We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several group-theoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2 .

We continue the study of robust reductions initiated by Gavald`a and Balc'azar. In particular, a 1991 paper of Gavald`a and Balc'azar [GB91] claimed an optimal separation between the power of robust and nondeterministic strong reductions. Unfortunately, their proof is invalid. We re-establish their theorem. Generalizing robust reductions, we note that robustly strong reductions are built from two restrictions, robust underproductivity and robust overproductivity, both of which have been separately studied before in other contexts. By systematically analyzing the power of these reductions, we explore the extent to which each restriction weakens the power of reductions. We show that one of these reductions yields a new, strong form of the Karp-Lipton Theorem. 1 Introduction The study of the relative power of reductions has long been one of central importance in computational complexity theory. Reductions are the key tools used in complexity theory to compare the difficulty of problems. ...

We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several group-theoretic problems are low for ZPP . We also

This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE � = coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets. 1