Abstract
The problem of representing Boolean functions by two-pole contact circuits that are irredundant and admit short fault detection or diagnostic tests of closures of at most k contacts for a given positive integer k is considered. The following assertions are proved: for almost every Boolean function of n variables, the minimal length of a fault detection (diagnostic) test is equal to 2 (does not exceed 2k + 2, respectively).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. B. Lupanov, Asymptotic Bounds on Complexity of Control Systems (Izd. Moskov. Univ., Moscow, 1984) [in Russian].
I. A. Chegis and S. V. Yablonskii, “Logical methods for monitoring the operation of electric circuits” in Trudy Mat. Inst. Steklova, Vol. 51: Collection of Papers on Mathematical Logics and Its Applications to Some Problems of Cybernetics (Izd. Akad. Nauk SSSR, Moscow, 1958), pp. 270–360 [in Russian].
S. V. Yablonskii, “Reliability and control system monitoring,” in Proceedings of All-Union Seminar on Discrete Mathematics and Its Applications, January 31-February 2, 1984, Moscow (Izd. Moskov. Univ., Moscow, 1986), pp. 7–12 [in Russian].
S. V. Yablonskii, “Some problems of reliability and monitoring of control systems,” in Mathematical Problems of Cybernetics, No. 1 (Nauka, Moscow, 1988), pp. 5–25 [in Russian].
N. P. Red’kin, Reliability and Diagnostics of Schemes (Izd. Moskov. Univ., Moscow, 1992) [in Russian].
S. S. Kolyada, Upper Bounds for the Length of Fault Detection Tests for Schemes of Functional Elements, Cand. Sci. (Phys.-Math.) Dissertation (Moskov. Univ., Moscow, 2013) [in Russian].
Kh. A. Madatyan, “Full test for repeating contact circuits,” in Problems of Cybernetics, No. 23 (Nauka, Moscow, 1970), pp. 103–118 [in Russian].
N. P. Red’kin, “On complete checking tests for contact circuits,” in Methods of Discrete Analysis in the Study of Extremal Structures, No. 39 (Inst. Math. Sib. Otdel. Akad. Nauk SSSR, Novosibisk, 1983), pp. 80–87 [in Russian].
N. P. Red’kin, “On checking tests of closure and opening,” in Methods of Discrete Analysis in Optimization of Controlling Systems, No. 40 (Inst. Math. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk, 1983), pp. 87–99 [in Russian].
D. S. Romanov, “On the synthesis of contact circuits that allow short fault detection tests,” in Uchen. Zap. Kazan. Univ.. Ser. Fiz.-Mat. Nauki (Izd. Kazan. Univ., Kazan, 2014), Vol. 156, pp. 110–115 [in Russian].
K. A. Popkov, “Tests of contact closure for contact circuits,” Diskret. Mat. 28 (1), 87–100 (2016) [Discrete Math. Appl. 26 (5), 299-308 (2016)].
K. A. Popkov, “On fault detection tests of contact break for contact circuits,” Diskret. Mat. 29 (4), 66–86 (2017) [Discrete Math. Appl. 28(6), 369-383(2017)].
K. A. Popkov, “On diagnostic tests of contact break for contact circuits,” Preprint of Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, No. 271 (2018).
N. P. Red’kin, “Diagnostic tests for contact circuits,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. No. 2, 35–37 (2019) [Moscow Univ. Math. Bull. 74, 62-64 (2019)].
S.V. Yablonskii, Introduction to Discrete Mathematics (Nauka, Moscow, 1986) [in Russian].
Funding
This work was supported by the Russian Science Foundation under grant 19-71-30004.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 4, pp. 591-603.
Rights and permissions
About this article
Cite this article
Popkov, K.A. Short Tests of Closures for Contact Circuits. Math Notes 107, 653–662 (2020). https://doi.org/10.1134/S0001434620030323
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620030323