TY - GEN

T1 - The semialgebraic orbit problem

AU - Almagor, Shaull

AU - Ouaknine, Joël

AU - Worrell, James

N1 - Publisher Copyright: © Shaull Almagor, Joël Ouaknine, and James Worrell.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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 ∈ Qd×d, and semialgebraic source and target sets S, T ⊆ Rd. 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 Rd 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.

AB - 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 ∈ Qd×d, and semialgebraic source and target sets S, T ⊆ Rd. 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 Rd 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.

KW - First order theory of the reals

KW - Linear dynamical systems

KW - Orbit problem

UR - http://www.scopus.com/inward/record.url?scp=85074905121&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.STACS.2019.6

DO - https://doi.org/10.4230/LIPIcs.STACS.2019.6

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019

A2 - Niedermeier, Rolf

A2 - Paul, Christophe

T2 - 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019

Y2 - 13 March 2019 through 16 March 2019

ER -