Reachability in Dynamical Systems with Rounding
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We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M ∈ℚ^d × d, an initial vector x∈ℚ^d, a granularity g∈ℚ_+ and a rounding operation [·] projecting a vector of ℚ^d onto another vector whose every entry is a multiple of g, we are interested in the behaviour of the orbit 𝒪=<[x], [M[x]],[M[M[x]]],…>, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability—whether a given target y ∈ℚ^d belongs to 𝒪—is PSPACE-complete for hyperbolic systems (when no eigenvalue of M has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.
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