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We consider two-person zero-sum stochastic games with signals, a standard model of stochastic games with imperfect information. The only source of information for the players consists of the signals they receive; they cannot directly observe the state of the game, nor the actions played by their opponent, nor their own actions. We are interested in the existence of almost-surely winning or positively winning strategies, under reachability, safety, Büchi, or co-Büchi winning objectives, and the computation of these strategies when the game has finitely many states and actions. We prove two qualitative determinacy results. First, in a reachability game, either player 1 can achieve almost surely the reachability objective, or player 2 can achieve surely the dual safety objective, or both players have positively winning strategies. Second, in a Büchi game, if player 1 cannot achieve almost surely the Büchi objective, then player 2 can ensure positively the dual co-Büchi objective. We prove that players only need strategies with finite memory . The number of memory states needed to win with finite-memory strategies ranges from one (corresponding to memoryless strategies) to doubly exponential, with matching upper and lower bounds. Together with the qualitative determinacy results, we also provide fix-point algorithms for deciding which player has an almost-surely winning or a positively winning strategy and for computing an associated finite-memory strategy. Complexity ranges from EXPTIME to 2EXPTIME, with matching lower bounds. Our fix-point algorithms also enjoy a better complexity in the cases where one of the players is better informed than their opponent. Our results hold even when players do not necessarily observe their own actions. The adequate class of strategies, in this case, is mixed or general strategies (they are equivalent). Behavioral strategies are too restrictive to guarantee determinacy: it may happen that one of the players has a winning general strategy but none of them has a winning behavioral strategy. On the other hand, if a player can observe their actions, then general, mixed, and behavioral strategies are equivalent. Finite-memory strategies are sufficient for determinacy to hold, provided that randomized memory updates are allowed.

Two-player, turn-based, stochastic games with reachability conditions are considered, where the maximizer has no information (he is blind) and is restricted to deterministic strategies whereas the minimizer is perfectly informed. We ask the question of whether the game has maxmin value of 1 in other words we ask whether for all \(\epsilon >0\) there exists a deterministic strategy for the (blind) maximizer such that against all the strategies of the minimizer, it is possible to reach the set of final states with probability larger than \(1-\epsilon \). This problem is undecidable in general, but we define a class of games, called leaktight half-blind games where the problem becomes decidable. We also show that mixed strategies in general are stronger for both players and that optimal strategies for the minimizer might require infinite-memory.

In a recent paper de Alfaro, Henzinger and Majumdar [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] observed that discounting successive payments, the procedure that is employed in the classical stochastic game theory since the seminal paper of Shapley [L.S. Shapley. Stochastic games. Proceedings Nat. Acad. of Science USA, 39:1095–1100, 1953], is also pertinent in the context of much more recent theory of stochastic parity games [L. de Alfaro and R. Majumdar. Quantitative solution to omega-regular games. In STOC'01, pages 675–683. ACM Press, 2001. final version to appear in Journal of Computer and System Sciences, L. de Alfaro, T.A. Henzinger, and O. Kupferman. Concurrent reachability games. In FOCS'98, pages 564–575. IEEE Computer Society Press, 1998, L. de Alfaro and T.A. Henzinger. Concurrent ω-regular games. In LICS'00, pages 142–154. IEEE Computer Society Press, 2000] which were proposed as a tool for verification of probabilistic systems. We show that, surprisingly perhaps, the particular discounting used in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] is in fact very close to the original ideas of Shapley. This observation allows to realize that the specific discounting of [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] suffers in fact from some needless restrictions. We advocate that dropping the constraints imposed in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] leads to a more general and elegant theory that includes parity and mean payoff games as particular limit cases.

