ON UNIFORMITY WITHIN NC1

In order to study circuit complexity classes within NC1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC1 uniformity, have appeared in recent research: Immerman's families of circuits defined by first-order formulas and a uniformity corresponding to Buss' deterministic log-time reductions. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC1 still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Thérien, and Thomas. A preliminary version of this work appeared in "Structure of Complexity Theory: Third Annual Conference" pp. 47-59, IEEE Comput. Soc., Washington, DC, 1988. © 1990.