We investigate left-shifts of variables in Boolean functions and stuck-at faults. Some bounds of Shannon functions are established, producing the minimum test lengths for detection and diagnosis of any *n*-variable Boolean function with respect to a given fault source. (The Shannon function of the test length is the maximum of the minimum test lengths over all *n-*variable Boolean functions.) A left-shift in an *n-*variable Boolean function simultaneously increments the indexes of all the variables by a natural number and substitutes some constants for variables with indexes greater than *n.* The Shannon function of the relevant test is shown to be exactly of length 2; the order of growth of the Shannon function of the length of the diagnosis test is \( \Theta \left(\sqrt{2^n}\right). \) With the variables left-shifted by *k* places, 1 ≤ *k* ≤ *n*, the Shannon function \( {L}^{diagn}\left({U}_{n,k}^{shifts},n\right) \) of the length of the complete diagnosis test has the following bounds: \( \min \left({2}^k-1,{2}^{n-k}\right)\le {L}^{diagn}\left({U}_{n,k}^{shifts},n\right)\le \min \left({2}^k,{2}^{n-k}+1\right). \) The article also considers local stuck-at faults of multiplicity *k* on the inputs of *k-*input, one-output circuits realizing a Boolean function. An asymptotic expression is derived for the log Shannon function of the diagnosis test length \( {\log}_2{L}^{diagn}\left({U}_k^{lc},n\right)\sim k\ \mathrm{as}\ n\to \infty, k=k(n)\to \infty, \kern1em 1\le k\le n/2,{\log}_2n=o(k). \) An asymptotic lower bound of the Shannon function of the diagnosis test length with stuck-at faults on circuit inputs is shown to be *L*^{diagn}(*U*^{c}, *n*) ≥ 2 ⋅ 2^{[n/2]} ⋅ (1 + *o*(1)) as *n* → ∞ .

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Translated from Prikladnaya Matematika i Informatika, No. 64, 2020, pp. 79–85.

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Antyufeev, G.V., Romanov, D.S. Tests with Stuck-At and Shift Faults on Circuit Inputs.
*Comput Math Model* **31, **494–500 (2020). https://doi.org/10.1007/s10598-021-09511-4

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### Keywords

- Boolean function
- test
- shift
- stuck-at faults on circuit inputs