On Uniformity within NC¹

In order to study circuit complexity classes within NC in a uniform setting we need a uniformity condition which is more restrictive than those in common use Two such conditions stricter than NC uniformity Ru Co have appeared in recent research Immerman s families of circuits de ned by rst order formulas Im a Im b and a unifor mity corresponding to Buss deterministic log time reductions Bu 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 Ba 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 STT A preliminary version of this work appeared as BIS