Abstract
In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the “space” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic log-space L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width.
Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of “read-once” certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs can be expressed as a read-once exponential sum over polynomials in VL, i.e. \({\sf VBP}\in{\it \Sigma}^R \cdot{\sf VL}\). We also show that \({\it \Sigma}^R \cdot {\sf VBP} ={\sf VBP}\), i.e. VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sums over a restricted class of constant-width arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property.
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
Valiant, L.G.: Completeness classes in algebra. In: Symposium on Theory of Computing STOC, pp. 249–261 (1979)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics. Springer, Heidelberg (2000)
Michaux, C.: Une remarque à propos des machines sur ℝ introduites par Blum, Shub et Smale. Comptes Rendus de l’Académie des Sciences de Paris 309(7), 435–437 (1989)
de Naurois, P.J.: A Measure of Space for Computing over the Reals. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 231–240. Springer, Heidelberg (2006)
Koiran, P., Perifel, S.: VPSPACE and a Transfer Theorem over the Reals. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 417–428. Springer, Heidelberg (2007)
Koiran, P., Perifel, S.: VPSPACE and a Transfer Theorem over the Complex Field. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 359–370. Springer, Heidelberg (2007)
Limaye, N., Mahajan, M., Rao, B.V.R.: Arithmetizing classes around NC\(^{\mbox{1}}\) and L. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 477–488. Springer, Heidelberg (2007); full version in ECCC TR07-087
Mahajan, M., Rao, B.V.R.: Arithmetic circuits, syntactic multilinearity and skew formulae. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 455–466. Springer, Heidelberg (2008); full version in ECCC TR08-048
Jansen, M., Rao, B.: Simulation of arithmetical circuits by branching programs preserving constant width and syntactic multilinearity. In: Frid, A.E., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) CSR 2009. LNCS, vol. 5675. Springer, Heidelberg (2009)
Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, New York (1999)
Arora, S., Barak, B.: Complexity Theory: A Modern Approach (to be published) (2009)
Barrington, D.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences 38(1), 150–164 (1989)
Venkateswaran, H.: Circuit definitions of nondeterministic complexity classes. SIAM Journal on Computing 21, 655–670 (1992)
Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 704–716. Springer, Heidelberg (2006)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1997)
Malod, G.: The complexity of polynomials and their coefficient functions. In: IEEE Conference on Computational Complexity, pp. 193–204 (2007)
Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC1 computation. Journal of Computer and System Sciences 57, 200–212 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mahajan, M., Rao, B.V.R. (2009). Small-Space Analogues of Valiant’s Classes. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds) Fundamentals of Computation Theory. FCT 2009. Lecture Notes in Computer Science, vol 5699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03409-1_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-03409-1_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03408-4
Online ISBN: 978-3-642-03409-1
eBook Packages: Computer ScienceComputer Science (R0)