Abstract
It is considered the realization of k-valued logic functions (k = 4 and k = 5) by the circuits of unreliable functional elements in a complete basis consisting of the Webb function. It is assumed that the basis elements pass into faulty states independently of each other and the faults are such that each of the incorrect values appears at the output of the basis element with the same probability.
It is shown that any function of k-valued logic (k = 4 and k = 5) can be realized by the reliable circuit, the upper bound is obtained, and moreover the probability of the appearance of the fault is bounded by a constant.
Similar content being viewed by others
References
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).
S. I. Ortyukov, “On the redundancy of the implementation of Boolean functions by the circuits of unreliable elements,” in Proceedings of the Workshop on Discrete Mathematics and its Applications (1989), pp. 166–168.
M. Alekhina, “On synthesis of reliable schemes consisting of functional elements-X/Y with similar constant incorrectedness on elements outputs,” Vestn. Mosk. Univ., Mat. Mekh. 5, 80–83 (1991).
M. Alekhina and A. Vasin, “On bases with unreliability coefficient 2,” Math. Notes 95, 147–173 (2014).
M. Alekhina and O. Barsukova, “Upper bound of the circuits unreliability in a complete finite basis (P 3)with arbitrary faults of elements,” Lobachevskii J. Math. 39 (1), 13–19 (2018).
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).
M. A. Alekhina, O. Yu. Barsukova, and A. V. Moiseev, “Asymptotically optimal reliable circuits in Rosser-Turkett basis (in P k),” Lobachevskii J. Math. 38 (1), 62–72 (2017).
S. V. Yablonskii, Introduction into Discrete Mathematics (Nauka, Moscow, 2001) [in Russian].
M. A. Alekhina, “Recurrence relations for the reliability of circuits in a basis consisting of the Webb function, in P 4 and P 5,” Izv. Vyssh. Uchebn. Zaved., Povolzh. Region, Fiz.-Mat. Nauki 3, 19–30 (2016).
N. P. Redkin, Reliability and Diagnostics of Circuits (Mosk. Gos. Univ., Moscow, 1992) [in Russian].
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by F. M. Ablaev)
Rights and permissions
About this article
Cite this article
Barsukova, O., Alekhina, M. Synthesis of Reliable Circuits in the Basis Consisting of the Webb Function in P4 and P5. Lobachevskii J Math 40, 140–148 (2019). https://doi.org/10.1134/S1995080219020045
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080219020045