Diagnostic tests under shifts with fixed filling tuple

Abstract

Abstract We consider a fault source under which the fault functions are obtained from the original function f(x̃n) ∈ P2n $\begin{array}{} \displaystyle P_2^n \end{array}$ by a left shift of values of the Boolean variables by at most n. For the vacant positions of the variables, the values are selected from a given filling tuple γ̃ = (γ1, γ2, …, γn) ∈ E2n $\begin{array}{} \displaystyle E^n_2 \end{array}$, which also moves to the left by the number of positions corresponding to a specific fault function. The problem of diagnostic of faults of this kind is considered. We show that the Shannon function Lγ~shifts,diagn(n), $\begin{array}{} \displaystyle L_{\tilde{\gamma}}^{\rm shifts, diagn}(n), \end{array}$ which is equal to the smallest necessary test length for diagnostic of any n-place Boolean function with respect to a described fault source, satisfies the inequality n2≤Lγ~shifts,diagn(n)≤n. $\begin{array}{} \displaystyle \left\lceil \frac{n}{2} \right\rceil \leq L_{\tilde{\gamma}}^{\rm shifts, diagn}(n) \leq n. \end{array}$