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 (Theory Comput. Syst. 41(4):753–777, 2007) proved that for any fixed F, the problem is coNP-complete, with a few exceptions where it is in P. Vollmer (Theory Comput. Syst. 44(1): 82–90, 2009) incorrectly claimed that the full problem CMP is also coNP-complete. We prove that CMP is in fact \(\boldsymbol {\Theta }^{\mathbf {P}}_{2}\)-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 \(\boldsymbol {\Theta }^{\mathbf {P}}_2\)-complete (possibly under randomized reductions), coDP-complete, or in P.

我们研究了布尔克隆隶属度问题（CMP）的复杂性：给定一组布尔函数F和一个布尔函数f，确定f是否在由F生成的克隆中，即，是否可以由具有F的电路表示门 在这里，f和F的元素在通常的De Morgan基础上作为电路或公式给出。伯乐和施诺尔（理论COMPUT SYST。41（4）：753-777，2007）证明了对于任何固定˚F，问题是Ç ö Ñ P -complete，除少数例外，其中它是在P。沃尔默（理论COMPUT SYST。44（1）：82-90，2009）错误地宣称，全问题CMP也Ç ö Ñ P -complete。我们证明CMP实际上是\（\ boldsymbol {\ Theta} ^ {\ mathbf {P}} _ {2} \）-完全，并且我们通过证明对于固定f，问题是N来对Böhler和Schnoor的结果进行补充。除非f是投影，否则P-完成。更笼统地说，我们研究问题B -CMP，其中F和f由使用B的门的电路给出。对于B的大多数选择，我们将B的复杂度分类-CMP为\（\ boldsymbol {\ Theta} ^ {\ mathbf {P}} _ 2 \）- complete（可能在随机归约的情况下），c o D P- complete或在P中。