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16
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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Cited by 36 (14 self)
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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 8 (5 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
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 5 (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 5 (0 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.
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 4 (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
Decision Problems for Nash Equilibria in Stochastic Games
, 2009
"... We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with ωregular objectives.While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restr ..."
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Cited by 3 (1 self)
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We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with ωregular objectives.While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restricting the search space to stationary, or pure stationary, equilibria results in problems that are typically contained in PSPace and NP, respectively. Second, we show that the existence of an equilibrium with a binary payoff (i.e. an equilibrium where each player either wins or loses with probability 1) is decidable. We also establish that the existence of a Nash equilibrium with a certain binary payoff entails the existence of an equilibrium with the same payoff in pure, finitestate strategies.
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 3 (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.
Borel Determinacy of Concurrent Games
"... Abstract—Just as traditional games are represented by trees, so distributed/concurrent games are represented by event structures. We show the determinacy of such concurrent games with Borel sets of configurations as winning conditions, provided the games are racefree and boundedconcurrent. Both re ..."
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Cited by 1 (1 self)
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Abstract—Just as traditional games are represented by trees, so distributed/concurrent games are represented by event structures. We show the determinacy of such concurrent games with Borel sets of configurations as winning conditions, provided the games are racefree and boundedconcurrent. Both restrictions are shown necessary. The determinacy proof proceeds via a reduction to the determinacy of tree games, and the determinacy of these in turn reduces to the determinacy of GaleStewart games. Keywords Concurrent games; Nondeterministic strategies; Winning conditions; Borel Determinacy; Event structures. I.
Playing stochastic games precisely
, 2012
"... Abstract. We study stochastic twoplayer games where the goal of one player is to achieve precisely a given expected value of the objective function, while the goal of the opponent is the opposite. Potential applications for such games include controller synthesis problems where the optimisation obj ..."
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Abstract. We study stochastic twoplayer games where the goal of one player is to achieve precisely a given expected value of the objective function, while the goal of the opponent is the opposite. Potential applications for such games include controller synthesis problems where the optimisation objective is to maximise or minimise a given payoff function while respecting a strict upper or lower bound, respectively. We consider a number of objective functions including reachability, ωregular, discounted reward, and total reward. We show that precise value games are not determined, and compare the memory requirements for winning strategies. For stopping games we establish necessary and sufficient conditions for the existence of a winning strategy of the controller for a large class of functions, as well as provide the constructions of compact strategies for the studied objectives. 1
Competitive Stochastic Systems
"... 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