Chonev, Ventsislav ;
Ouaknine, Joël ;
Worrell, James
On the Skolem Problem for Continuous Linear Dynamical Systems
Abstract
The Continuous Skolem Problem asks whether a realvalued function satisfying a linear differential 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 continuoustime Markov chains. Decidability of the problem is currently open — indeed decidability is open even for the subproblem 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 decidability 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 Diophantineapproximation types of all real algebraic numbers.
BibTeX  Entry
@InProceedings{chonev_et_al:LIPIcs:2016:6235,
author = {Ventsislav Chonev and Jo{\"e}l Ouaknine and James Worrell},
title = {{On the Skolem Problem for Continuous Linear Dynamical Systems}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {100:1100:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770132},
ISSN = {18688969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6235},
URN = {urn:nbn:de:0030drops62357},
doi = {10.4230/LIPIcs.ICALP.2016.100},
annote = {Keywords: differential equations, reachability, Baker’s Theorem, Schanuel’s Conjecture, semialgebraic sets}
}
23.08.2016
Keywords: 

differential equations, reachability, Baker’s Theorem, Schanuel’s Conjecture, semialgebraic sets 
Seminar: 

43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Issue date: 

2016 
Date of publication: 

23.08.2016 