Circuits over PP and PL

C.B. Wilson´s (1985) model of oracle gates provides a framework for considering reductions whose strength is intermediate between truth-table and Turing. Improving on a stream of results by previous authors, we prove that PL and PP are closed under NC₁ reductions. This answers an open problem of M. Ogihara (1996). More generally, we show that NC_k+1^PP=AC_k^PP and NC_k+1 ^PL=AC_k^PL for all k⩾0. On the other hand, we construct an oracle A such that NC_k(PP^A )≠NC_k+1(PP^A) for all integers k⩾1. Slightly weaker than NC₁ reductions are Boolean formula reductions. We ask whether PL and PP are closed under Boolean formula reductions. This is a nontrivial question despite NC₁=BF, because that equality is easily seen not to relativize. We prove that P _log2nloglogn-T/^PP⊆BF^PP ⊆PrTIME(n^O(logn)). Because P_log2nloglogn-T/^PP⊄PP relative to an oracle, we think it is unlikely that PP is closed under Boolean formula reductions. We also show that PL is unlikely to be closed under BF reductions