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arXiv · 1901.11023

The Semialgebraic Orbit Problem

Shaull Almagor·Joël Oukanine·James Worrell
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Abstract

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 $d\in\mathbb{N}$, a square matrix $A\in\mathbb{Q}^{d\times d}$, and semialgebraic source and target sets $S,T\subseteq \mathbb{R}^d$. The question is whether there exists $x\in S$ and $n\in\mathbb{N}$ such that $A^nx \in T$. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension $d\leq 3$. 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 $\mathbb{R}^d$ 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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Shaull Almagor, Joël Oukanine, James Worrell. 2019-01-30. The Semialgebraic Orbit Problem. https://arxiv.org/abs/1901.11023

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