On dlogtime and polylogtime reductions

We investigate properties of the relativized NC and AC hierarchies in their DLOGTIME-. respectively, ALOGTIME-uniform setting and show that these hierarchies can be characterized in terms of adaptive reducibility in logarithmic or polylogarithmic time, i.e. O (log_n)〗⁡2 for i ≥ 0. As a corollary, the relationship between AC^i and NC^i+1 reducibility is clarified by the result stating that if DLOGTIME-uniform AC' and ALOGTIME-u11iform NC•+1 reducibility coincide for i = o when applied to an arbitrary function class F, then they coincide on F for all i 2 O. Our result.substantially generalize various previous results (Wi 90), (ABJ 91), (Ba 91)