Skip to main content
SpringerLink
Log in

Threshold circuits for iterated multiplication: Using AC0 for free

  • Conference paper
  • First Online:
STACS 93 (STACS 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 665))

Included in the following conference series:

Abstract

We investigate small-depth threshold circuits for iterated multiplication and related problems. One result is that we can solve this problem with an ACo-connection of TC o3 -languages, i.e. an ACo-connection of languages recognizable by depth-3 threshold circuits. This can be compared to the best known construction, which uses four levels of threshold gates (but no ACo-circuitry). Similarly, we design small-depth circuits for powering, division and logarithm. Iterated multiplication is then considered in the context of finite fields. Finally, we look at circuits of quasipolynomial size and we establish various normal forms.

Supported by NSERC and FCAR scholarships.

Supported by NSERC grant A4546 and by FCAR grant 89-EQ-2933.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barrington, D.A.M.: Quasipolynomial size circuit classes. In Proc. of the 7th Ann. Conf. on Structure in Complexity Theory, 1992, 86–93

    Google Scholar 

  2. Beame, P.W., Cook, S.A., Hoover, H.J.: Log depth circuits for division and related problems. SIAM J. on Computing 15:4 (1986) 994–1003

    Google Scholar 

  3. Beigel, R., Reingold, N., Spielman, D.: The Perceptron strikes back. In Proc. of the 6th Ann. Conf. on Structure in Complexity Theory, 1991, 286–291

    Google Scholar 

  4. Beigel, R., Tarui, J.: On ACC. In Proc. of the 32th Ann. IEEE Symp. on Foundations of Computer Sc., 1991, 783–792

    Google Scholar 

  5. Chandra, A.K., Stockmeyer, L., Vishkin, U.: Constant depth reducibility. SIAM J. on Computing 13:2 (1984) 423–439

    Google Scholar 

  6. Fagin, R., Klawe, M., Pippenger, N.J., Stockmeyer, L.: Bounded-depth polynomial-size circuits for symmetric functions. Theoretical Computer Sc. 36 (1985) 239–250

    Google Scholar 

  7. Goldmann, M., Håstad, J., Razborov, A.: Majority gates vs. general weighted threshold gates. In Proc. of the 7th Ann. Conference on Structure in Complexity Theory, 1992, 2–13. To appear in Computational Complexity.

    Google Scholar 

  8. Goldmann, M., Karpinski, M.: Simulating threshold circuits by majority circuits. Manuscript, 1992.

    Google Scholar 

  9. Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold circuits of bounded depth. In Proc. of the 28th Ann. IEEE Symp. on Foundations of Computer Sc., 1987, 99–110

    Google Scholar 

  10. Hofmeister, T., Hohberg, W., Köhling, S.: Some notes on threshold circuits, and multiplication in depth 4. Information Processing Letters 39 (1991) 219–225

    Google Scholar 

  11. Immerman, N., Landau, S.: The complexity of iterated multiplication. In Proc. of the 4th Ann. Conference on Structure in Complexity Theory, 1989, 104–111

    Google Scholar 

  12. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd ed., (Grad. Texts in Math., 84) Springer-Verlag, 1990

    Google Scholar 

  13. Lidl, R., Niederreiter, H.: Finite Fields, (Enc. of Math. and its App., 20) Addison-Wesley, 1983

    Google Scholar 

  14. Reif, J.H.: On threshold circuits and polynomial computation. In Proc. of the 2nd Ann. Conference on Structure in Complexity Theory, 1987, 118–123

    Google Scholar 

  15. Siu, K.-Y., Roychowdhury, V.: On optimal depth threshold circuits for multiplication and related problems. Manuscript, 1992. To appear in SIAM J. on Discrete Math.

    Google Scholar 

  16. Tarui, J.: Randomized polynomials, threshold circuits, and the polynomial hierarchy. In Proc. of the 8th Ann. Symp. on Theoretical Aspects of Computer Sc., (LNCS, 480) Springer-Verlag, 1991, 238–250. To appear in Theoretical Computer Sc. under the title: Probabilistic polynomials, ACo functions, and the polynomial-time hierarchy.

    Google Scholar 

  17. Yao, A.C.-C.: Separating the polynomial-time hierarchy by oracles. In Proc. of the 26th Ann. IEEE Symp. on Foundations of Computer Sc., 1985, 1–10

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, H3A 2A7, Montréal, Québec, Canada

    Alexis Maciel & Denis Thérien

Authors
  1. Alexis Maciel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denis Thérien
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. Enjalbert A. Finkel K. W. Wagner

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Maciel, A., Thérien, D. (1993). Threshold circuits for iterated multiplication: Using AC0 for free. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_54

Download citation

  • DOI: https://doi.org/10.1007/3-540-56503-5_54

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56503-1

  • Online ISBN: 978-3-540-47574-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation