On Uniformity and Circuit Lower Bounds

Abstract—We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is “medium uniform ” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: • For all k, P is not contained in P-uniform SIZE(n k). That is, for all k there is a language Lk ∈ P that does not have O(n k)-size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in P-uniform SIZE(n k) for any fixed k