Abstract
We investigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functionsf k that can be computed in depthk and linear size ⋎, ⋏-circuits but require exponential size to compute by a depthk−1 monotone weighted threshold circuit.
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References
M. Ajtai. ∑ 11 on finite structures.Annals of Pure and Applied Logic, 24:1–48, 1983.
E. Allender. A note on the power of threshold circuits.Proceedings 30'th Annual Symposium on Foundations of Computer Science, pages 580–584, 1989.
N. Alon and R. B. Boppana. The monotone circuit complexity of boolean functions.Combinatorica, 7:1–22, 1987.
A. E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions.Dokl. Ak. Nauk. SSSR 282, pages 1033–1037, 1985. English translation inSov. Math. Dokl., 31:530–534, 1985.
L. Babai, N. Nisan, and M. Szegedy. Multipary protocols and logspacehard pseudorandom sequences.Proceedings of 21 st Annual ACM Symposium on Theory of Computing, pages 1–11, 1989.
P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems.Proceedings 25'th Annual Symposium on Foundations of Computer Science, pages 1–6, 1984.
A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility.SIAM J. on Computing, 13:423–439, 1984.
B. Chor and O. Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity.SIAM J. on Computing, 17:230–261, 1988.
M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial time hierarchy.Math. System Theory, 17:13–27, 1984.
J. Hastad.Computational Limitations of Small-Depth Circuits. MIT PRESS, 1986.
J. Hastad and M. Goldmann. On the power of small-depth threshold circuits.Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science, pages 610–618, 1990.
A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turán. Threshold circuits of bounded depth.Proceedings 28th Annual IEEE Symposium on Foundation of computer science, pages 99–110, 1987.
M. Karchmer and A. Wigderson. Monotone circuits for connectivity require super-logarithmic depth. InProceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988.
R. Raz and A. Wigderson. Probabilistic communication complexity of boolean relations. InProceedings of the 30th Annual IEEE Symposium on Foundation of computer science, pages 562–567, 1989.
R. Raz and A. Wigderson. Monotone circuits for matching require linear depth.22nd annual ACM Symposium on Theory of Computing, pages 287–292, 1990.
A. A. Razborov. Lower bounds on monotone network complexity of the logical permanent.Matem. Zam., 37(6):887–900, 1985. English translation inMath. Notes of the Academy of Sciences of the USSR, 37:485–493, 1985.
A. A. Razborov. Lower bounds on the size of bounded-depth networks over a complete basis with logical addition.Mathematical Notes of the Academy of Sciences of the USSR, 41(4):598–607, 1987. English translation in 41:4, pages 333–338.
M. Sipser. Borel sets and circuit complexity. InProceedings of 15th Annual ACM Symposium on Theory of Computing, pages 61–69, 1983.
R. Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity.Proceedings of 19th Annual ACM Symposium on Theory of Computing, pages 77–82, 1987.
S. Toda. On the computational power ofPP and ⊕P. Proceedings 30th Annual IEEE Symposium on Foundations of Computer Science, pages 514–519, 1989.
A. Yao. Separating the polynomial-time hierarchy by oracles.Proceedings 26th Annual IEEE Symposium on Foundations of Computer Science, pages 1–10, 1985.
A. Yao. Circuits and local computation.Proceedings of 21 st Annual ACM Symposium on Theory of Computing, pages 186–196, 1989.
A. Yao. OnACC and threshold circuits.Proceedings 31 st Annual IEEE Symposium on Foundations of Computer Science, pages 619–627, 1990.
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Håstad, J., Goldmann, M. On the power of small-depth threshold circuits. Comput Complexity 1, 113–129 (1991). https://doi.org/10.1007/BF01272517
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DOI: https://doi.org/10.1007/BF01272517