Undecidability Over Continuous-time JERZY MYCKA, Institute of Mathematics, University of Maria Curie-Sklodowska, Lublin, Poland, E-mail: Jerzy.Mycka@umcs.lublin.pl JOS ´ EF ´ ELIX COSTA, Department of Mathematics, I.S.T., Universidade T´ ecnica de Lisboa, Lisboa, Portugal, E-mail: fgc@math.ist.utl.pt Abstract Since 1996, some models of recursive functions over the real numbers have been analyzed by several researchers. It could be expected that they exhibit a computational power much greater than that of Turing machines (as other well known models of computation over the real numbers already considered in the past fifteen years, like neural net models with real weights). The fact is that they have not got such a power. Although they decide the classical halting problem of Turing machines, they have almost the same limitations of Turing machines. Our profit on them has been to represent classical complexity classes (like P or NP ) by analytical means, and possibly relate them by unusual ways. Keywords : decidability, analog computation, real recursive functions 1 Introduction and motivation The first presentation of the theory of recursive functions over the reals, analogous to Kleene’s classical theory, was attempted by Cristopher Moore [9]. Real recursive functions are generated by a fundamental operator, called differential recursion. The other fundamental operator is the taking of infinite limits [12]. Between 1996 (since Moore’s seminal paper) and 2002, we have been working with the single concept of differential recursion. In [4] it is shown that a linearization of the differential recursion scheme gives rise to an analog characterization of the class of (Kalmar’s) elementary functions and of the Grzegorczyk hierarchy. In [3] and [8] it is proved that the GPAC (General Purpose Analog Computer) of Claude Shannon [18] is not closed under iteration and that a subclass of real recursive functions coincides with the class of GPAC-computable functions (as in [18, 16]). In [12] we finally show how to capture higher computational classes through the limit operator. Manuel Campagnolo, a PhD student of one of the authors, showed also in [2] that other computational complexity classes can be captured through appropriate structured differential schemata or by adding simple (bounded) integration. In [13] we try to show that our framework is versatile: from a careful and not so complex definition of the (countable) set of recursive functions over the reals we show by means of the toolbox of Analysis that: (a) Laplace transform can be used to quickly obtain useful real recursive functions and to measure their rate of growth, (b) the embedding of Turing machines into real recursive functions is trivial, (c) a (limit) 1 L. J. of the IGPL, Vol. 0 No. 0, pp. 1–10 0000 c  Oxford University Press