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. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Theory of computation → Logic and verification