On the Depth of a Multiplexer Function with a Small Number of Select Lines

This paper continues the research on the circuit synthesis problem for a multiplexer function of logic algebra, which is a component of many integrated circuits and is also used in theoretical study. The exact value of the depth of a multiplexer with $n$ select lines in the standard basis is found under the assumption that the conjunction and disjunction gates are of depth 1 and the negation gate is of depth 0; the depth equals $n+2$ if $10 \leqslant n \leqslant 19$. Thus, it follows from previous results that the exact depth value equals $n+2$ for all positive integers $n$ such that either $2 \leqslant n \leqslant 5$ or $n \geqslant 10$. Moreover, for $n=1$, this value equals 2, and for $6 \leqslant n \leqslant 9$, it equals either $n+2$ or $n+3$. Similar results are also obtained for a basis consisting of all elementary conjunctions and elementary disjunctions of two variables.