Generating Boolean $\mu$-expressions

Abstract.

In this paper, we consider the class of Boolean $\mu$-functions, which are the Boolean functions definable by $\mu$-expressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formula $E$ into an equivalent $\mu$-expression--if possible--in time linear in$\Vert E\Vert$ times $2^{n_m}$, where $\Vert E \Vert$ is the size of$E$ and$n_m$ is the number of variables that occur more than once in $E$. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing $\mu$-functions from $k$-formulas [17]. Furthermore, we show that recognizing Boolean $\mu$-functions is co-NP-complete for functions essentially dependent on all variables and we give a bound close to co-NP for the general case.