The Complexity of Problems Defined by Boolean Circuits

We study the complexity of circuit-based combinatorial problems (e.g., the circuit value problem and the satisfiability problem) defined by boolean circuits w ith gates from an arbitrary finite base of boolean functions. Special cases have been investigated in the literature. We give a complete characterization of their complexity depending on the base . For example, for the satisfiability problem for boolean circuits with gates from we present a complete collection of (decidable) criteria which tell us for which this problem is in , is complete for , is complete for , is complete for , or is complete for . Our proofs make substantial use of the characterization of all closed classes of boolean functions given by E.L. POST already in the twenties.

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