Abstract
We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class \({\sf PAR}_{\mathbb{R}}\) of decision problems that can be solved in parallel polynomial time over the real numbers collapses to P ℝ. As a result, one must first be able to show that there are \({\sf VPSPACE}\) families which are hard to evaluate in order to separate \(\sf{P}_{\mathbb{R}}\) from \(\sf{NP}_{\mathbb{R}}\), or even from \(\sf{PAR}_{\mathbb{R}}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blum, L., et al.: Complexity and Real Computation. Springer, Heidelberg (1998)
Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society 21(1), 1–46 (1989)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer, Heidelberg (2000)
Chapuis, O., Koiran, P.: Saturation and stability in the theory of computation over the reals. Annals of Pure and Applied Logic 99, 1–49 (1999)
Charbit, P., et al.: Finding a vector orthogonal to roughly half a collection of vectors. Accepted for publication in Journal of Complexity (2006), Available from http://perso.ens-lyon.fr/pascal.koiran/publications.html
Cole, R.: Parallel merge sort. SIAM J. Comput. 17(4), 770–785 (1988)
Cucker, F., Grigoriev, D.: On the power of real Turing machines over binary inputs. SIAM Journal on Computing 26(1), 243–254 (1997)
Grigoriev, D.: Complexity of deciding Tarski algebra. Journal of Symbolic Computation 5, 65–108 (1988)
Grigoriev, D.: Topological complexity of the range searching. Journal of Complexity 16, 50–53 (2000)
Koiran, P.: Valiant’s model and the cost of computing integers. Computational Complexity 13, 131–146 (2004)
Koiran, P., Perifel, S.: Valiant’s model: from exponential sums to exponential products. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 596–607. Springer, Heidelberg (2006)
Koiran, P., Perifel, S.: VPSPACE and a Transfer Theorem over the Reals (2006), Available from http://prunel.ccsd.cnrs.fr/ensl-00103018
Malod, G.: Polynômes et coefficients. PhD thesis, Université Claude Bernard Lyon 1 (July 2003)
Milnor, J.: On Betti numbers of real varieties. Proceedings of the American Mathematical Society 15(2), 275–280 (1964)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Poizat, B.: Les petits cailloux. Aléas, Lyon (1995)
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, part 1. Journal of Symbolic Computation 13, 255–299 (1992)
Valiant, L.G.: Completeness classes in algebra. In: Proc. 11th ACM Symposium on Theory of Computing, pp. 249–261. ACM Press, New York (1979)
Valiant, L.G., et al.: Fast parallel computation of polynomials using few processors. SIAM Journal on Computing 12(4), 641–644 (1983)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Koiran, P., Perifel, S. (2007). VPSPACE and a Transfer Theorem over the Reals. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-70918-3_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70917-6
Online ISBN: 978-3-540-70918-3
eBook Packages: Computer ScienceComputer Science (R0)