Check and diagnostic tests for AND, OR, and NOT gates

We examine problems of operability checking and state diagnosis of $N$ functional elements realizing given Boolean function $f\in\{x_1\&\ldots\&x_n, x_1\vee\ldots\vee x_n, \overline{x_1}\}$ in their perfect states and including not more than $k$ faulty elements by means of composition of one-output circuits from them and observation of values produced by these circuits on any value set of input variables. Arbitrary constant faults on outputs of functional elements are permitted. One has to minimize the number of circuits required for operability checking and determination of states of all elements. Lower bounds of the form $ck(\log_2 N-\log_2 k)$ are obtained for the number of indicated circuits.