Skip to main content
Springer Nature Link
Log in

Circuits constructed with MOD q gates cannot compute “AND” in sublinear size

  • Published:
computational complexity Aims and scope Submit manuscript

Abstract

Algebraic techniques are used to prove that any circuit constructed with MOD q gates that computes the AND function must use Ω(n) gates at the first level. The best bound previously known to be valid for arbitraryq was Ω(logn).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Ajtai, 387-1 on finite structures.Annals of Pure and Applied Logic 24 (1983), 1–48.

    Google Scholar 

  2. D. A. Mix Barrington, Some problems involving Razborov-Smolensky polynomials. In M. S. Patterson, ed.,Boolean Function Complexity, London Mathematical Society Lecture Notes Series 169, Cambridge University Press, 1992, 109–128.

  3. D. A. Mix Barrington, H. Straubing, andD. Thérien, Non-uniform automata over groups.Inform. and Comput. 89 (2) (1990) 109–132.

    Google Scholar 

  4. L. Euler,Opusc. analytics Vol. 2 (1785), p. 241.

    Google Scholar 

  5. M. Furst, J.B. Saxe, andM. Sipser, Parity, circuits and the polynomial-time hierarchy.Math. Systems Theory 18 (1984), 13–27.

    Google Scholar 

  6. J. Håstad, Computational limitations on Small Depth Circuits. Ph.D. Thesis, Massachusetts Institute of Technology, 1986.

  7. J.D. Kahn, andR. Meshulam, On mod p Traversals.Combinatorica 11 (1991), 17–22.

    Google Scholar 

  8. A.A. Razborov, Lower bounds on the size of bounded-depth networks over a complete basis with logical additions.Mat. Zametki 41 (4) (1987), 598–607 (in Russian). English translation inMathematical Notes of the Academy of Sciences of the USSR 41 (4) (1987), 333–338.

    Google Scholar 

  9. M. Szegedy, Algebraic Methods in Lower Bounds for Computational Models with Limited Communication. Ph.D. Thesis, University of Chicago, 1989.

  10. R. Smolensky, Algebraic methods in the theory of lower bounds for boolean circuit complexity.In Proc. Nineteenth Ann. ACM Symp. Theor. Comput. (1987), 77–82.

  11. R. Smolensky, On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates.In Proc. 31st Ann. Symp. Found. Comput. Sci. (1990), 628–631.

  12. A. Yao, Separating the polynomial-time hierarchy.In Proc. 26th Ann. Symp. Found. Comput. Sci. (1985), 1–10.

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, 3480 University Street, H3A 2A7, Montréal, Québec, Canada

    Denis Thérien

Authors
  1. Denis Thérien
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thérien, D. Circuits constructed with MOD q gates cannot compute “AND” in sublinear size. Comput Complexity 4, 383–388 (1994). https://doi.org/10.1007/BF01263425

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01263425

Subject classifications