On the skolem problem for continuous linear dynamical systems Conference Paper

Author(s): Chonev, Ventsislav; Ouaknine, Joël; Worrell, James
Title: On the skolem problem for continuous linear dynamical systems
Title Series: LIPIcs
Affiliation IST Austria
Abstract: The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differen- tial equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuous- time Markov chains. Decidability of the problem is currently open – indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel’s Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots. We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that de- cidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.
Keywords: reachability; Differential equations; Baker’s theorem; Schanuel’s conjecture; semialgebraic sets
Conference Title: ICALP: Automata, Languages and Programming
Volume: 55
Conference Dates: July 11-15, 2016
Conference Location: Rome, IT
Publisher: Springer  
Date Published: 2016-01-01
Start Page: Article Number: 100
Copyright Statement: CC BY
Sponsor: Ventsislav Chonev is supported by Austrian Science Fund (FWF) NFN Grant No S11407-N23 (RiSE/SHiNE), ERC Start grant (279307: Graph Games), and ERC Advanced Grant (267989: QUAREM).
DOI: 10.4230/LIPIcs.ICALP.2016.100
Open access: yes (OA journal)
IST Austria Authors
  1. Ventsislav Chonev
    3 Chonev