Minimal contact circuits for characteristic functions of spheres

Abstract

Abstract We study the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex σ̃ = (σ1, …, σn), σ1, …, σn ∈ {0, 1}, we mean the Boolean function φσ~(n) $\begin{array}{} \varphi^{(n)}_{\tilde\sigma} \end{array} $(x1, …, xn) which is equal to 1 on those and only those tuples of values that differ from the tuple σ̃ only in one digit. It is shown that the number 3n − 2 of contacts is necessary and sufficient for implementation of φσ~(n) $\begin{array}{} \varphi^{(n)}_{\tilde\sigma} \end{array} $(x̃) by a contact circuit.