Blackwell-Optimal Strategies in Priority Mean-Payoff Games

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

Games where you can play optimally without any memory

International audienceReactive 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

Optimal Strategies in Infinite-state Stochastic Reachability Games

We consider perfect-information reachability stochastic games for 2 players on infinite graphs. We identify a subclass of such games, and prove two interesting properties of it: first, Player Max always has optimal strategies in games from this subclass, and second, these games are strongly determined. The subclass is defined by the property that the set of all values can only have one accumulation point -- 0. Our results nicely mirror recent results for finitely-branching games, where, on the contrary, Player Min always has optimal strategies. However, our proof methods are substantially different, because the roles of the players are not symmetric. We also do not restrict the branching of the games. Finally, we apply our results in the context of recently studied One-Counter stochastic games

Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Perfect Information Stochastic Priority Games

International audienceWe 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

Distributed Synthesis in Continuous Time

We introduce a formalism modelling communication of distributed agents strictly in continuous-time. Within this framework, we study the problem of synthesising local strategies for individual agents such that a specified set of goal states is reached, or reached with at least a given probability. The flow of time is modelled explicitly based on continuous-time randomness, with two natural implications: First, the non-determinism stemming from interleaving disappears. Second, when we restrict to a subclass of non-urgent models, the quantitative value problem for two players can be solved in EXPTIME. Indeed, the explicit continuous time enables players to communicate their states by delaying synchronisation (which is unrestricted for non-urgent models). In general, the problems are undecidable already for two players in the quantitative case and three players in the qualitative case. The qualitative undecidability is shown by a reduction to decentralized POMDPs for which we provide the strongest (and rather surprising) undecidability result so far

Deterministic Priority Mean-payoff Games as Limits of Discounted Games

International audienceInspired 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

Computer aided synthesis: a game theoretic approach

In this invited contribution, we propose a comprehensive introduction to game theory applied in computer aided synthesis. In this context, we give some classical results on two-player zero-sum games and then on multi-player non zero-sum games. The simple case of one-player games is strongly related to automata theory on infinite words. All along the article, we focus on general approaches to solve the studied problems, and we provide several illustrative examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language Theory" (DLT 2017

Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use of higher-order stacks, that is, a nested "stack of stacks" structure. These systems may be used to model higher-order programs and are closely related to the Caucal hierarchy of infinite graphs and safe higher-order recursion schemes. We consider the backwards-reachability problem over higher-order Alternating PDSs (APDSs), a generalisation of higher-order PDSs. This builds on and extends previous work on pushdown systems and context-free higher-order processes in a non-trivial manner. In particular, we show that the set of configurations from which a regular set of higher-order APDS configurations is reachable is regular and computable in n-EXPTIME. In fact, the problem is n-EXPTIME-complete. We show that this work has several applications in the verification of higher-order PDSs, such as linear-time model-checking, alternation-free mu-calculus model-checking and the computation of winning regions of reachability games