Abstract
We consider the problems of fault check and diagnostics of the state of N logic gates, which, in the operable condition, implement a given Boolean function f(x 1,..., x n ), by constructing out of them one-output logic circuits and observing the output values of these circuits for arbitrary input values of the variables. Random constant faults are permitted at the outputs of logic gates under the assumption that at most k gates are faulty, where a given natural k is at most N. It is required to minimize the number of circuits necessary to check the operability and determine the state of all logic gates. A lower bound is obtained on this number when k is close to N. As a corollary of this estimate, it is established that, under some conditions on N and k belonging to a certain segment, the number of required circuits is at least ck, where c > 1 is a constant independent of the choice of k from this segment.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. F. Beckenbach and R. Bellman, An Introduction to Inequalities (Random House, New York, 1961; Mir, Moscow, 1965).
O. B. Lupanov, Asymptotic Bounds on Complexity of Control Systems (Moskov. Gos. Univ., Moscow, 1984) [in Russian].
K. A. Popkov, “Bounds on Lengths of Check and Diagnostic Tests for Logic Gates,” Diskretn. Anal. Issled. Oper. 21 (6), 73–89 (2014).
K. A. Popkov, “Fault Detection and Diagnostic Tests for AND,OR, and NOT Gates,” VestnikMoskov. Univ. Ser. 1, Mat. Mekh. No. 6, 40–44 (2014) [Moscow Univ.Math. Bull. 69 (6), 267–271 (2014)].
K. A. Popkov, “Fault Detection and Diagnostic Tests for Logic Gates,” Diskretn. Mat. 26 (2), 83–99 (2014) [DiscreteMath. Appl. 24 (4), 213–225 (2014)].
K. A. Popkov, “On Single Tests for Logic Gates,” Diskretn. Mat. 27 (2), 73–93 (2015).
N. P. Red’kin, “On Complete Checking Tests for Circuits of Functional Elements,” Vestnik Moskov. Univ. Ser. 1, Mat. Mekh. No. 1, 72–74 (1986).
N. P. Red’kin, “On Complete Checking Tests for Circuits of Functional Elements,” in Mathematical Problems of Cybernetics, Vol. 2, Ed. by S. V. Yablonskii (Nauka, Moscow, 1989), pp. 198–222.
N. P. Red’kin, Reliability and Diagnosis of Circuits (Moskov. Gos. Univ.,Moscow, 1992) [in Russian].
D. S. Romanov, “On the Synthesis of Circuits Admitting Complete Fault Detection Test Sets of Constant Length under Arbitrary Constant Faults at theOutputs of the Gates,” Diskretn. Mat. 25 (2), 104–120 (2013) [DiscreteMath. Appl. 23 (3–4), 343–362 (2013)].
D. S. Romanov, “Method of Synthesis of Easily Testable Circuits Admitting Single Fault Detection Tests of Constant Length,” Diskretn. Mat. 26 (2), 100–130 (2014) [DiscreteMath. Appl. 24 (4), 227–251 (2014)].
I. A. Chegis and S. V. Yablonskii, “LogicalMethods for Control of Electric Circuits,” TrudyMIAN SSSR 51, 270–360 (1958).
S. V. Yablonskii, “Reliability and Monitoring of Control Systems,” in Proceedings of the All-Union Seminar on DiscreteMathematics and Its Applications (Moskov. Gos. Univ.,Moscow, 1986), pp. 7–12.
S. V. Yablonskii, “Some Questions of Reliability and Monitoring of Control Systems,” in Mathematical Problems of Cybernetics, Ed. by S. V. Yablonskii, Vol. 1 (Nauka, Moscow, 1988), pp. 5–25.
S. M. Reddy, “Easily Testable Realization for Logic Functions,” IEEE Trans. Comput. 21 (1), 124–141 (1972).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © K.A. Popkov, 2015, published in Diskretnyi Analiz i Issledovanie Operatsii, 2015, Vol. 22, No. 5, pp. 52–68.
Rights and permissions
About this article
Cite this article
Popkov, K.A. Estimations of the lengths of tests for logic gates in presence of many permissible faults. J. Appl. Ind. Math. 9, 559–569 (2015). https://doi.org/10.1134/S1990478915040122
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478915040122