, the cardinality of V a r . W e propose also the following deenitions: Deenition 4 . 4 A hybrid system H is time-deterministic 1] if for every l 2 Loc and every v 2 V, system H, and note dim(H)

, Act(l) and every tt t 0 2 R +, A hybrid system H is discrete-deterministic if for every l 2 Loc

R. Alur, C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P. H. Ho et al., The algorithmic analysis of hybrid systems, In Theoretical Computer Science, vol.138, pp.3-34, 1995.

R. Alur, C. Courcoubetis, T. A. Henzinger, and P. , Hybrid Automata: An algorithmic Approa c h t o t h e S p eciication and Veriication of Hybrid Systems, Lecture Notes in Computer Science, vol.630, pp.340-354, 1992.

E. Asarin, O. Maler, and A. Pnueli, Reachability analysis of dynamical systems having piecewise-constant derivatives, In Theoretical Computer Science, vol.138, pp.33-65, 1995.

E. Asarin and O. Maler, On some Relations between Dynamical Systems and Transition Systems

J. L. Balcc-azar, J. Dii, and J. Gabarrr, Structural Complexity I. EATCS Monographs on Theoretical Computer Science, 1988.

C. H. , Bennett Logical Reversibility of computation, IBM J.Res.Develop, vol.6, pp.525-532, 1973.

L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers : NP-completeness, recursive functions and universal machines, Bulletin of the American Mathematical Society, vol.21, issue.1, pp.1-46, 1989.

M. S. Branicky, Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems, Theoretical Computer Science, vol.138, pp.67-100, 1995.

K. Cerans, Decidability of Bissimulation Equivalences for Parallel Timer Processes, Proc. 4th Ann. Workshop on Computer-Aided Veriication, vol.663, pp.269-300, 1992.

J. E. Hopcroft and J. D. , Ullman Introduction to Automata Theory Languages and Computation Addison-Wesley, 1979.
DOI : 10.1145/568438.568455

Y. Kesten, A. Pnuelli, J. Sifakis, and S. Yovine, Integration Graphs: A Class of Decidable Hybrid Systems, Hybrid Systems, vol.736, pp.179-208, 1993.

P. Koiran, Puissance d e c alcul des r eseaux de neurones artiiciels, 1993.

P. Koiran, Computing over the reals with addition and order, Theoretical Computer Science, vol.133, pp.35-47, 1994.
DOI : 10.1016/0304-3975(93)00063-b

URL : https://doi.org/10.1016/0304-3975(93)00063-b

P. Koiran, On the relations between dynamical systems and boolean circuits
URL : https://hal.archives-ouvertes.fr/hal-02101931

P. Koiran, M. Cosnard, and M. Garzon, Computability with low-dimensional dynamical systems, In Theoretical Computer Science, v ol, vol.132, pp.113-128, 1994.
DOI : 10.1016/0304-3975(94)90229-1

URL : https://doi.org/10.1016/0304-3975(94)90229-1

O. Maler, Z. Manna, and A. Pnuelli, From Timed to Hybrid Systems, Proc. of the REX Workshop \Real-Time: Theory in Practice, vol.600, pp.447-484, 1992.
DOI : 10.1007/bfb0032003

O. Maler and A. Pnuelli, Reachability Analysis of Planar Multi-linear Systems, Proc. of the 5th Workshop on Computer-Aided Veriication, vol.697, pp.194-209
DOI : 10.1007/3-540-56922-7_17

URL : https://link.springer.com/content/pdf/10.1007%2F3-540-56922-7_17.pdf

K. Meer and C. Michaux, A survey on real Structural Complexity Theory Preprint

C. Moore, Unpredictability and Undecidability in Dynamical Systems, In Physical Review Letters, vol.20, p.2354, 1990.
DOI : 10.1103/physrevlett.64.2354

X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine, An A p p r oach to the Description and Analysis of Hybrid Systems, Lect. Notes in Comp. Sci, vol.736, pp.149-178, 1993.

X. Nicollin, J. Sifakis, and S. Yovine, From ATP Timed G r aphs and Hybrid Systems, In Acta Inform, vol.30, pp.181-202, 1993.
DOI : 10.1007/bfb0032007

H. T. Siegelman, The Simple Dynamics of Super Turing Theories

H. T. Siegelman and E. D. Sontag, Analog c omputation via neural networks, Proc. 2 nd Israeli Symposium on Theory of Computing and Systems, 1993.