpha(n)) time, where alpha is the inverse of the Ackermann function, and at all times uses O(n^epsilon) space. The input graph G is not considered in the space complexity.

Krishnendu Chatterjee, Rasmus Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Optimal Reachability and a Space-Time Tradeoff for Distance Queries in Constant-Treewidth Graphs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 28:1-28:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatterjee_et_al:LIPIcs.ESA.2016.28, author = {Chatterjee, Krishnendu and Rasmus Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas}, title = {{Optimal Reachability and a Space-Time Tradeoff for Distance Queries in Constant-Treewidth Graphs}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.28}, URN = {urn:nbn:de:0030-drops-63797}, doi = {10.4230/LIPIcs.ESA.2016.28}, annote = {Keywords: Graph algorithms, Constant-treewidth graphs, Reachability queries, Distance queries} }

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**Published in:** Dagstuhl Reports, Volume 5, Issue 2 (2015)

In this report, the program, research issues, and results of Dagstuhl Seminar 15061 "Non-Zero-Sum-Games and Control" are described. The area of non-zero-sum games is addressed in a wide range of topics: multi-player games, partial-observation games, quantitative game models, and - as a special focus - connections with control engineering (supervisory control).

Krishnendu Chatterjee, Stéphane Lafortune, Nicolas Markey, and Wolfgang Thomas. Non-Zero-Sum-Games and Control (Dagstuhl Seminar 15061). In Dagstuhl Reports, Volume 5, Issue 2, pp. 1-25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@Article{chatterjee_et_al:DagRep.5.2.1, author = {Chatterjee, Krishnendu and Lafortune, St\'{e}phane and Markey, Nicolas and Thomas, Wolfgang}, title = {{Non-Zero-Sum-Games and Control (Dagstuhl Seminar 15061)}}, pages = {1--25}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {5}, number = {2}, editor = {Chatterjee, Krishnendu and Lafortune, St\'{e}phane and Markey, Nicolas and Thomas, Wolfgang}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.2.1}, URN = {urn:nbn:de:0030-drops-50424}, doi = {10.4230/DagRep.5.2.1}, annote = {Keywords: non-zero-sum games, infinite games, multi-player games, partial-observation games, quantitative games, controller synthesis, supervisory control} }

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**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

We consider partially observable Markov decision processes (POMDPs) with omega-regular conditions specified as parity objectives. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal EXPTIME-complete complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finite-memory strategies. We also establish optimal (exponential) memory bounds.

Krishnendu Chatterjee, Martin Chmelik, and Mathieu Tracol. What is Decidable about Partially Observable Markov Decision Processes with omega-Regular Objectives. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 165-180, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2013.165, author = {Chatterjee, Krishnendu and Chmelik, Martin and Tracol, Mathieu}, title = {{What is Decidable about Partially Observable Markov Decision Processes with omega-Regular Objectives}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {165--180}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.165}, URN = {urn:nbn:de:0030-drops-41962}, doi = {10.4230/LIPIcs.CSL.2013.165}, annote = {Keywords: POMDPs, Omega-regular objectives, Parity objectives, Qualitative analysis} }

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**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

We study two-player zero-sum games over infinite-state graphs equipped with omega-B and finitary conditions.
Our first contribution is about the strategy complexity, i.e the memory required for winning strategies: we prove that over general infinite-state graphs, memoryless strategies are sufficient for finitary Büchi, and finite-memory suffices for finitary parity games.
We then study pushdown games with boundedness conditions, with two contributions. First we prove a collapse result for pushdown games with omega-B-conditions, implying the decidability of solving these games. Second we consider pushdown games with finitary parity along with stack boundedness conditions, and show that solving these games is EXPTIME-complete.

Krishnendu Chatterjee and Nathanaël Fijalkow. Infinite-state games with finitary conditions. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 181-196, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2013.181, author = {Chatterjee, Krishnendu and Fijalkow, Nathana\"{e}l}, title = {{Infinite-state games with finitary conditions}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {181--196}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.181}, URN = {urn:nbn:de:0030-drops-41970}, doi = {10.4230/LIPIcs.CSL.2013.181}, annote = {Keywords: Two-player games, Infinite-state systems, Pushdown games, Bounds in omega-regularity, Synthesis} }

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**Published in:** Dagstuhl Reports, Volume 2, Issue 11 (2013)

This report documents the program and the outcomes of the Dagstuhl Seminar 12461 "Games and Decisions for Rigorous Systems Engineering". The seminar brought together researchers working in rigorous software engineering, with a special focus on the interaction between synthesis and automated deduction. This event was the first seminar of this kind and a kickoff of a series of seminars organised on rigorous systems engineering. The theme of the seminar was close in spirit to many events that have been held over the last decades. The talks scheduled during the seminar naturally reflected fundamental research
themes of the involved communities.

Nikolaj Bjorner, Krishnendu Chatterjee, Laura Kovacs, and Rupak M. Majumdar. Games and Decisions for Rigorous Systems Engineering (Dagstuhl Seminar 12461). In Dagstuhl Reports, Volume 2, Issue 11, pp. 45-65, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@Article{bjorner_et_al:DagRep.2.11.45, author = {Bjorner, Nikolaj and Chatterjee, Krishnendu and Kovacs, Laura and Majumdar, Rupak M.}, title = {{Games and Decisions for Rigorous Systems Engineering (Dagstuhl Seminar 12461)}}, pages = {45--65}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {2}, number = {11}, editor = {Bjorner, Nikolaj and Chatterjee, Krishnendu and Kovacs, Laura and Majumdar, Rupak M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.11.45}, URN = {urn:nbn:de:0030-drops-39092}, doi = {10.4230/DagRep.2.11.45}, annote = {Keywords: Systems Engineering, Software Verification, Reactive Synthesis, Automated Deduction} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We consider Markov decision processes (MDPs) with specifications given as Büchi (liveness) objectives. We consider the problem of computing the set of almost-sure winning vertices from where the objective can be ensured with probability 1. We study for the first time the average case complexity of the classical algorithm for computing the set of almost-sure winning vertices for MDPs with Buchi objectives. Our contributions are as follows:
First, we show that for MDPs with constant out-degree the expected number of iterations is at most logarithmic and the average case running time is linear (as compared to the worst case linear number of iterations and quadratic time complexity). Second, for the average case analysis over all MDPs we show that the expected number of iterations is constant and the average case running time is linear (again as compared to the worst case linear number of iterations and
quadratic time complexity). Finally we also show that given that all MDPs are equally likely, the probability that the classical algorithm requires more than constant number of iterations is exponentially small.

