On the skolem problem for continuous linear dynamical systems

Chonev, Ventsislav and Ouaknine, Joël and Worrell, James (2016) On the skolem problem for continuous linear dynamical systems. In: ICALP: International Colloquium on Automata, Languages, and Programming , July 11-15, 2016, Rome, IT.

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Official URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.100

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.

Item Type: Conference or Workshop Item (Paper)
Uncontrolled Keywords: reachability, differential equations, Baker's theorem, Schanuel's conjecture, semialgebraic sets
Subjects: 000 Computer science, knowledge & general works > 000 Computer science, knowledge & systems > 004 Data processing & computer science
000 Computer science, knowledge & general works > 000 Computer science, knowledge & systems > 006 Special computer methods
Research Group: Chatterjee Group
SWORD Depositor: Sword Import User
Depositing User: Sword Import User
Date Deposited: 01 Mar 2017 08:46
Last Modified: 30 Aug 2017 14:40
URI: https://repository.ist.ac.at/id/eprint/778

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