On recurrent reachability for continuous linear dynamical systems

Conference item

The continuous evolution of a wide variety of systems, including continous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution x(t) of a system of linear differential equations dx=dt = Ax reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function f : R≥0 -> R satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most 7, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order 9 (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.

Authors: Ouaknine, J; Chonev, V; Worrell, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Association for Computing Machinery
• Host title: LICS 2016 (Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science)
• Journal: Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science
• Publication date: 2016-04-01
• Acceptance date: 2016-04-04
• DOI: 10.1145/2933575.2934548
• Keywords: linear dynamical systems; Diophantine Approximation; reachability; Skolem problem; differential equations