We show that all sets that arecomplete for NP under non-uniform AC are isomorphic under non-uniform AC -computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC reductions.

The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.

We prove that the Berman-Hartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 -computable many-one reductions. Isomorphism: The sets complete for C under AC 0 reductions are all isomorphic under isomorphisms computable and invertible by AC 0 circuits of depth three. Gap: The sets that are complete for C under AC 0 and NC 0 reducibility coincide. Stop Gap: The sets that are complete for C under AC 0 [mod 2] and AC 0 reducibility do not coincide. (These theorems hold both in the non-uniform and P-uniform settings.) To prove the second theorem for P-uniform settings, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.) 1 Introduction The notion of complete sets in complexity classes provides one of ...

It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a non-trivial ≤1−NL m-degree that coincides with a ≤1−L m-degree. 3. For every class C closed under log-lin reductions, the ≤1−NL m coincides with the ≤1−L m-complete degree of C. 1

this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture.

The development of Small Span Theorems for various complexity classes and reducibilities plays a basic role in (resource bounded) measure-theoretic investigations of e cient reductions. A Small Span Theorem for a complexity class C and reducibility r is the assertion that, for all sets A in C, at least one of the cones below or above A is a negligible small class with respect to C, where the cones below orabove A refer to the sets fB: B r Ag and fB: A r Bg, respectively. That is, a Small Span Theorem rules out one of the four possibilities of the size of upper and lower cones for a set in C. Here we use the recent formulation of resource-bounded measure of Allender and Strauss which allows meaningful notions of measure on polynomial-time complexity classes. We showtwo Small Span Theorems for polynomial-time complexity classes and sublinear-time reducibilities, namely a Small Span Theorem for P and Dlogtimeuniform NC0-computable reductions, and for PNP and Dlogtime-transformations. Furthermore, we showthat, for every xed k, the hard set for P under Dlogtimeuniform AC 0-reductions of depth k and size n k is a small class. In contrast, we show that every upper cone under P-uniform NC 0-reductions is not small. 1

In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1