, HyperJumpg] is a sampling of H up to local dimension

, Then for all k 2 N, x k is of local dimension (d 0 + 1 ) , w h e r e HyperJumpg] ( k QQ tt x) = ( t k x k ), t k 2 R x k 2 R d . For all t 2 R, for all x 2 R d, Assume HyperJumpg] is Zeno for some Q 2 P t2 Rx2 R d

, There exists a xed rst order formula F such that for all k 2 N Q2 P t2 Rx2 R d , dHyperJumpg](k + 1 Q t x )e is deenable by formula F from relation R g (HyperJumpg](k QQ tt x)) and from some recursive relations

, There exists clearly a xed rst order formula G such that for all real sequence (g 0 (k 0 Q x )) k 0 2N converging to some g 0 (QQ x) 2 R d , dg 0 (QQ x)e is deenable by f o r m ula G from relation R g 0 (x), where R g 0 denotes the relation associated to g 0 corresponding to x. As a consequence, the last assertion is clear, since deenition 5.8 can be translated directly into a xed rst order formula F that, there exists a xed rst order formula over relation R g (tt x) that tells if g is Zeno for QQ tt x, vol.42

, We prove n o w that HyperJumpg] is a sampling of H up to local dimension

, t k 2 R x k 2 R d , f o r all k 2 N. Let be the trajectory of H starting from x at time t. From the fact that g is a sampling it is easy to show b y induction over k that for all k 2 N (t k ) = x k . N o w, if there is some k 0 with x k0 2 Q NoEvolution(H), since g is a sampling, it is clear than x k = x k0 t k = t k0 for all k k 0 . If for all kkx k 6 2 Q NoEvolution(H), QQ tt x) = ( t k x k )

, QQ tt x. F or all k 2 N, g must be Zeno for QQ t k;1 x k;1 : hence, (t k x k ) is the limit of g(k 0 Q t k;1 x k;1 ) k 0 2 N. Since g is a sampling up to local dimension (d 0 ) + , the local dimension of x k must be >

, By lemma 5.4, the local dimension of x is > (d 0 + 1 ) + . T h i s proves the rst assertion, Denote t = sup k2N t k and x = = ( t )

. N-p-r-r-d-!-r-r,

, some x 2 Q d , a r ational polyhedron F not intersecting Q, s u c h that Cycle(x k;1 z 2 H F x ) is true, x k;1 6 2 QQ z 2 6 2 Q, z 2 6 2 F c

, QQ tt x) = Cycle

, r(m) where r(m) i s a n a n H(y + 0 z k;1 + 0 1) d(ttx)e -recursive index of R gk, p.1

, By lemma 5.11, there exists a xed rst order formula F such that for all n 2 N Q2 P t2 Rx2 R d

, By lemma 5.2, there exists y F 2 OOjy F j < ! and a recursive g that maps r(m) t o g(r(m)), where g(r(m)) is an H, nn QQ tt x)) and from some recursive relations

, ))), where r 0 (g(r(m))) is an H(y + 0 z k;1 + 0 1 + o y F ) d(ttx)e -recursively enumerable index of HyperJumpg k

, 0 z k;1 + 0 1 + o y F for all n 2 N. As a consequence, for all n 2 N, R <n HyperJumpgk;1] is semi-recognized by the machine with oracle H d(ttx)e (h(n ; 1)) that on input <n Q P> , compute for i = 1 : : : n ; 1 a n H d(ttx)e (h(i ; 1))-recursively enumerable index m i of HyperJumpg k;1 ] ( ii QQ tt x) from the H d(ttx)e (h(i;2))-recursively enumerable index m i, Denote by h : N ! O the recursive mapping such that r(0) = 1 r (n + 1 ) = r(n) +

, Assume we h a ve H(y) d(ttx)e -recursively enumerable index m of CycleFreeg k;1 ](nn QQ tt x), where m 2 N y2 O. By lemma 5.6, there exists a recursive r that maps m to r(m) where r(m) i s a n H(y + 0 h(n ; 1) + 0 1) d(ttx)e -recursive i n d e x o f R <n HyperJumpgk;1] (CycleFreeg k;1 ](nn QQ tt x)). By lemma 5, vol.12

, As before, by lemma 5.2, and by lemma 5.6, there exists some recursive g and r 0 that maps m to r 0 (g(r(m))) an H(y+ 0 h(n;1)+ 0 1+ o y G ) d(ttx)e -recursively enumerable index of CycleFreeg k, QQ tt x)) and from some recursive relations, vol.1

E. Asarin and O. Maler, On some Relations between Dynamical Systems and Transition Systems, Proceedings of ICALP, p.820, 1994.

E. Asarin and O. Maler, Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy, Proceedings of FSTTCS, p.1026, 1995.

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

L. Blum and S. Smale, On a Theory of Computation and Complexity o ver the Real Numbers: NP-completeness, Recursive F unctions and Universal Machines, Bulletin of the American Mathematical Society, vol.21, issue.1, pp.1-46, 1989.

O. Bournez, Some bounds on the computational power of piecewise constant derivative systems, Proceeding of ICALP'97, 1997.

O. Bournez, Some bounds on the computational power of purely rational piecewise constant derivative systems, LIP ENS-Lyon, 1997.

O. Bournez and M. Cosnard, On the computational power of hybrid and dynamical systems, Theoretical Computer Science, vol.168, issue.2, pp.417-459, 1996.

M. S. Branicky, Universal computation and other capabilities of hybrid and continuous dynamical systems, Theoretical Computer Science, vol.138, pp.67-100, 1995.

J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory Languages and Computation, 1979.

P. Koiran, Computing over the reals with addition and order, Theoretical Computer Science, vol.133, pp.35-47, 1994.

P. Koiran, A Weak Version of the Blum Shub Smale Model, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, pp.486-495, 1993.

K. Meer and C. Michaux, A Survey on real Structural Complexity Theory, Bulletin of the Belgian Mathematical Society -Simon Stevin

K. Meer, A note on a P 6 = NPResult for a Restricted Class of Real Machines, Journal of Complexity, vol.8, pp.451-453, 1992.

C. Moore, Recursion theory on the reals and continuous-time computation, Theoretical Computer Science, vol.162, pp.23-44, 1996.

P. Odifreddi, Classical Recursion Theory, v olume 125 of Studies in Logic and the foundations of mathematics, 1992.

H. Rogers, Theory of Recursive Functions and EEective Computability, 1967.