conference paper published yes Ventsislav K Chonev author 36CBE2E6-F248-11E8-B48F-1D18A9856A87 Joël Ouaknine author James Worrell author KrCh department ICALP: Automata, Languages and Programming Rigorous Systems Engineering project Quantitative Graph Games: Theory and Applications project Quantitative Reactive Modeling project 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. https://research-explorer.ista.ac.at/download/1069/5213/IST-2017-778-v1+1_LIPIcs-ICALP-2016-100.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2016Rome, Italy eng 10.4230/LIPIcs.ICALP.2016.100 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, 100. V.K. Chonev, J. Ouaknine, J. Worrell, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Chonev, Ventsislav K., et al. <i>On the Skolem Problem for Continuous Linear Dynamical Systems</i>. Vol. 55, 100, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016, doi:<a href="https://doi.org/10.4230/LIPIcs.ICALP.2016.100">10.4230/LIPIcs.ICALP.2016.100</a>. Chonev VK, Ouaknine J, Worrell J. On the skolem problem for continuous linear dynamical systems. In: Vol 55. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2016. doi:<a href="https://doi.org/10.4230/LIPIcs.ICALP.2016.100">10.4230/LIPIcs.ICALP.2016.100</a> Chonev, Ventsislav K, Joël Ouaknine, and James Worrell. “On the Skolem Problem for Continuous Linear Dynamical Systems,” Vol. 55. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. <a href="https://doi.org/10.4230/LIPIcs.ICALP.2016.100">https://doi.org/10.4230/LIPIcs.ICALP.2016.100</a>. Chonev, V. K., Ouaknine, J., &#38; 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 für Informatik. <a href="https://doi.org/10.4230/LIPIcs.ICALP.2016.100">https://doi.org/10.4230/LIPIcs.ICALP.2016.100</a> 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. 10692018-12-11T11:49:59Z2025-06-03T11:32:08Z