International audience; The theory of graph games is the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic processes, we use 2-1/2-player games where some transitions of the game graph are controlled by two adversarial players, the System and the Environment, and the other transitions are determined probabilistically. We consider 2-1/2-player games where the objective of the System is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). We establish that the problem of deciding whether the System can ensure that the probability to satisfy the mean-payoff parity objective is at least a given threshold is in NP ∩ coNP, matching the best known bound in the special case of 2-player games (where all transitions are deterministic) with only parity objectives, or with only mean-payoff objectives. We present an algorithm running in time O(d * n^{2d}*MeanGame) to compute the set of almost-sure winning states from which the objective can be ensured with probability 1, where n is the number of states of the game, d the number of priorities of the parity objective, and MeanGame is the complexity to compute the set of almost-sure winning states in 2-1/2-player mean-payoff games. Our results are useful in the synthesis of stochastic reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective).

The value 1 problem is a natural decision problem in algorithmic game theory. For partially observable Markov decision processes with reachability objective, this problem is defined as follows: are there observational strategies that achieve the reachability objective with probability arbitrarily close to 1? This problem was shown undecidable recently. Our contribution is to introduce a class of partially observable Markov decision processes, namely \(\sharp-acyclic\) partially observable Markov decision processes, for which the value 1 problem is decidable. Our algorithm is based on the construction of a two-player perfect information game, called the knowledge game, abstracting the behaviour of a \(\sharp\)-acyclic partially observable Markov decision process \({\mathcal M}\) such that the first player has a winning strategy in the knowledge game if and only if the value of \({\mathcal M}\) is 1.

This paper introduces and investigates decision problems for numberless probabilistic automata, i.e. probabilistic automata where the support of each probabilistic transitions is specified, but the exact values of the probabilities are not. A numberless probabilistic automaton can be instantiated into a probabilistic automaton by specifying the exact values of the non-zero probabilistic transitions.

International audience; This paper reports about the robotic installation set up by the Rhoban Project in the French pavilion of the Expo 2012 of Yeosu, Korea ([6]). The installation has consisted in a humorous show involving humanoid robots and anthropomorphic arms, with the illusion of life as a guideline. We emphasized natural compliant motion and physical interaction in order to make the show attractive. The design raised some issues dealing with robustness of robots, but also with the realism of the motions and the synchronization of the robots with the music.

We consider the task of controlling in a distributed way a Zielonka asynchronous automaton. Every process of a controller has access to its causal past to determine the next set of actions it proposes to play. An action can be played only if every process controlling this action proposes to play it. We consider reachability objectives: every process should reach its set of final states. We show that this control problem is decidable for tree architectures, where every process can communicate with its parent, its children, and with the environment. The complexity of our algorithm is l-fold exponential with l being the height of the tree representing the architecture. We show that this is unavoidable by showing that even for three processes the problem is EXPTIME-complete, and that it is non-elementary in general.

We study turn-based quantitative multiplayer non zero-sum games played on finite graphs with reachability objectives. In such games, each player aims at reaching his own goal set of states as soon as possible. A previous work on this model showed that Nash equilibria (resp. secure equilibria) are guaranteed to exist in the multiplayer (resp. two-player) case. The existence of secure equilibria in the multiplayer case remained and is still an open problem. In this paper, we focus our study on the concept of subgame perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. We also introduce the new concept of subgame perfect secure equilibrium. We prove the existence of subgame perfect equilibria (resp. subgame perfect secure equilibria) in multiplayer (resp. two-player) quantitative reachability games. Moreover, we provide an algorithm deciding the existence of secure equilibria in the multiplayer case., Comment: 32 pages. Full version of the FoSSaCS 2012 proceedings paper

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton A, are there words accepted by A with probability arbitrarily close to 1? This problem was proved undecidable recently. We sharpen this result, showing that the undecidability result holds even if the probabilistic automata have only one probabilistic transition. Our main contribution is to introduce a new class of probabilistic automata, called leaktight automata, for which the value 1 problem is shown decidable (and PSPACE-complete). We construct an algorithm based on the computation of a monoid abstracting the behaviours of the automaton, and rely on algebraic techniques developed by Simon for the correctness proof. The class of leaktight automata is decidable in PSPACE, subsumes all subclasses of probabilistic automata whose value 1 problem is known to be decidable (in particular deterministic automata), and is closed under two natural composition operators., Comment: arXiv admin note: significant text overlap with arXiv:1104.3054

International audience; We examine perfect information stochastic mean-payoff games - a class of games containing as special sub-classes the usual mean-payoff games and parity games. We show that deterministic memoryless strategies that are optimal for discounted games with state-dependent discount factors close to 1 are optimal for priority mean-payoff games establishing a strong link between these two classes.

