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33
Quantitative languages
"... Quantitative generalizations of classical languages, which assign to each word a real number instead of a boolean value, have applications in modeling resourceconstrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitativ ..."
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Quantitative generalizations of classical languages, which assign to each word a real number instead of a boolean value, have applications in modeling resourceconstrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of the languageinclusion problem remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted
Expressiveness and closure properties for quantitative languages
 In Proc. of LICS: Logic in Computer Science. IEEE Comp. Soc
, 2009
"... Abstract. Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit avera ..."
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Cited by 20 (9 self)
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Abstract. Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be nonωregular for deterministic limitaverage and discountedsum automata, while this set is always ωregular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the ωregular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limitaverage case, but not in the discountedsum case. Third, for quantitative languages L1 and L2, we consider the operations max(L1, L2), min(L1, L2), and 1−L1, which generalize the boolean operations on languages, as well as the sum L1 +L2. We establish the closure properties of all classes of quantitative languages with respect to these four operations. 1
A.: Automatic verification of competitive stochastic systems
, 2011
"... Abstract. We present automatic verification techniques for the modelling and analysis of probabilistic systems that incorporate competitive behaviour. These systems are modelled as turnbased stochastic multiplayer games, in which the players can either collaborate or compete in order to achieve a p ..."
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Abstract. We present automatic verification techniques for the modelling and analysis of probabilistic systems that incorporate competitive behaviour. These systems are modelled as turnbased stochastic multiplayer games, in which the players can either collaborate or compete in order to achieve a particular goal. We define a temporal logic called rPATL for expressing quantitative properties of stochastic multiplayer games. This logic allows us to reason about the collective ability of a set of players to achieve a goal relating to the probability of an event’s occurrence or the expected amount of cost/reward accumulated. We give a model checking algorithm for verifying properties expressed in this logic and implement the techniques in a probabilistic model checker, based on the PRISM tool. We demonstrate the applicability and efficiency of our methods by deploying them to analyse and detect potential weaknesses in a variety of large case studies, including algorithms for energy management and collective decision making for autonomous systems. 1
Solving Simple Stochastic Tail Games
 "SODA'10 (SYMPOSIUM ON DISCRETE ALGORITHMS), UNITED STATES (2010)"
, 2010
"... Stochastic games are a natural model for open reactive processes: one player represents the controller and his opponent represents a hostile environment. The evolution of the system depends on the decisions of the players, supplemented by random transitions. There are two main algorithmic problems o ..."
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Cited by 13 (1 self)
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Stochastic games are a natural model for open reactive processes: one player represents the controller and his opponent represents a hostile environment. The evolution of the system depends on the decisions of the players, supplemented by random transitions. There are two main algorithmic problems on such games: computing the values (quantitative analysis) and deciding whether a player can win with probability 1 (qualitative analysis). In this paper we reduce the quantitative analysis to the qualitative analysis: we provide an algorithm for computing values which uses qualitative analysis as a subprocedure. The correctness proof of this algorithm reveals several nice properties of perfectinformation stochastic tail games, in particular the existence of optimal strategies. We apply these results to games whose winning conditions are boolean combinations of meanpayoff and Büchi conditions.
Qualitative Concurrent Stochastic Games with Imperfect Information
, 2009
"... We study a model of games that combines concurrency, imperfect information and stochastic aspects. Those are finite states games in which, at each round, the two players choose, simultaneously and independently, an action. Then a successor state is chosen accordingly to some fixed probability dist ..."
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Cited by 8 (1 self)
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We study a model of games that combines concurrency, imperfect information and stochastic aspects. Those are finite states games in which, at each round, the two players choose, simultaneously and independently, an action. Then a successor state is chosen accordingly to some fixed probability distribution depending on the previous state and on the pair of actions chosen by the players. Imperfect information is modeled as follows: both players have an equivalence relation over states and, instead of observing the exact state, they only know to which equivalence class it belongs. Therefore, if two partial plays are indistinguishable by some player, he should behave the same in both of them. We consider reachability (does the play eventually visit a final state?) and Büchi objective (does the play visit infinitely often a final state?). Our main contribution is to prove that the following problem is complete for 2ExpTime: decide whether the first player has a strategy that ensures her to almostsurely win against any possible strategy of her oponent. We also characterise those strategies needed by the first player to almostsurely win.
