Skip to main content
SpringerLink
Log in

Closure operators with positive connectives and quantifiers

  • Published:
Moscow University Computational Mathematics and Cybernetics Aims and scope Submit manuscript

Abstract

Closure operators based on the logical connectives & and ∨ and the quantifiers ∃ and ∀ and extend the superposition operator are classified. The (&∃∀)-closure operator, which uses only the connective & and both quantifiers, is considered. The basic properties of the (&∃∀)-closure operator are determined. All of the 15 (&∃∀)-closed classes of Boolean functions are found.

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

References

  1. E. L. Post, “Introduction to a general theory of elementary propositions,” Am._J. Math. 43 (4), 163–185 (1921).

    MathSciNet  MATH  Google Scholar 

  2. E. L. Post, “Two-valued iterative systems of mathematical logic,” in Annals of Mathematical Studies (Princeton Univ. Press, Princeton, 1941), Vol. 5, pp. 1–122.

    MathSciNet  Google Scholar 

  3. Yu. I. Yanov and A. A. Muchnik, “Existence of k-valued closed classes having no finite basis,” Dokl. Akad. Nauk SSSR 127, 44–46 (1959).

    MathSciNet  MATH  Google Scholar 

  4. A. V. Kuznetsov, “Means for detection of nondeducibility and inexpressibility,” in Logical Inference, Ed. by V. A. Smirnov (Nauka, Moscow, 1979), pp. 5–33 [in Russian].

    Google Scholar 

  5. A. F. Danil’chenko, “On parametric expressibility of the function of three-valued logic,” Algebra Logika 16, 397–416 (1977).

    MathSciNet  Google Scholar 

  6. S. S. Marchenkov, “On expressibility of functions of multiple-valued logic in some logic-functional languages,” Diskret.Mat. 11 (4), 110–126 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. S. Marchenkov, Closed Classes of Boolean Functions (Fizmatlit, Moscow, 2000) [in Russian].

    MATH  Google Scholar 

  8. S. S. Marchenkov, Principles of the Theory of Boolean Functions (Fizmatlit, Moscow, 2014) [in Russian].

    MATH  Google Scholar 

  9. A. F. Danil’chenko, “Parametrically closed classes of three-valued logic function,” Izv. Akad. Nauk MSSR 2, 13–20 (1978).

    Google Scholar 

  10. A. F. Danil’čenko, “On parametrical expressibility of the functions of k-valued logic,” Colloq. Math. Soc. J. Bolyai. 28, 147–159 (1981).

    MathSciNet  Google Scholar 

  11. S. Barris and R. Willard, “Finitely many primitive positive clones,” Proc. Am. Math. Soc. 101, 427–430 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. S. Marchenkov, “Definition of positively closed classes by endomorphism semigroups,” Diskret. Mat. 24 (4), 19–26 (2012).

    Article  MATH  Google Scholar 

  13. S. S. Marchenkov, Closure Operators of Logic-Functional Type (MAKS Press, Moscow, 2012) [in Russian].

    MATH  Google Scholar 

  14. S. S. Marchenkov, “Positively closed classes of three-valued logic,” J. Appl. Ind.Math. 21, 67–83 (2014).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    S. S. Marchenkov

Authors
  1. S. S. Marchenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. S. Marchenkov.

Additional information

Original Russian Text © S.S. Marchenkov, 2017, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika, 2017, No. 1, pp. 33–37.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marchenkov, S.S. Closure operators with positive connectives and quantifiers. MoscowUniv.Comput.Math.Cybern. 41, 32–37 (2017). https://doi.org/10.3103/S027864191701006X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S027864191701006X

Keywords

Search

Navigation