On Uniformity Within

1 Abstract In order to study circuit complexity classes within NC 1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC 1 uniformity Ru81,Co85], have appeared in recent research: Immerman's families of circuits deened by rst-order formulas Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions Bu87]. 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 NC 1 Ba89] 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 sub-classes of the regular languages based on their logical expressibility, extending recent work of Straubing, Th erien, and Thomas STT88]. A preliminary version of this work appeared as BIS88]. 2 Introduction 2.1 Circuit Complexity Computer scientists have long tried to classify problems (deened as Boolean predicates or functions) by the size or depth of Boolean circuits needed to solve them. This eeort has