TY - CONF
AB - Energy parity games are infinite two-player turn-based games played on weighted graphs. The objective of the game combines a (qualitative) parity condition with the (quantitative) requirement that the sum of the weights (i.e., the level of energy in the game) must remain positive. Beside their own interest in the design and synthesis of resource-constrained omega-regular specifications, energy parity games provide one of the simplest model of games with combined qualitative and quantitative objective. Our main results are as follows: (a) exponential memory is sufficient and may be necessary for winning strategies in energy parity games; (b) the problem of deciding the winner in energy parity games can be solved in NP ∩ coNP; and (c) we give an algorithm to solve energy parity by reduction to energy games. We also show that the problem of deciding the winner in energy parity games is polynomially equivalent to the problem of deciding the winner in mean-payoff parity games, which can thus be solved in NP ∩ coNP. As a consequence we also obtain a conceptually simple algorithm to solve mean-payoff parity games.
AU - Chatterjee, Krishnendu
AU - Doyen, Laurent
ID - 3851
TI - Energy parity games
VL - 6199
ER -
TY - CONF
AB - We introduce two-level discounted games played by two players on a perfect-information stochastic game graph. The upper level game is a discounted game and the lower level game is an undiscounted reachability game. Two-level games model hierarchical and sequential decision making under uncertainty across different time scales. We show the existence of pure memoryless optimal strategies for both players and an ordered field property for such games. We show that if there is only one player (Markov decision processes), then the values can be computed in polynomial time. It follows that whether the value of a player is equal to a given rational constant in two-level discounted games can be decided in NP intersected coNP. We also give an alternate strategy improvement algorithm to compute the value.
AU - Chatterjee, Krishnendu
AU - Majumdar, Ritankar
ID - 3852
TI - Discounting in games across time scales
VL - 25
ER -
TY - CONF
AB - Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic meanpayoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.
AU - Chatterjee, Krishnendu
AU - Doyen, Laurent
AU - Edelsbrunner, Herbert
AU - Henzinger, Thomas A
AU - Rannou, Philippe
ID - 3853
TI - Mean-payoff automaton expressions
VL - 6269
ER -
TY - CONF
AB - Graph games of infinite length provide a natural model for open reactive systems: one player (Eve) represents the controller and the other player (Adam) represents the environment. The evolution of the system depends on the decisions of both players. The specification for the system is usually given as an ω-regular language L over paths and Eve’s goal is to ensure that the play belongs to L irrespective of Adam’s behaviour. The classical notion of winning strategies fails to capture several interesting scenarios. For example, strong fairness (Streett) conditions are specified by a number of request-grant pairs and require every pair that is requested infinitely often to be granted infinitely often: Eve might win just by preventing Adam from making any new request, but a “better” strategy would allow Adam to make as many requests as possible and still ensure fairness. To address such questions, we introduce the notion of obliging games, where Eve has to ensure a strong condition Φ, while always allowing Adam to satisfy a weak condition Ψ. We present a linear time reduction of obliging games with two Muller conditions Φ and Ψ to classical Muller games. We consider obliging Streett games and show they are co-NP complete, and show a natural quantitative optimisation problem for obliging Streett games is in FNP. We also show how obliging games can provide new and interesting semantics for multi-player games.
AU - Chatterjee, Krishnendu
AU - Horn, Florian
AU - Löding, Christof
ID - 3854
TI - Obliging games
VL - 6269
ER -
TY - CONF
AB - We study observation-based strategies for partially-observable Markov decision processes (POMDPs) with parity objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider qualitative analysis problems: given a POMDP with a parity objective, decide whether there exists an observation-based strategy to achieve the objective with probability 1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis problem for POMDPs with parity objectives and its subclasses: safety, reachability, Büchi, and coBüchi objectives. We establish several upper and lower bounds that were not known in the literature. Second, we give optimal bounds (matching upper and lower bounds) for the memory required by pure and randomized observation-based strategies for each class of objectives.
AU - Chatterjee, Krishnendu
AU - Doyen, Laurent
AU - Henzinger, Thomas A
ID - 3855
TI - Qualitative analysis of partially-observable Markov Decision Processes
VL - 6281
ER -