Abstract
Asymptotically tight bounds are obtained for the complexity of computation of the classes of (m, n)-matrices with entries from the set {0, 1,..., q − 1} by rectifier circuits of bounded depth d, under some relations between m, n, and q. In the most important case of q = 2, it is shown that the asymptotics of the complexity of Boolean (m, n)-matrices, log n = o(m), logm = o(n), is achieved for the circuits of depth 3.
Similar content being viewed by others
References
S. B. Gashkov and V. V. Kochergin, “On Addition Chains of Vectors, Gate Circuits, and the Complexity of Computations of Powers,” in Methods of Discrete Analysis in Graph Theory and Complexity, Vol. 52 (Izd. Inst. Mat., Novosibirsk, 1992), pp. 22–40 [Siberian Adv. Math. 4 (4), 1–16 (1994)].
S. B. Gashkov and I. S. Sergeev, “An Application of the Method of Additive Chains to Inversion in Finite Fields,” Diskretn. Mat. 18 (4), 56–72 (2006) [Discrete Math. Appl. 16 (6), 601–618 (2006)].
V. V. Kochergin, “On the Complexity of Calculating Systems of Monomials with Restrictions on the Powers of Variables,” Diskretn. Mat. 10 (3), 27–34 (1998) [Discrete Math. Appl. 8 (4), 375–382 (1998)].
V. V. Kochergin, “The Theory of Rectifier Circuits (the Present State),” in Discrete Mathematics and Its Applications, Vol. 7 (Izd. Inst. Prikl.Mat.,Moscow, 2013), pp. 23–40.
O. B. Lupanov, “On Rectifier and Switching-and-Rectifier Schemes,” Dokl. Akad. Nauk SSSR 111 (6), 1171–1174 (1956).
E. I. Nechiporuk, “Rectifier Networks,” Dokl. Akad. Nauk SSSR 148 (1), 50–53 (1963) [Sov. Phys. Dokl. 8, 5–7 (1963)].
E. I. Nechiporuk, “On the Topological Principles of Self-Correction,” in Problems of Cybernetics, Vol. 21 (Nauka, Moscow, 1969), pp. 5–102 [Systems Theory Res. 21, 1–99 (1970)].
A. Brauer, “On Addition Chains,” Bull. AMS 45, 736–739 (1939).
S. Jukna and I. S. Sergeev, “Complexity of Linear Boolean Operators,” Found. Trends Theor. Comput. Sci. 9 (1), 1–123 (2013).
N. Pippenger, “The Minimum Number of Edges in Graphs with Prescribed Paths,” Math. Systems Theory 12, 325–346 (1979).
N. Pippenger, “On the Evaluation of Powers and Monomials,” SIAM J. Comput. 9 (2), 230–250 (1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.S. Sergeev, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 1, pp. 120–141.
Rights and permissions
About this article
Cite this article
Sergeev, I.S. Rectifier Circuits of Bounded Depth. J. Appl. Ind. Math. 12, 153–166 (2018). https://doi.org/10.1134/S1990478918010143
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478918010143