On Approximability of Boolean Formula Minimization

Abstract

For a Boolean function $f:\left\{ {0,1} \right\}^n \to \left\{ {0,1} \right\}$ given by a Boolean formula (or a binary circuit) S we discuss the problem of building a Boolean formula (binary circuit) of minimal size, which computes the function g equivalent to $f$, or ∈-equivalent to $f$, i.e., $Pr_{x \in \left\{ {0,1} \right\}^n } \left\{ {g\left( x \right) \ne f\left( x \right)} \right\} \leqslant \varepsilon$. In this paper we prove that if P ≠ NP then this problem can not be approximated with a “good” approximation ratio by a polynomial time algorithm.