Church Synthesis Problem for Noisy Input

We study two variants of infinite games with imperfect information. In the first variant, in each round player-1 may decide to hide his move from player-2. This captures situations where the input signal is subject to fluctuations (noises), and every error in the input signal can be detected by the controller. In the second variant, all of player- 1 moves are visible to player-2; however, after the game ends, player-1 may change some of his moves. This captures situations where the input signal is subject to fluctuations; however, the controller cannot detect errors in the input signal. We consider several cases, according to the amount of errors allowed in the input signal: a fixed number of errors, finitely many errors and the case where the rate of errors is bounded by a threshold. For each of these cases we consider games with regular and mean-payoff winning conditions. We investigate the decidability of these games. There is a natural reduction for some of these games to (perfect information) multidimensional mean-payoff games recently considered in [7]. However, the decidability of the winner of multidimensional mean-payoff games was stated as an open question. We prove its decidability and provide tight complexity bounds.