Probabilistic Modal µCalculus with Independent Product
 In Foundations of Software Science and Computation Structures
, 2011
"... Vol. 8(4:18)2012, pp. 1–36 ..."
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Gist: A solver for probabilistic games
, 2010
"... Abstract. Gistis atool that (a) solves thequalitative analysis problem of turnbased probabilistic games with ωregular objectives; and (b) synthesizesreasonable environmentassumptionsforsynthesisofunrealizable specifications. Our tool provides the first and efficient implementations of several redu ..."
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Cited by 6 (0 self)
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Abstract. Gistis atool that (a) solves thequalitative analysis problem of turnbased probabilistic games with ωregular objectives; and (b) synthesizesreasonable environmentassumptionsforsynthesisofunrealizable specifications. Our tool provides the first and efficient implementations of several reductionbased techniques to solve turnbased probabilistic games, and uses the analysis of turnbased probabilistic games for synthesizing environment assumptions for unrealizable specifications. 1
Nash Equilibrium in Generalised Muller Games
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... We suggest that extending Muller games with preference ordering for players is a natural way to reason about unbounded duration games. In this context, we look at the standard solution concept of Nash equilibrium for nonzero sum games. We show that Nash equilibria always exists for such generalised ..."
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Cited by 6 (0 self)
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We suggest that extending Muller games with preference ordering for players is a natural way to reason about unbounded duration games. In this context, we look at the standard solution concept of Nash equilibrium for nonzero sum games. We show that Nash equilibria always exists for such generalised Muller games on finite graphs and present a procedure to compute an equilibrium strategy profile. We also give a procedure to compute a subgame perfect equilibrium when it exists in such games.
A Survey of PartialObservation Stochastic Parity Games
"... We consider twoplayer zerosum stochastic games on graphs with ωregular winning conditions specified as parity objectives. These games have applications in the design and control of reactive systems. We survey the complexity results for the problem of deciding the winner in such games, and in cla ..."
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Cited by 5 (2 self)
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We consider twoplayer zerosum stochastic games on graphs with ωregular winning conditions specified as parity objectives. These games have applications in the design and control of reactive systems. We survey the complexity results for the problem of deciding the winner in such games, and in classes of interest obtained as special cases, based on the information and the power of randomization available to the players, on the class of objectives and on the winning mode. On the basis of information, these games can be classified as follows: (a) partialobservation (both players have partial view of the game); (b) onesided partialobservation (one player has partialobservation and the other player has completeobservation); and (c) completeobservation (both players have complete view of the game). The onesided partialobservation games have two important subclasses: the oneplayer games, known as partialobservation Markov decision processes (POMDPs), and the blind oneplayer games, known as probabilistic automata. On the basis of randomization, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. Finally, various classes of games are obtained by restricting the parity objective to a reachability, safety, Büchi, or coBüchi condition. We also consider several winning modes, such as surewinning (i.e., all outcomes of a strategy have to satisfy the winning condition), almostsure winning (i.e., winning with probability 1), limitsure winning (i.e., winning with probability arbitrarily close to 1), and valuethreshold winning (i.e., winning with probability at least ν, where ν is a given rational).
The Complexity of Partialobservation Stochastic Parity Games With Finitememory Strategies⋆
"... Abstract. We consider twoplayer partialobservation stochastic games on finitestate graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are ωregular conditions specified as parity objectives. The qualitativeanalysis problem given a p ..."
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Abstract. We consider twoplayer partialobservation stochastic games on finitestate graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are ωregular conditions specified as parity objectives. The qualitativeanalysis problem given a partialobservation stochastic game and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). These qualitativeanalysis problems are known to be undecidable. However in many applications the relevant question is the existence of finitememory strategies, and the qualitativeanalysis problems under finitememory strategies was recently shown to be decidable in 2EXPTIME. We improve the complexity and show that the qualitativeanalysis problems for partialobservation stochastic parity games under finitememory strategies are EXPTIMEcomplete; and also establish optimal (exponential) memory bounds for finitememory strategies required for qualitative analysis. 1