International audience; We study turn-based quantitative multiplayer non zero-sum games played on finite graphs with reachability objectives. In such games, each player aims at reaching his own goal set of states as soon as possible. A previous work on this model showed that Nash equilibria (resp. secure equilibria) are guaranteed to exist in the multiplayer (resp. two-player) case. The existence of secure equilibria in the multiplayer case remained, and is still an open problem. In this paper, we focus our study on the concept of subgame perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. We also introduce the new concept of subgame perfect secure equilibrium. We prove the existence of subgame perfect equilibria (resp. subgame perfect secure equilibria) in multiplayer (resp. two-player) quantitative reachability games. Moreover, we provide an algorithm deciding the existence of secure equilibria in the multiplayer case.

International audience; This paper tackles three algorithmic problems for probabilis- tic automata on finite words: the Emptiness Problem, the Isolation Prob- lem and the Value 1 Problem. The Emptiness Problem asks, given some probability 0 ≤ λ ≤ 1, whether there exists a word accepted with prob- ability greater than λ, and the Isolation Problem asks whether there exist words whose acceptance probability is arbitrarily close to λ. Both these problems are known to be undecidable [8, 2, 3]. About the Empti- ness problem, we provide a new simple undecidability proof and prove that it is decidable for automata with one probabilistic transition and undecidable for automata with as few as two probabilistic transitions. The Value 1 Problem is the special case of the Isolation Problem when λ = 1 or λ = 0. The decidability of the Value 1 Problem was an open question. We show that the Value 1 Problem is undecidable. Moreover, we introduce a new class of probabilistic automata, ♯-acyclic automata, for which the Value 1 Problem is decidable.

International audience; Asynchronous automata are parallel compositions of finite- state processes synchronizing over shared variables. A deep theorem due to Zielonka says that every regular trace language can be represented by a deterministic asynchronous automaton. In this paper we improve the construction, in that the size of the obtained asynchronous automaton is polynomial in the size of a given DFA and simply exponential in the number of processes. We show that our construction is optimal within the class of automata produced by Zielonka-type constructions. In particular, we provide the first non trivial lower bound on the size of asynchronous automata.

Simple stochastic games are two-player zero-sum stochastic games with turn-based moves, perfect information, and reachability winning conditions. We present two new algorithms computing the values of simple stochastic games. Both of them rely on the existence of optimal permutation strategies, a class of positional strategies derived from permutations of the random vertices. The "permutation-enumeration" algorithm performs an exhaustive search among these strategies, while the "permutation-improvement" algorithm is based on successive improvements, \`a la Hoffman-Karp. Our algorithms improve previously known algorithms in several aspects. First they run in polynomial time when the number of random vertices is fixed, so the problem of solving simple stochastic games is fixed-parameter tractable when the parameter is the number of random vertices. Furthermore, our algorithms do not require the input game to be transformed into a stopping game. Finally, the permutation-enumeration algorithm does not use linear programming, while the permutation-improvement algorithm may run in polynomial time.

We present a new algorithm for solving Simple Stochastic Games (SSGs). This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the f-strategies. The running time is O( |VR|! ċ (|V ||E| + |p|) ), where |V |, |VR|, |E| and |p| are respectively the number of vertices, random vertices and edges, and the maximum bit-length of a transition probability. Our algorithm improves existing algorithms for solving SSGs in three aspects. First, our algorithm performs well on SSGs with few random vertices, second it does not rely on linear or quadratic programming, third it applies to all SSGs, not only stopping SSGs.

