E. Asarin, O. Maler, **On some Relations between Dynamical
Systems and Transition Systems.** In this paper we define a
precise notion of abstraction relation between continuous
dynamical systems and discrete state-transition systems. Our main
result states that every Turing Machine can be realized by a
dynamical system with piecewise-constant derivatives in a 3-dimensional
space and thus the reachability problem for such systems is
undecidable for 3 dimensions. A decision procedure for 2-dimensional
systems has been recently reported by Maler and Pnueli. On the
other hand we show that some non-deterministic finite automata
cannot be realized by any continuous dynamical system with less
than 3 dimensions. [Postscript]