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Short Complete Diagnostic Tests for Circuits Implementing Linear Boolean Functions

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Abstract

We prove that each of the Boolean functions \(x_1\oplus\dots\oplus x_n\), \(x_1\oplus\dots\oplus x_n\oplus 1\) can be implemented by a logic circuit in each of the bases \(\{x\oplus y,1\}\), \(\{x\&\overline y,x\vee y,\overline x\}\), \(\{x\&y,x\vee y,\overline x\}\), allowing a complete diagnostic test of length not exceeding \(\lceil\log_2(n+1)\rceil\) (for the first two bases) or not exceeding \(n\) (for the third basis) relative to one-type stuck-at faults at outputs of gates. We also establish that each of the functions \(x_1\oplus\dots\oplus x_n\), \(x_1\oplus\dots\oplus x_n\oplus 1\) can be implemented by a logic circuit in the basis \(\{x\oplus y,1\}\) allowing a complete diagnostic test of length not exceeding \(\lceil\log_2(n+1)\rceil+1\) relative to arbitrary stuck-at faults at outputs of gates.

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Funding

This work was supported by the Moscow Center for Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education of the Russian Federation no. 075-15-2022-283.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, 125047, Russia

    K. A. Popkov

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  1. K. A. Popkov
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Correspondence to K. A. Popkov.

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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 75–89 https://doi.org/10.4213/mzm13639.

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Popkov, K.A. Short Complete Diagnostic Tests for Circuits Implementing Linear Boolean Functions. Math Notes 113, 80–92 (2023). https://doi.org/10.1134/S0001434623010091

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