Abstract
Suppose that a Boolean functionf is computed by a constant depth circuit with 2m AND-, OR-, and NOT-gates—andm majority-gates. We prove thatf is computed by a constant depth circuit with\(2^{m^{O(1)} }\) AND-, OR-, and NOT-gates—and a single majority-gate, which is at the root.
One consequence is that if a Boolean functionf is computed by an AC0 circuit plus polylog majority-gates, thenf is computed by a probabilistic perceptron having polylog order. Another consequence is that iff agrees with the parity function on three-fourths of all inputs, thenf cannot be computed by a constant depth circuit with AND-, OR-, and NOT-gates, and\(2^{n^{O(1)} } \) majority-gates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Allender, A note on the power of threshold circuits. InProc. 30th Ann. Symp. Found. Comput. Sci., 1989, 580–584.
E. Allender and U. Hertrampf, Depth reduction for circuits of unbounded fanin.Inform. and Comput. 108 (1994). To appear.
A. Amir, R. Beigel, and W. I. Gasarch, Some connections between bounded query classes and nonuniform complexity. InProc. 5th Ann. Conf. Structure in Complexity Theory, 1990, 232–243.
J. Aspnes, R. Beigel, M. Furst, and S. Rudich, The expressive power of voting polynomials. InProc. 23rd Ann. ACM Symp. Theor. Comput., 1991, 402–409. A revised version is to appear inCombinatorica.
L. Babai, A random oracle separates PSPACE from the polynomial hierarchy.Inform. Process. Lett. 26 (1987), 51–53.
R. Beigel and J. Tarui, On ACC. This Journal.
R. Beigel, N. Reingold, and D. Spielman, The perceptron strikes back. InProc. 6th Ann. Conf. Structure in Complexity Theory, 1991, 286–291.
R. Beigel, N. Reingold, and D. Spielman, PP is closed under intersection.J. Comput. System Sci. 48 (1994). To appear.
J.-Y. Cai, With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy.J. Comput. System Sci. 38 (1989), 68–85.
L. Fortnow and N. Reingold, PP is closed under truth-table reductions. InProc. 6th Ann. Conf. Structure in Complexity Theory, 1991, 13–15.
J. T. Håstad,Computational Limitations for Small-Depth Circuits. ACM Doctoral Dissertation Award. MIT Press, Cambridge, MA, 1986.
J. Hastad, Almost optimal lower bounds for small depth circuits. InAdvances in Computing Research 5: Randomness and Computation, ed.S. Micali, Greenwich, CT, 1989, JAI Press, 143–170.
D. J. Newman, Rational approximation to |x|.Michigan Mathematical Journal 11 (1964), 11–14.
C. H. Papadimitriou and S. K. Zachos, Two remarks on the complexity of counting. InProc. of the 6th GI Conference on Theoretical Computer Science. Springer-Verlag, 1983, 269–276. Volume 145 ofLecture Notes in Computer Science.
R. Paturi and M. E. Saks, Approximating threshold circuits by rational functions.Inform. and Comput. 109 (1994). To appear.
K.-Y. Siu, V. Roychowdhury, and T. Kailath, Computing with almost optimal size neural networks. InAdvances in Neural Information Processing Systems, ed.S. J. Hancock, J. Cowan, and L. Giles, vol. 5. Morgan Kaufman, 1993.
J. Tarui, Probabilistic polynomials, AC0 functions, and the polynomial-time hierarchy.Theoret. Comput. Sci. 113 (1993), 167–183.
A. C. Yao, Separating the polynomial-time hierarchy by oracles. InProc. 26th Ann. Symp. Found. Comput. Sci., 1985, 1–10.
A. C.-C. Yao, On ACC and threshold circuits. InProc. 31st Ann. Symp. Found. Comput. Sci., 1990, 619–627.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beigel, R. When do extra majority gates help? Polylog (N) majority gates are equivalent to one. Comput Complexity 4, 314–324 (1994). https://doi.org/10.1007/BF01263420
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01263420