Krishnendu Chatterjee, Manas Joglekar, and Nisarg Shah. Average Case Analysis of the Classical Algorithm for Markov Decision Processes with Büchi Objectives. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 461-473, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2012.461, author = {Chatterjee, Krishnendu and Joglekar, Manas and Shah, Nisarg}, title = {{Average Case Analysis of the Classical Algorithm for Markov Decision Processes with B\"{u}chi Objectives}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {461--473}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.461}, URN = {urn:nbn:de:0030-drops-38817}, doi = {10.4230/LIPIcs.FSTTCS.2012.461}, annote = {Keywords: MDPs, Buchi objectives, Average case analysis} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with Büchi fairness constraints; our algorithm requires O(n^3 * m) time as compared to the previous known O(n^6)-time algorithm, where n is the number of states and m is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires O(m^2)-time as compared to the previous known O((n*m)^2)-time algorithm, where n is the number of states and m is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.

Krishnendu Chatterjee, Siddhesh Chaubal, and Pritish Kamath. Faster Algorithms for Alternating Refinement Relations. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 167-182, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2012.167, author = {Chatterjee, Krishnendu and Chaubal, Siddhesh and Kamath, Pritish}, title = {{Faster Algorithms for Alternating Refinement Relations}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {167--182}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.167}, URN = {urn:nbn:de:0030-drops-36713}, doi = {10.4230/LIPIcs.CSL.2012.167}, annote = {Keywords: Simulation and fair simulation, Alternating simulation, Graph games} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Generalized mean-payoff and energy games replace individual weights by tuples, and the limit average (resp. running sum) of each coordinate must be (resp. remain) nonnegative. These games have applications in the synthesis of resource-bounded processes with multiple resources.
We prove the finite-memory determinacy of generalized energy games and show the inter-reducibility of generalized mean-payoff and energy games for finite-memory strategies. We also improve the computational complexity for solving both classes of games with finite-memory strategies: while the previously best known upper bound was EXPSPACE, and no lower bound was known, we give an optimal coNP-complete bound. For memoryless strategies, we show that the problem of deciding
the existence of a winning strategy for the protagonist is NP-complete.

Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, and Jean-François Raskin. Generalized Mean-payoff and Energy Games. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 505-516, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2010.505, author = {Chatterjee, Krishnendu and Doyen, Laurent and Henzinger, Thomas A. and Raskin, Jean-Fran\c{c}ois}, title = {{Generalized Mean-payoff and Energy Games}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {505--516}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.505}, URN = {urn:nbn:de:0030-drops-28484}, doi = {10.4230/LIPIcs.FSTTCS.2010.505}, annote = {Keywords: mean-payoff games, energy games, finite memory strategies, determinacy} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Simulation and bisimulation metrics for stochastic systems provide a
quantitative generalization of the classical simulation and
bisimulation relations.
These metrics capture the similarity of states with respect to
quantitative specifications written in the quantitative $\mu$-calculus
and related probabilistic logics.
We present algorithms for computing the metrics on Markov
decision processes (MDPs), turn-based stochastic games, and concurrent
games.
For turn-based games and MDPs, we provide a polynomial-time algorithm
based on linear programming
for the computation of the one-step metric distance between states.
The algorithm improves on the
previously known exponential-time algorithm based on a reduction to the theory of
reals.
We then present PSPACE algorithms for both the decision problem and the
problem of approximating the metric distance between two states,
matching the best known bound for Markov chains.
For the bisimulation kernel of the metric, which corresponds to probabilistic
bisimulation, our algorithm works in time $\calo(n^4)$ for both
turn-based games and MDPs; improving the previously best known
$\calo(n^9\cdot\log(n))$ time algorithm for MDPs.
For a concurrent game $G$, we show that computing the exact distance
between states is at least as hard as computing the value of
concurrent reachability games and
the square-root-sum problem in computational geometry.
We show that checking whether the metric distance is bounded by a
rational $r$, can be accomplished via a reduction to the theory of
real closed fields, involving a formula with three quantifier
alternations, yielding $\calo(|G|^{\calo(|G|^5)})$ time
complexity, improving the previously known reduction with
$\calo(|G|^{\calo(|G|^7)})$ time complexity.
These algorithms can be iterated to approximate the metrics using
binary search.

Krishnendu Chatterjee, Luca de Alfaro, Rupak Majumdar, and Vishwanath Raman. Algorithms for Game Metrics. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 107-118, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2008.1745, author = {Chatterjee, Krishnendu and de Alfaro, Luca and Majumdar, Rupak and Raman, Vishwanath}, title = {{Algorithms for Game Metrics}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {107--118}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1745}, URN = {urn:nbn:de:0030-drops-17455}, doi = {10.4230/LIPIcs.FSTTCS.2008.1745}, annote = {Keywords: Algorithms, Metrics, Kernel, Simulation, Bisimulation, Linear Programming, Theory of Reals} }

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