Recursion Theory on the Reals and Continuous-Time Computation

We de ne a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time This class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense We stratify this class of functions into a hierarchy according to the number of uses of the zero nding operator At the lowest level are continuous functions that are di erentially algebraic and computable by Shannon s General Purpose Analog Computer At higher levels are in creasingly discontinuous and complex functions We relate this hierarchy to the Arithmetical and Analytical Hierarchies of classical recursion the