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On the Complexity of the Clone Membership Problem

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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.

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Acknowledgements

I would like to thank the anonymous referees for helpful comments.

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  1. Institute of Mathematics, The Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Emil Jeřábek

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  1. Emil Jeřábek
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Supported by grant 19-05497S of GA ČR. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO: 67985840.

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Jeřábek, E. On the Complexity of the Clone Membership Problem. Theory Comput Syst 65, 839–868 (2021). https://doi.org/10.1007/s00224-020-10016-7

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