Computing over the Reals (or an Arbitrary Ring)

Abstract

Classically, the theories of computation and computational complexity deal with discrete problems, for example over the integers, about graphs, etc.. On the other hand, most computational problems that arise in numerical analysis and scientific computation, in optimization theory and more recently in robotics and computational geometry have as natural domains the reals, or complex numbers. A variety of ad hoc methods and models have been employed to analyze complexity issues in this realm, but unlike the classical case, a natural and invariant theory has not yet emerged. One would like to develop theoretical foundations for a theory of computational complexity for numerical analysis and scientific computation that might embody some of the naturalness and strengths of the classical theory.