Computing over the Reals (or an Arbitrary Ring)
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.
KeywordsClassical Theory Optimization Theory Invariant Theory Computational Geometry Computational Problem
- L. Blum, Lectures on a theory of computation and complexity over the reals (or an arbitrary ring), TR-89-065, International Computer Science Institute, December 1989, 49 pp. (To appear in Lectures in the Sciences of Complexity II, Addison Wesley, 1990.)Google Scholar
- L. Blum, M. Shub and S. Smale, On a theory of computation and complexity over the real numbers: NP completeness, recursive functions and universal machines, The Bulletin of the American Mathematical Society, Vol. 21, No.1, (July 1989):l–46.Google Scholar
- L. Blum And S. Smale, The Godel incompleteness theorem and decidability over a ring, in preparation.Google Scholar