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Prikl. Diskr. Mat., 2017, Number 38, Pages 66–88 (Mi pdm602)  

This article is cited in 2 scientific papers (total in 2 papers)

Mathematical Backgrounds of Computer and Control System Reliability

Single fault detection tests for logic networks of AND, NOT gates

K. A. Popkov

Keldysh Institute of Applied Mathematics, Moscow, Russia

Abstract: Let $D_1(f)$ ($D_0(f)$) be the least length of a fault detection test for irredundant logic networks consisting of logic gates in the basis $\{&,\neg\}$, implementing a given Boolean function $f(x_1,\ldots,x_n)$, and having at most one stuck-at-1 (stuck-at-0 respectively) fault on outputs of the logic gates. Let $f_1=x_i$; $f_2=0, \overline{x_i}$, or $x_{i_1}^{\sigma_1}& x_{i_2}…& x_{i_k}$; $f_3=\overline{x_{i_1}}&\overline{x_{i_2}}& x_{i_3}&…& x_{i_k}$ or $\underbrace{(…((}_{k-1}x_{i_1}^{\sigma_1}& x_{i_2})^{\sigma_2}& x_{i_3})^{\sigma_3}&…& x_{i_k})^{\sigma_k}$; $f_4=x_{i_1}^{\sigma_1}&…& x_{i_k}^{\sigma_k}$; $f_5=\underbrace{(…((}_{k-1}x_{i_1}^{\sigma_1}& x_{i_2}^{\sigma_2})^{\delta_1}& x_{i_3}^{\sigma_3})^{\delta_2}&…& x_{i_k}^{\sigma_k})^{\delta_{k-1}}$, where $2\leqslant k\leqslant n$ for $f_2$, $f_3$, and $f_5$; $1\leqslant k\leqslant n$ for $f_4$; $\sigma_1,…,\sigma_k,\delta_1,…,\delta_{k-1}\in\{0,1\}$; $i,i_1,…,i_k\in\{1,…,n\}$; indices $i_1,…,i_k$ are pairwise different; for $f_3$, at least one of numbers $\sigma_2,…,\sigma_k$ equals $0$ and if $k=2$, then assume $x_{i_3}&…&x_{i_k}\equiv1$; for $f_5$, at least one of numbers $\delta_1,…,\delta_{k-1}$ equals $0$. It is proved that, for each Boolean function $f(x_1,…,x_n)\not\equiv1$,
$$ D_1(f)=\begin{cases} 0,&iff the function $f$ is representable in the form of $f_1$, 1,&iff the function $f$ is representable in the form of $f_2$, 2,&iff the function $f$ is representable in the form of $f_3$, 3&otherwise. \end{cases} $$
If $f\equiv1$ then the value $D_1(f)$ is undefined. Also, it is proved that, for each Boolean function $f(x_1,…,x_n)$ which is different from constants,
$$ D_0(f)=\begin{cases} 0,&iff the function $f$ is representable in the form of $f_1$, 1,&iff the function $f$ is representable in the form of $f_4$ but not of $f_1$, 2,&iff the function $f$ is representable in the form of $f_5$, 3&otherwise. \end{cases} $$
If $f\equiv1$ or $f\equiv0$ then the value $D_0(f)$ is undefined.

Keywords: logic network, stuck-at fault, single fault detection test.

Funding Agency Grant Number
Russian Science Foundation 14-21-00025 П


DOI: https://doi.org/10.17223/20710410/38/5

Full text: PDF file (850 kB)
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Bibliographic databases:

UDC: 519.718.7

Citation: K. A. Popkov, “Single fault detection tests for logic networks of AND, NOT gates”, Prikl. Diskr. Mat., 2017, no. 38, 66–88

Citation in format AMSBIB
\Bibitem{Pop17}
\by K.~A.~Popkov
\paper Single fault detection tests for logic networks of AND, NOT gates
\jour Prikl. Diskr. Mat.
\yr 2017
\issue 38
\pages 66--88
\mathnet{http://mi.mathnet.ru/pdm602}
\crossref{https://doi.org/10.17223/20710410/38/5}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. K. A. Popkov, “Short single tests for circuits with arbitrary stuck-at faults at outputs of gates”, Discrete Math. Appl., 29:5 (2019), 321–333  mathnet  crossref  crossref  mathscinet  isi  elib
    2. K. A. Popkov, “O skhemakh, dopuskayuschikh korotkie edinichnye proveryayuschie testy pri proizvolnykh neispravnostyakh funktsionalnykh elementov”, PDM, 2021, no. 51, 85–100  mathnet  crossref
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