The semialgebraic orbit problem

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 Aⁿx ∈ 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.