Skip to main content
SpringerLink
Log in

Formula Complexity of a Linear Function in a \(k\)-ary Basis

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The Khrapchenko method of finding a lower bound for the complexity of binary formulas is extended to formulas in \(k\)-ary bases. The resulting extension makes it possible to evaluate the complexity of linear Boolean functions and a majority function of \(n\) variables when realized by formulas in the basis of all \(k\)-ary monotone functions and negation as \(\Omega(n^{g(k)})\), where \(g (k)=1+\Theta(1/\ln k)\). For a linear function, the complexity bound in this form is unimprovable. For \(k=3\), the sharper lower bound \(\Omega(n^{1.53})\) is proved.

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

Access this article

Log in via an institution

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

Notes

  1. Bases \(B_1\) and \(B_2\) are said to be equivalent if \(L_{B_1}(f)=\Theta(L_{B_2}(f))\) for all functions \(f\) expressed in them.

  2. The bases \(B_1\) and \(B_2\) are said to be strictly equivalent if \(L_{B_1}(f)=L_{B_2} (f)\) for any function \(f\) represented in them.

  3. The expression \(f\preceq g\) means \(f=O(g)\).

  4. The maximum exists, because the set of suitable numbers \(\chi\) is bounded and closed.

  5. Otherwise, we consider the subgraph \(G'\subset G\) for which \(s(G)\) is attained and its cover by the graphs \(G_1\cap G'\) and \(G_2\cap G'\). Then it follows from \(s(G_1\cap G')+s(G_2\cap G')\ge s(G')\) that \(s(G_1)+s(G_2)\ge s(G)\).

References

  1. V. M. Khrapchenko, “Method of determining lower bounds for the complexity of \(P\)-schemes,” Math. Notes 10 (1), 474–479 (1971).

    Article  MathSciNet  Google Scholar 

  2. B. A. Subbotovskaya, “Realization of linear functions by formulas using \(\vee\), \(\&\), \(^-\),” Dokl. Akad. Nauk SSSR 136 (3), 553–555 (1961).

    MathSciNet  Google Scholar 

  3. B. A. Muchnik, “Estimation of the complexity of the realization of a linear function by formulae in certain bases,” Kibernetika (Kiev) 4, 29–38 (1970).

    MathSciNet  Google Scholar 

  4. N. A. Peryazev, “Complexity of representations of Boolean functions by formulas in nonmonolinear bases,” in Diskret. Mat. Inform. (Izd. Irkutsk Univ., Irkutsk, 1995), Vol. 2.

  5. D. Yu. Cherukhin, “On the complexity of the realization of a linear function by formulas in finite Boolean bases,” Discrete Math. Appl. 10 (2), 147–157 (2000).

    Article  MathSciNet  Google Scholar 

  6. D. Yu. Cherukhin, “On circuits of functional elements of finite depth of branching,” Discrete Math. Appl. 16 (6), 577–587 (2006).

    Article  MathSciNet  Google Scholar 

  7. H. Chockler and U. Zwick, “Which bases admit nontrivial shrinkage of formulae?,” Comput. Complexity 10, 28–40 (2001).

    Article  MathSciNet  Google Scholar 

  8. D. Yu. Cherukhin, “Realization of linear functions by formulas in various bases,” Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., No. 6, 15–19 (2001).

    MathSciNet  MATH  Google Scholar 

  9. K. Ueno, “Formula complexity of ternary majorities,” in Computing and Combinatorics, Lecture Notes in Comput. Sci. (Springer, Heidelberg, 2012), Vol. 7434, pp. 433–444.

    Article  MathSciNet  Google Scholar 

  10. I. S. Sergeev, “Upper bounds for the formula size of symmetric Boolean functions,” Russian Math. (Iz. VUZ) 58 (5), 30–42 (2014).

    Article  MathSciNet  Google Scholar 

  11. I. S. Sergeev, “Complexity and depth of formulas for symmetric Boolean functions,” Moscow University Mathematics Bulletin 71 (3), 127–130 (2016).

    Article  MathSciNet  Google Scholar 

  12. M. M. Rokhlina, “The schemes that increase reliability,” in Problemy Kibernet. (Nauka, Moscow, 1970), Vol. 23, pp. 295-301.

  13. A. Gupta and S. Mahajan, “Using amplification to compute majority with small majority gates,” Comput. Complexity 6, 46–63 (1996).

    Article  MathSciNet  Google Scholar 

  14. S. Jukna, Boolean Function Complexity, in Algorithms Combin. (Springer-Verlag, Berlin, 2012), Vol. 27.

    Book  Google Scholar 

  15. I. Wegener, Complexity of Boolean functions (B. G. Teubner, Stuttgart, 1987).

    MATH  Google Scholar 

  16. K. L. Rychkov, “Modification of the Khrapchenko method and its application to bounds for complexity for \(\Pi\)-schemes for code functions,” in Methods of Discrete Analysis in Graph and Circuit Theory (IM SO AN USSR, Novosibirsk, 1985), Vol. 42, pp. 91–98.

  17. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934).

    MATH  Google Scholar 

  18. M. S. Paterson, N. Pippenger, and U. Zwick, “Optimal carry save networks,” in Boolean Function Complexity, London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1992), Vol. 169, pp. 174–201.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute “Kvant”, Moscow, 125438, Russia

    I. S. Sergeev

Authors
  1. I. S. Sergeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. S. Sergeev.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 419-435 https://doi.org/10.4213/mzm12802.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sergeev, I.S. Formula Complexity of a Linear Function in a \(k\)-ary Basis. Math Notes 109, 445–458 (2021). https://doi.org/10.1134/S0001434621030123

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434621030123

Keywords

Search

Navigation