About a mathematical model of faults in contact networks (Q1345676)

The author deals with contact schemes (switching circuits) with brief- and open-circuit faults on the set of contacts. The author also uses a special condition (very artificial in the reviewer's opinion) on the set of the same type contacts. Let \(S\) be a contact scheme (with the author's condition!) and \(f(x_ 1,\dots, x_ n)\) be a correct boolean function of \(S\), \(\{g_ k\}\) be the set of all noncorrect nontrivial \((g_ k\neq f)\) boolean functions. A set \(T\), \(T\subset \{0, 1\}^ n\), where \(n\) is the number of different types of contacts in \(S\), is a complete test for \(S\) iff for all \(g_ k\) exists \(\widetilde {\alpha}\in T\) and \(g_ k (\widetilde {\alpha})\neq f(\widetilde {\alpha})\). Let \(D(S)= \min\{| T|\): \(T\) is a complete test for \(S\}\), \(D(f)= \min\{ D(S)\): \(f\) is realized by \(S\}\) and \(D(n)= \max\{ D(f)\): \(f\) is a boolean function with \(n\) variables\}. The main result of the paper is the following: for \(n=2^ t+t+1\),\ \ \(2n-2t-1\leq D(n)\leq 2n\); for \(2^ t+t+1< n\leq 2^{t+1}+ t+1,\;\;2n- 2t- 2\leq D(n)\leq 2n\).