Abstract
We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We prove that the languages recognized by PCD systems in dimension d in finite continuous time are precisely the languages of the d–2 th level of the arithmetical hierarchy. Hence we provide a precise characterization of the computational power of purely rational PCD systems in continuous time according to their dimension and we solve a problem left open by [2].
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Eugene Asarin and Oded Maler. On some Relations between Dynamical Systems and Transition Systems. In Proceedings of ICALP, pages 59–72, 1994. Lecture Notes in Computer Science, 820.
Eugene Asarin and Oded Maler. Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy. In Proceedings of FSTTCS, pages 471–483, 1995. Lecture Notes in Computer Science, 1026.
Eugene Asarin, Oded Maler, and Amir Pnueli. Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science, 138:33–65, 1995.
Olivier Bournez and Michel Cosnard. On the computational power of hybrid and dynamical systems. Theoretical Computer Science, 168(2):417–459, 1996.
Michael S. Branicky. Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoretical Computer Science, 138:67–100, 1995.
Cristopher Moore. Recursion theory on the reals and continuous-time computation. Theoretical Computer Science, 162:23–44, 1996.
P. Odifreddi. Classical Recursion Theory, volume 125 of Studies in Logic and the foundations of mathematics. Elsevier, 1992.
H. R.ogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bournez, O. (1997). Some bounds on the computational power of piecewise constant derivative systems (extended abstract). In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds) Automata, Languages and Programming. ICALP 1997. Lecture Notes in Computer Science, vol 1256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63165-8_172
Download citation
DOI: https://doi.org/10.1007/3-540-63165-8_172
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63165-1
Online ISBN: 978-3-540-69194-5
eBook Packages: Springer Book Archive