2015/06/01 by Ventsislav Chonev, Joël Ouaknine, Chonev, Ventsislav +3
Computer Science · #Algorithms and Data Compression #F.1.1 #F.2.1 #FOS: Computer and information sciences #FOS: Electrical engineering #Logic, programming, and type systems #Symbolic Computation (cs.SC) #Systems and Control (eess.SY) #electronic engineering #information engineering #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1506.00695
openalex publication_date 2015/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The Continuous Skolem Problem asks whether a real-valued 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 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 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 Diophantine-approximation types of all real algebraic numbers.