Skip to main content
SpringerLink
Log in

Representations of Normalized Formulas

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

A class of objects called \( \Pi \)-partitions is defined. In a certain well-defined sense, these objects are the equivalents of formulas in a basis consisting of disjunction, conjunction, and negation in which negations are possible only over variables (normalized formulas). \( \Pi \)-partitions are viewed as representations of formulas, just as \( \Pi \)-schemes can be viewed as equivalents and graphical representations of the same formulas. Some theory of such representations is developed, which is essentially a mathematical apparatus focused on describing a class of minimal normalized formulas implementing linear Boolean functions. REMOVE— \( \Pi \)-scheme, \( \Pi \)-partition—REMOVE

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

REFERENCES

  1. V. M. Khrapchenko, “Complexity of the realization of a linear function in the class of \( \Pi \)-circuits,” Mat. Zametki 9 (1), 35–40 (1971) [Math. Notes Acad. Sci. USSR 9 (1), 21–23, (1971)].

    MathSciNet  MATH  Google Scholar 

  2. S. V. Yablonskii, “Realization of a linear function in the class of \( \pi \)-circuits,” Dokl. Akad. Nauk SSSR. Nov. Ser. 94 (5), 805–806 (1954).

    MATH  Google Scholar 

  3. K. L. Rychkov, “Lower bounds on the complexity of parallel-sequential switching circuits that realize linear Boolean functions,” Sib. Zh. Issled. Oper. 1 (4), 33–52 (1994) English translation in Discrete Analysis and Operation Research, A. D. Korshunov, Ed. (Kluwer, Dordrecht, 1996), Math. Its Appl. 355, 217–234.

  4. D. Yu. Cherukhin, “To the question of a logical representation for the parity counter,” Neform. Nauka (2), 14–23 (2009).

  5. K. L. Rychkov, “Lower bounds on the formula complexity of a linear Boolean function,” Sib. Elektron. Mat. Izv. 11 , 165–184 (2014).

    MathSciNet  MATH  Google Scholar 

  6. K. L. Rychkov, “Complexity of the realization of a linear Boolean function in the class of \( \pi \)-schemes,” Diskr. Anal. Issled. Oper. 25 (3), 36–94 (2018) [J. Appl. Ind. Math. 12 (3), 540–576 (2018)].

    Article  MathSciNet  MATH  Google Scholar 

  7. K. L. Rychkov, “On minimal \( \pi \)-schemes for linear Boolean functions,” Metody Diskr. Anal. Sinteze Realizatsii Bulevykh Funktsii (Izd. Inst. Mat., Novosibirsk, 1991) 51 (), 84-104 [Sib. Adv. Math. 3 (3), 172-185 (1993)].

  8. K. L. Rychkov, “Sufficient conditions for the minimal \( \pi \)-schemes for linear Boolean functions to be locally non-repeating,” Diskr. Anal. Issled. Oper. 22 (5), 71–85 (2015) [J. Appl. Ind. Math. 9 (4), 580–587 (2015)].

    Article  MathSciNet  Google Scholar 

  9. K. L. Rychkov, “On the perfectness of minimal regular partitions of the edge set of the \( n \)-dimensional cube,” Diskr. Anal. Issled. Oper. 26 (4), 74–107 (2019) [J. Appl. Ind. Math. 13 (4), 717–739 (2019)].

  10. V. M. Khrapchenko, “A method of determining lower bounds for the complexity of \( \Pi \)-schemes,” Mat. Zametki 10 (1), 83–92 (1971) [Math. Notes Acad. Sci. USSR 10 (1), 474–479 (1971)].

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. V. M. Khrapchenko, “A simplified proof of a lower complexity estimate,” Diskr. Mat. 25 (2), 82–84 (2013) [Discr. Math. Appl. 23 (2), 171–174 (2013)].

    MathSciNet  MATH  Google Scholar 

  13. S. V. Avgustinovich, Yu. L. Vasil’ev, and K. L. Rychkov, “The computation complexity in the class of formulas,” Diskr. Anal. Issled. Oper. 19 (3), 3–12 (2012) [J. Appl. Ind. Math. 6 (4), 403–409 (2012)].

    Article  MathSciNet  Google Scholar 

  14. Yu. L. Vasil’ev and K. L. Rychkov, “A lower bound on formula size of a ternary linear function,” Diskr. Anal. Issled. Oper. 20 (4), 15–26 (2013) [J. Appl. Ind. Math. 7 (4), 490–499 (2013)].

    Article  Google Scholar 

  15. I. S. Sergeev, “Formula complexity of a linear function in a \( k \)-ary basis,” Mat. Zametki 109 (3), 419–435 (2021) [Math. Notes 109 (3), 445–458 (2021)].

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Razborov, “Applications of matrix methods to the theory of lower bounds in computational complexity,” Combinatorica 10 (1), 81–93 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Karchmer and A. Wigderson, “Monotone circuits for connectivity require super-logarithmics depth,” SIAM J. Discr. Math. 3 (2), 255–265 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Håstad, “The shrinkage exponent is 2,” SIAM J. Comput. 27 , 48–64 (1998).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research was carried out within the framework of the state order for the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project no. FWNF–2022–0017.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    K. L. Rychkov

Authors
  1. K. L. Rychkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. L. Rychkov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rychkov, K.L. Representations of Normalized Formulas. J. Appl. Ind. Math. 16, 760–775 (2022). https://doi.org/10.1134/S1990478922040160

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478922040160

Keywords

Search

Navigation