Some Bounds on the Computational Power of Piecewise Constant Derivative Systems

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 relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the (d-2) th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is Σd-2 -complete.

[1]  Eugene Asarin,et al.  Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy , 1995, J. Comput. Syst. Sci..

[2]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[3]  Amir Pnueli,et al.  Reachability Analysis of Dynamical Systems Having Piecewise-Constant Derivatives , 1995, Theor. Comput. Sci..

[4]  P. A. Smith.  Review: M. H. A. Newman, Elements of the Topology of Plane Sets of Points , 1939 .

[5]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[6]  W. Rudin.  Real and complex analysis , 1968 .

[7]  Michael S. Branicky,et al.  Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems , 1995, Theor. Comput. Sci..

[8]  Eugene Asarin,et al.  On some Relations between Dynamical Systems and Transition Systems , 1994, ICALP.

[9]  M. H. A. Newman,et al.  Topology . Elements of the topology of plane sets of points , 1939 .

[10]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Pascal Koiran.  Computing over the Reals with Addition and Order , 1994, Theor. Comput. Sci..

[13]  Olivier Bournez.  Achilles and the Tortoise Climbing up the Hyper-Arithmetical Hierarchy , 1999, Theor. Comput. Sci..

[14]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[15]  C. Michaux,et al.  A survey on real structural complexity theory , 1997 .

[16]  Société mathématique de Belgique.  Bulletin of the Belgian Mathematical Society, Simon Stevin , 1994 .

[17]  Klaus Meer.  A note on a P NP result for a restricted class of real machines , 1992, J. Complex..

[18]  Pascal Koiran.  A weak version of the Blum, Shub and Smale model , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[19]  P. Odifreddi.  Classical recursion theory , 1989 .

[20]  Olivier Bournez.  Some Bounds on the Computational Power of Piecewise Constant Derivative Systems (Extended Abstract) , 1997, ICALP.

[21]  José L. Balcázar,et al.  Structural Complexity II , 2012, EATCS.

[22]  Cristopher Moore,et al.  Recursion Theory on the Reals and Continuous-Time Computation , 1996, Theor. Comput. Sci..