Circuits Over PP and PL

Wilson’s [19] 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 Beigel, Reingold, Spielman, Fortnow, and Ogihara [6, 9, 14], we prove that PL and PP are closed under NC1 reductions. This answers an open problem of Ogihara [14]. More generally, we show that NC k+1 = AC k and NC k+1 = AC k for all k 0. On the other hand, we construct an oracle A such that NC A k 6= NCPPA k+1 for all integers k 1. Slightlyweaker 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 PPP log2 n= log logn-T BF PrTIME(nO(logn)). Because PPP log2 n= log logn-T 6 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. Both authors are visiting the University of Maryland Dept. of Computer and UMIACS for 1996–97. Address: Dept. of Computer Science, University of Maryland at College Park, College Park, MD 20742-3251. Research supported in part by the National Science Foundation under grants CCR-8958528 and CCR-9415410. yAlso supported in part by NASA under grant NAG 52895. Email: [email protected] zDepartment of Computer Science, Yale University, New Haven,CT06520 and UMIACS, University of Maryland. Email: [email protected]