# Parity, circuits, and the polynomial-time hierarchy

Article

- 745 Downloads
- 361 Citations

## Abstract

A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.

## Keywords

Computational Mathematic Programmable Logic Parity Function Transitive Closure Closure Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D. Angluin, Counting problems and the polynomial-time hierarchy.
*Theoretical Computer Science*, to appear.Google Scholar - 2.N. Blum, A 2.75
*n*lower bound for the combinational complexity of boolean functions. University of Saarbrucken, Technical Report.Google Scholar - 3.T. Baker, J. Gill, and R. Solovay, Relativizations of the
*P*=^{?}*NP*question.*SIAM Journal of Computing*, 4, 4, 1975.Google Scholar - 4.T. Baker and A. Selman, A second step toward the polynomial hierarchy.
*Theoretical Computer Science*, 8, 2, 1979, pp. 177–187.Google Scholar - 5.A. Chandra, D. Kozen, and L. Stockmeyer, Alternation.
*Journal of the ACM*, 28, 1, January 1981.Google Scholar - 6.Digital Equipment Corporation,
*Decsystem 10 Assembly Language Handbook.*Third Edition, 1973, pp. 51–52.Google Scholar - 7.M. Furst, Bounded width computation DAG's. In preparation, 1982.Google Scholar
- 8.M. Furst, J. B. Saxe, M. Sipser, Parity, circuits and the polynomial-time hierarchy. 22ND
*Symposium on the Foundations of Computer Science*, 1981, pp. 260–270.Google Scholar - 9.M. Furst, J. B. Saxe, M. Sipser, Depth 3 circuits require Ω(
*n*^{clogn}) gates to compute parity: a geometric argument. In preparation.Google Scholar - 10.V. Krapchenko, Complexity of the realization of a linear function in the class of II-circuits. English translation in
*Math. Notes Acad. Sci., USSR*, 1971, pp. 21–23; orig. in*Mat. Zamet*, 9, 1, pp. 35–40.Google Scholar - 11.V. Krapchenko, A method of obtaining lower bounds for the complexity of II-schemes. English translation in
*Math. Notes Acad. Sci USSR*, 1972, pp. 474–479; orig. in*Mat. Zamet*, 10, 1, pp. 83–92.Google Scholar - 12.O. Lupanov, Implementing the algebra of logic functions in terms of constant-depth formulas in the basis +,
^{*}, -. English translation in*Sov. Phys.-Dokl.*, 6, 2, 1961; orig. in*Dokla. Akad. Nauk SSSR*, 136, 5.Google Scholar - 13.R. Ladner and N. Lynch, Relativization of questions about log space computability.
*Mathematical Systems Theory*, 10, 1, 1976.Google Scholar - 14.C. Mead and L. Conway,
*Introduction to VLSI Systems*. Addison-Wesley, Reading, Mass. 1980.Google Scholar - 15.W. Paul, A 2.5
*N*lower bound for the combinational complexity of boolean functions. 7th*Annual ACM Symposium on Theory of Computing*, 1975, pp. 27–36.Google Scholar - 16.J. Savage,
*The Complexity of Computing*. John Wiley and Sons, New York, 1976, Sect. 2.4.Google Scholar - 17.C. P. Schnorr, A 3
*n*lower bound on the network complexity of boolean functions.*Theoretical Computer Science*, 10, 1, 1980, p. 83.Google Scholar - 18.L. J. Stockmeyer, The polynomial-time hierarchy.
*Theoretical Computer Science*, 3, 1, 1976, pp. 1–22.Google Scholar - 19.S. Skyum and L. G. Valiant, A complexity theory based on boolean algebra. 22nd
*Symposium on the Foundations of Computer Science*, 1981, pp. 244–253.Google Scholar - 20.M. Sipser, On polynomial vs exponential growth. In preparation.Google Scholar
- 21.L. Stockmeyer and A. Meyer, Word problems requiring exponential time, preliminary report. 5th
*Annual ACM Symposium on Theory of Computing*, 1973.Google Scholar

## Copyright information

© Springer-Verlag New York Inc 1984