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Certain sufficient conditions of uniformity for systems of functions of many-valued logic

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Abstract

For any finite system A of functions of the k-valued logic taking values in the set E s = {0,1,…, s − 1}, k ≥ s ≥ 2, such that the closed class generated by restriction of functions from A on the set E s contains a near-unanimity function, it is proved that there exist constants c and d such that for an arbitrary function f ∈ [A] the depth D A (f) and the complexity L A (f) of f in the class of formulas over A satisfy the relation D A (f) ≤ clog2 L A (f) + d.

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References

  1. S. V. Yablonskii, Introduction to Discrete Mathematics (Vysshaya Shkola, Moscow, 2001) [in Russian].

    Google Scholar 

  2. D. Lau, Function Algebras on Finite Sets (Springer-Verlag, Berlin, Heidelberg, 2006).

    MATH  Google Scholar 

  3. A. B Ugol’nikov, “Depth and Polynomial Equivalence of Formulas for Closed Classes of Two-Valued Logic,” Matem. Zametki 42(4), 603 (1987).

    MathSciNet  Google Scholar 

  4. S. V. Yablonskii and V. P. Kozyrev, “Mathematical Problems of Cybernetics,” in Information Materials of Scientific Council for Integrated Problem “Cybernetics” Akad. Sci. SSSR (Moscow, 1968), Issue 19 a, pp. 3–15.

    Google Scholar 

  5. P. M. Spira, “On Time-Hardware Complexity Tradeoffs for Boolean Functions,” in Proc. 4th Hawai Symposium, on System, Sciences (Western Periodicals Company, North Hollywood, 1971). pp. 525–527.

    Google Scholar 

  6. V. M. Khrapchenko. “Relation Between the Complexity and Depth of Formulas,” in Methods of Discrete Analysis in, Synthesis of Control Systems (IM SB AS SSSR, Novosibirsk, 1978), Issue 32, pp. 76–94.

    Google Scholar 

  7. I. Wegener, “Relating Monotone Formula Size and Monotone Depth of Boolean Functions,” Inform. Proces. Let. 16, 41 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. B. Ugol’nikov, “Relations Between the Depth and Complexity of Formulas for Complete Classes of Two-Valued logic,” in IV All-Union Conf. “Application of Methods of Mathematical Logic, ” Theses of Reports (Tallinn, 1986), p. 184.

    Google Scholar 

  9. A. B. Ugol’nikov, “Polynomial Equivalence of Formulas for Closed Classes of Two-Valued Logic,” in VII All Union Conf. “Problems of Theoretical Cybernetics,” Reports. Part 1 (Irkutsk, Irkutsk State Univ., 1985), pp. 194–195.

    Google Scholar 

  10. M. E. Ragaz, “Parallelizable Algebras,” Arch. fur math. Log. und Grundlagenforsch. 26, 77 (1986/87).

    Article  MathSciNet  Google Scholar 

  11. L. I. Akhmetova, “Depth of Formulas for Precomplete Classes of Three-Valued Logic,” in Methods and Systems of Technical Diagnostics, Issue 18 (Saratov, Saratov State Univ., 1993), pp. 19–20.

    Google Scholar 

  12. R. F. Safin, “Depth and Complexity of Formulas in Certain Classes of k-Valued Logic,” Vestn. Mosk. Univ., Matem. Mekhan., No. 6, 65 (2000).

    Google Scholar 

  13. R. F. Safin, “Relations Between the Depth and Complexity of Formulas for Precomplete Classes of k-Valued Logic,” in Mathematical Problems of Cybernetics (Fizmatlit, Moscow, 2004), pp. 223–278.

    Google Scholar 

  14. R. F. Safin, “Uniformity of Systems of Monotone Functions,” Vestn. Mosk. Univ., Matem. Mekhan., No. 2, 15 (2003).

    Google Scholar 

  15. L. Rosenberg, “La Structure des Fonctions de Plusieurs Variables sur un Ensemble Fini,” C. r. Acad. sci. Paris. Ser. A 260, 3817 (1965).

    MATH  Google Scholar 

  16. O. S. Dudakova, “Classes of Functions of the k-Valued Logic Monotone with Respect to Sets of Width Two,” Vestn. Mosk. Univ., Matem. Mekhan., No. 1, 31 (2008).

    Google Scholar 

  17. P. B. Tarasov, “Uniformity of a Certain Systems of Functions of Many-Valued Logic,” Vestn. Mosk. Univ., Matem. Mekhan., No. 2, 61 (2013).

    Google Scholar 

  18. A. B. Ugol’nikov, “Closed Post Classes,” Izv. Vuzov, Matem. No. 7, 79 (1988).

    Google Scholar 

  19. A. B. Ugol’nikov, Post Classes (Moscow State Univ., Mech. Math. Dept., Moscow, 2008) [in Russian].

    Google Scholar 

  20. E. L. Post, “Two-Valued Rerative Systems of Mathematical Logic,” Ann. Math. Stud. 5, 1941.

  21. S. V. Yablonskii, G. P. Gavrilov, and V. B. Kudryavtsev, The Functions of Algebra of Logic and Post Classes (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  22. S. S. Marchenkov, “On the id-Decompositions of the Class Pk over Precomplete Classes,” Diskret. Matem. 5(2), 98 (1993).

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    P. B. Tarasov

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  1. P. B. Tarasov
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Original Russian Text © P.B. Tarasov, 2013, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2013, Vol. 67, No. 5, pp. 41–46.

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Tarasov, P.B. Certain sufficient conditions of uniformity for systems of functions of many-valued logic. Moscow Univ. Math. Bull. 68, 253–257 (2013). https://doi.org/10.3103/S0027132213050082

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