The reliability of networks in the basis \((x \&{} y, x{\vee{}}y, {\bar x})\) in the case of constant faults of the same type at the outputs of the gates (Q1803080)

It is proved that using gates which may pass to a faulty state with the probability \(\gamma\), any Boolean function may be realised in the bases \(\{x\& y, x\vee y,\overline{x}\}\), by a network \(S\) having the unrealibility \(P(S)\leq\gamma+ \gamma^ 2+10\cdot\gamma^ 3\). Also, it is shown that for any network \(S\), which realises a function \(f\not\equiv 0\) (\(f\not\equiv 1\)), \(P(S)\geq\gamma\) holds true at \(\gamma\leq 1/11\) \((\leq 1)\). As mentioned in a remark the presented results are also valid -- in the basis \(\{x\& y, x\vee y\}\) -- for monotonic functions.