On the complexity of the clone membership problem

TL;DR

This paper analyzes the computational complexity of the Boolean clone membership problem, establishing its precise position in the polynomial hierarchy and classifying various cases based on the function sets involved.

Contribution

It proves that CMP is $ ext{Theta}_2^P$-complete and classifies the complexity for fixed functions and different gate sets, clarifying previous misconceptions.

Findings

CMP is $ ext{Theta}_2^P$-complete.

For fixed $f$, CMP is NP-complete unless $f$ is a projection.

Most gate set variants of CMP are either $ ext{Theta}_2^P$-complete, coDP-complete, or in P.

Abstract

We investigate the complexity of the Boolean clone membership problem (CMP): given a set of Boolean functions $F$ and a Boolean function $f$, determine if $f$ is in the clone generated by $F$, i.e., if it can be expressed by a circuit with $F$-gates. Here, $f$ and elements of $F$ are given as circuits or formulas over the usual De Morgan basis. B\"ohler and Schnoor (2007) proved that for any fixed $F$, the problem is coNP-complete, with a few exceptions where it is in P. Vollmer (2009) incorrectly claimed that the full problem CMP is also coNP-complete. We prove that CMP is in fact $\Theta^P_2$-complete, and we complement B\"ohler and Schnoor's results by showing that for fixed $f$, the problem is NP-complete unless $f$ is a projection. More generally, we study the problem $B$-CMP where $F$ and $f$ are given by circuits using gates from $B$. For most choices of $B$, we classify the complexity of $B$-CMP as being $\Theta^P_2$-complete (possibly under randomized reductions), coDP-complete, or in P.