Chonev, Ventsislav KIST Austria ; Ouaknine, Joël ; Worrell, James
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.
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).
ICALP: Automata, Languages and Programming
2016-07-12 – 2016-07-15
Chonev VK, Ouaknine J, Worrell J. On the skolem problem for continuous linear dynamical systems. In: Vol 55. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik; 2016. doi:10.4230/LIPIcs.ICALP.2016.100
Chonev, V. K., Ouaknine, J., & Worrell, J. (2016). On the skolem problem for continuous linear dynamical systems (Vol. 55). Presented at the ICALP: Automata, Languages and Programming, Rome, Italy: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik. https://doi.org/10.4230/LIPIcs.ICALP.2016.100
Chonev, Ventsislav K, Joël Ouaknine, and James Worrell. “On the Skolem Problem for Continuous Linear Dynamical Systems,” Vol. 55. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik, 2016. https://doi.org/10.4230/LIPIcs.ICALP.2016.100.
V. K. Chonev, J. Ouaknine, and J. Worrell, “On the skolem problem for continuous linear dynamical systems,” presented at the ICALP: Automata, Languages and Programming, Rome, Italy, 2016, vol. 55.
Chonev VK, Ouaknine J, Worrell J. 2016. On the skolem problem for continuous linear dynamical systems. ICALP: Automata, Languages and Programming, LIPIcs, vol. 55.
Chonev, Ventsislav K., et al. On the Skolem Problem for Continuous Linear Dynamical Systems. Vol. 55, 100, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik, 2016, doi:10.4230/LIPIcs.ICALP.2016.100.