The Semialgebraic Orbit Problem
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The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d∈N, a square matrix A∈Q^d× d, and semialgebraic source and target sets S,T⊆R^d. The question is whether there exists x∈ S and n∈N such that A^nx ∈ T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d≤ 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory---Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.
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