Boolean Circuit Complexity and Two-Dimensional Cover Problems

We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudl\'ak, R\"odl, and Savick\'y (1988). For convenience, we formalize these ideas in the more general setting of "discrete complexity", i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures. We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity. Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of "cyclic" Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.