Low uniform versions of NC

In the setting known as DLOGTIME-uniformity, the fundamental complexity classes AC ⊂ ACC ⊆ TC ⊆ NC have several robust characterizations. In this paper we refine uniformity further and examine the impact of these refinements on NC and its subclasses. When applied to the logarithmic circuit depth characterization of NC, some refinements leave NC unchanged while others collapse NC to NC. Thus we study refinements of other circuit characterizations of NC. In the case of the AC(A5) characterization of NC , where A5 is the NC-complete word problem of the group A5, our refinements collapse NC to a subset of the regular languages. For the AC(D+) characterizations of NC, where D+ is the NC-complete language capturing the formula value problem, interestingly, these refinements scale down to circuits with linear fan-in. In particular, the latter refinements bring to the fore two classes, denoted FO[<]-uniform AC(D+)LIN and FO[<]uniform TCLIN , whose separation may be within the reach of current lower bound techniques, and whose separation would amount to distinguishing the power of a MAJ gate from that of a D+ gate.