Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 13–19 | Cite as

Upper Bound of the Circuits Unreliability in a Complete Finite Basis (in P3) with Arbitrary Faults of Elements



In this paper it is considered the implementation of ternary logic functions by the circuits of unreliable functional elements in an arbitrary complete finite basis. It is assumed that all the circuit elements pass to fault states independently of each other, and the faults can be arbitrary (for example, inverse or constant). Previously known class of ternary logic functions is extended, the circuits of these functions can be used to raise the reliability of the original circuits. With inverse faults at the outputs of basis elements it is constructively proved using functions of this class (we denote by G it) that a function which differs from any one of the variables can be implemented by a reliable circuit, and the probability of the inverse fault is bounded above by a constant. In particular if the basis under consideration contains at least one of the class G functions then for any function which differs from any one of the variables the constructed circuit is not only reliable, but this one is asymptotically optimal by reliability (we remind that function which is equal to one of the variables can be implemented absolutely reliably, not using functional element).

Keywords and phrases

ternary logics functions unreliable functional elements synthesis of circuits composed of unreliable elements asymptotically optimal by reliability circuits inverse faults at the outputs of elements 


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  1. 1.
    Yu. Vinogradov, “Synthesis of triple-MOS circuits,” in Mathematical Problems of Cybernetics, Collection of Articles, Ed. by S. V. Yablonskii (Nauka, Moscow, 1991), Vol. 3, pp. 187–198 [in Russian].Google Scholar
  2. 2.
    M. Alekhina and O. Barsukova, “On unreliability of circuits realizing ternary logic functions,” Discrete Anal. Operat. Res. 21-4 (118), 12–24 (2014).MATHGoogle Scholar
  3. 3.
    J. von Neuman, “Probabilistic logics and the synthesis of reliable organisms from unreliable components,” in Automata studies, Ed. by C. Shannon and J. McCarthy (Princeton Univ. Press, Princeton, 1956), pp. 43–98.Google Scholar
  4. 4.
    M. Alekhina, “On reliability of circuits over an arbitrary complete finite basis under single-type constant faults at outputs of elements,”’ DiscreteMath. Appl. 22, 383–391 (2012).MathSciNetMATHGoogle Scholar
  5. 5.
    M. Alekhina, “Synthesis and complexity of asymptotically optimal circuits with unreliable gates,” Fundam. Inform. 104, 219–225 (2010).MathSciNetMATHGoogle Scholar
  6. 6.
    M. Alekhina and A. Vasin, “On bases with unreliability coefficient 2,” Math. Notes 95, 147–173 (2014).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. Alekhina and A. Vasin, “Sufficient conditions for realizability of Boolean functions by asymptotically optimal circuits with the unreliability 2ε,” Russ. Math. 54 (5), 68–70 (2010).CrossRefMATHGoogle Scholar
  8. 8.
    M. Alekhina and S. Grabovskaya, “Reliability of nonbranching programs in an arbitrary complete finite basis,” Russ. Math. 56 (2), 10–18 (2012).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Alekhina and S. Kargin, “Asymptotically optimal reliable circuits in RosserЦTurkett basis in P 4,” Izv. Vyssh. Uchebn. Zaved., Povolzh. Region, Fiz. Mat. Nauki 1, 37–53 (2015).Google Scholar
  10. 10.
    M. Alekhina, O. Barsukova, and A. Moiseev, “Asymptotically optimal reliable circuits in Rosser-Turkett basis (in P k),” Lobachevskii J. Math. 38, 62–72 (2017).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    M. Alekhina and O. Barsukova, “Upper bound of unreliability of circuits in a basis consisting of Webb functions,” Russ. Math. 59 (3), 13–24 (2015).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    O. Barsukova, “The synthesis of reliable circuitswhich realizes binary and ternary logics functions,” Extended Abstract of Cand. Sci. (Phys. Math.) Dissertation (Kazan State Univ., Kazan, 2014).Google Scholar
  13. 13.
    O. Lupanov, Asymptotic Estimates of the Complexity of Control Systems (Mosk. Gos. Univ., Moscow, 1984) [in Russian].Google Scholar
  14. 14.
    S. Yablonskiy, “Asymptotically the best method of synthesis of reliable circuits from unreliable elements,” Banach Center 7, 11–19 (1982).MathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Yablonskii, Introduction into Discrete Mathematics (Nauka, Moscow, 2001) [in Russian].Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Penza State Technological UniversityPenzaRussia
  2. 2.Penza State UniversityPenzaRussia

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