International audience; Markov decision processes (MDPs) are controllable discrete event systems with stochastic transitions. The payoff received by the controller can be evaluated in different ways, depending on the payoff function the MDP is equipped with. For example a \emph{mean--payoff} function evaluates average performance, whereas a \emph{discounted} payoff function gives more weights to earlier performance by means of a discount factor. Another well--known example is the \emph{parity} payoff function which is used to encode logical specifications~\cite{dagstuhl}. Surprisingly, parity and mean--payoff MDPs share two non--trivial properties: they both have pure stationary optimal strategies~\cite{CourYan:1990,neyman} and they both are approximable by discounted MDPs with multiple discount factors (multi--discounted MDPs)~\cite{dealf:2003,neyman}. In this paper we unify and generalize these results. We introduce a new class of payoff functions called the priority weighted payoff functions, which are generalization of both parity and mean--payoff functions. We prove that priority weighted MDPs admit optimal strategies that are pure and stationary, and that the priority weighted value of an MDP is the limit of the multi--discounted value when discount factors tend to $0$ simultaneously at various speeds.

International audience; Markov decision processes (MDPs) are controllable discrete event systems with stochastic transitions. Performances of an MDP are evaluated by a payoff function. The controller of the MDP seeks to optimize those performances, using optimal strategies. There exists various ways of measuring performances, i.e. various classes of payoff functions. For example, average performances can be evaluated by a mean-payoff function, peak performances by a limsup payoff function, and the parity payoff function can be used to encode logical specifications. Surprisingly, all the MDPs equipped with mean, limsup or parity payoff functions share a common non-trivial property: they admit pure stationary optimal strategies. In this paper, we introduce the class of prefix-independent and submixing payoff functions, and we prove that any MDP equipped with such a payoff function admits pure stationary optimal strategies. This result unifies and simplifies several existing proofs. Moreover, it is a key tool for generating new examples of MDPs with pure stationary optimal strategies.

International audience; We introduce stochastic priority games - a new class of perfect information stochastic games. These games can take two different, but equivalent, forms. In stopping priority games a play can be stopped by the environment after a finite number of stages, however, infinite plays are also possible. In discounted priority games only infinite plays are possible and the payoff is a linear combination of the classical discount payoff and of a limit payoff evaluating the performance at infinity. Shapley games and parity games are special extreme cases of priority games.

International audience; Inspired by the paper of de Alfaro, Henzinger and Majumdar about discounted $\mu$-calculus we show new surprising links between parity games and different classes of discounted games.

International audience; Reactive systems are often modelled as two person antagonistic games where one player represents the system while his adversary represents the environment. Undoubtedly, the most popular games in this context are parity games and their cousins (Rabin, Streett and Muller games). Recently however also games with other types of payments, like discounted or mean-payoff , previously used only in economic context, entered into the area of system modelling and verification. The most outstanding property of parity, mean-payoff and discounted games is the existence of optimal positional (memoryless) strategies for both players. This observation raises two questions: (1) can we characterise the family of payoff mappings for which there always exist optimal positional strategies for both players and (2) are there other payoff mappings with practical or theoretical interest and admitting optimal positional strategies. This paper provides a complete answer to the first question by presenting a simple necessary and sufficient condition on payoff mapping guaranteeing the existence of optimal positional strategies. As a corollary to this result we show the following remarkable property of payoff mappings: if both players have optimal positional strategies when playing solitary one-player games then also they have optimal positional strategies for two-player games.

International audience; This paper examines two players' turn-based perfect-information games played on infinite graphs. Our attention is focused on the classes of games where winning conditions are boolean combinations of the following two conditions: (1) the first one states that an infinite play is won by player $0$ if during the play infinitely many different vertices were visited, (2) the second one is the well known parity condition generalized to a countable number of priorities. We show that, in most cases, both players have positional winning strategies and we characterize their respective winning sets. In the special case of pushdown graphs, we use these results to show that the sets of winning positions are regular and we show how to compute them as well as positional winning strategies in exponential time.