Reasoning about Knowledge and Strategies under Hierarchical Information

Two distinct semantics have been considered for knowledge in the context of strategic reasoning, depending on whether players know each other's strategy or not. The problem of distributed synthesis for epistemic temporal specifications is known to be undecidable for the latter semantics, already on systems with hierarchical information. However, for the other, uninformed semantics, the problem is decidable on such systems. In this work we generalise this result by introducing an epistemic extension of Strategy Logic with imperfect information. The semantics of knowledge operators is uninformed, and captures agents that can change observation power when they change strategies. We solve the model-checking problem on a class of "hierarchical instances", which provides a solution to a vast class of strategic problems with epistemic temporal specifications on hierarchical systems, such as distributed synthesis or rational synthesis

Perspective Games with Notifications

A reactive system has to satisfy its specification in all environments. Accordingly, design of correct reactive systems corresponds to the synthesis of winning strategies in games that model the interaction between the system and its environment. The game is played on a graph whose vertices are partitioned among the players. The players jointly generate a path in the graph, with each player deciding the successor vertex whenever the path reaches a vertex she owns. The objective of the system player is to force the computation induced by the generated infinite path to satisfy a given specification. The traditional way of modelling uncertainty in such games is observation-based. There, uncertainty is longitudinal: the players partially observe all vertices in the history. Recently, researchers introduced perspective games, where uncertainty is transverse: players fully observe the vertices they own and have no information about the behavior of the computation between visits in such vertices. We introduce and study perspective games with notifications: uncertainty is still transverse, yet a player may be notified about events that happen between visits in vertices she owns. We distinguish between structural notifications, for example about visits in some vertices, and behavioral notifications, for example about the computation exhibiting a certain behavior. We study the theoretic properties of perspective games with notifications, and the problem of deciding whether a player has a winning perspective strategy. Such a strategy depends only on the visible history, which consists of both visits in vertices the player owns and notifications during visits in other vertices. We show that the problem is EXPTIME-complete for objectives given by a deterministic or universal parity automaton over an alphabet that labels the vertices of the game, and notifications given by a deterministic satellite, and is 2EXPTIME-complete for LTL objectives. In all cases, the complexity in the size of the graph and the satellite is polynomial - exponentially easier than games with observation-based partial visibility. We also analyze the complexity of the problem for richer types of satellites

Strategy Logic with Imperfect Information

We introduce an extension of Strategy Logic for the imperfect-information setting, called SLii, and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of "hierarchical instances" for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SLii restricted to hierarchical instances is decidable. To establish this result we go through QCTL, an intermediary, "low-level" logic much more adapted to automata techniques. QCTL is an extension of CTL with second-order quantification over atomic propositions. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. While the model-checking problem of QCTLii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable. We apply our result to solve complex strategic problems in the imperfect-information setting. We first show that the existence of Nash equilibria for deterministic strategies is decidable in games with hierarchical information. We also introduce distributed rational synthesis, a generalisation of rational synthesis to the imperfect-information setting. Because it can easily be expressed in our logic, our main result provides solution to this problem in the case of hierarchical information.Comment: arXiv admin note: text overlap with arXiv:1805.1259

Asynchronous Omega-Regular Games with Partial Information

We address the strategy problem for ω-regular two-player games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effect for a player are hidden completely from that player. We generalize the usual powerset construction for eliminating partial information to arbitrary, not necessarily observation based, winning conditions, both in the synchronous and in the asynchronous case, and we show that this generalized construction effectively preserves ω-regular winning conditions. From this we infer decidability of the strategy problem for arbitrary ω-regular winning conditions, in both cases. We also show that our ω-regular framework is sufficient for reasoning about synchronous and asynchronous knowledge by proving that any formula of the epistemic temporal specification formalism ETL can be effectively translated into an S1S-formula defining the same specification