On the complexity of some problems for the Blum, Shub & Smale model
We show some problems deriving from real algebra and semialgebraic geometry to be NP-complete or coNP-complete for the Blum, Shub and Smale model of computation. We also introduce a class of languages R lying between P and NP that uses probabilistic machines, and several problems from the same area are classified as “probably noncomplete” by showing their membership in R.
Unable to display preview. Download preview PDF.
- J.L. Balcázar, J. Díaz and J. Gabarró; Structural Complexity, vol.1, EATCS Monographs of Theoretical Computer Science, Springer Verlag, 1988.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”. Bulletin of the Amer. Math. Soc., vol.21, n.1, pp.1–46, 1989. A preliminary version appeared in 29th Found. of Comp. Sc. pp.387–397, 1988.Google Scholar
- J. Bochnak, M. Coste and M.-F. Roy; Géométrie algébrique réelle. Ergebnisse der Math., 3.Folge, Band 12, Springer Verlag, 1987.Google Scholar
- D. Grigor′; “Complexity of deciding Tarski algebra”. J. of Symb. Comp., 5, pp.65–108, 1988.Google Scholar
- J. Heintz, T. Krick, M.-F. Roy and P. Solerno;“Single exponential time algorithms for basic constructions in elementary geometry”, Proceedings of the 10th Int. Conf. of the Chilean Comp. Sc. Soc., Santiago de Chile, 1990.Google Scholar
- J. Heintz, M.-F. Roy and P. Solerno; “Sur la complexité du principe de Tarski-Seidenberg”. Bull. Soc. Math. France, 118, pp.101–126, 1990.Google Scholar
- M. Mignote; “Some useful bounds” in Computer Algebra, Symbolic and Algebraic Computation, pp.259–263, Springer Verlag, 1982.Google Scholar
- J. Renegar; “On the computational complexity and geometry of the first order thery of the reals”, parts I, II and III. Cornell University, Technical Reports 853,854 and 856, 1989.Google Scholar