A robot game, also known as a Z-VAS game, is a two-player vector addition game played on the integer lattice Z n , where one of the players, Adam, aims to avoid the origin while the other player, Eve, aims to reach the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the two-dimensional robot game closing the gap between undecidable and decidable cases. We also prove that deciding the winner in a robot game with states in dimension one is EXPSPACE-complete and study a subclass of robot games where deciding the winner is in EXPTIME.
We study the reachability problems in various nondeterministic polynomial maps in Z n. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is PSPACE-hard for two-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form ±x + b is lower. In this case the reachability problem is PSPACE-complete in general, and NP-hard for any fixed dimension. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps.
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