# On the complexity of the clone membership problem

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. Böhler 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 Θ^P_2-complete, and we complement Böhler 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 Θ^P_2-complete (possibly under randomized reductions), coDP-complete, or in P.

READ FULL TEXT