Tests with Stuck-At and Shift Faults on Circuit Inputs

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 →  ∞ .