Minimum ε-equivalent Circuit Size Problem

Abstract

For a Boolean function f given by its truth table (of length $2^n$) and a parameter s the problem considered is whether there is a Boolean function g $\epsilon$-equivalent to f, i.e., $Pr_{x\in {\{0,1\}}^n}\{g(x) \ne f(x)\} \le \epsilon$, and computed by a circuit of size at most s. In this paper we investigate the complexity of this problem and show that for specific values of $\epsilon$ it is unlikely to be in P/poly. Under the same assumptions we also consider the optimization variant of the problem and prove its inapproximability.