We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

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.

We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman-Hartmanis isomorphism conjecture [BH77] holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP^A-complete sets are polynomial-time isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.

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 ...

We show that for most complexity classes of interest, all sets complete under first-order projections are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds.

We construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NP-complete sets are polynomial-time isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP- complete under reductions that make one query to L but not under traditional many-one reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the sur...

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomial-size circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.

Abstract. For any class C und closed under NC 1 reductions, it is shown that all sets complete for C under first-order (equivalently, Dlogtimeuniform AC 0) reductions are isomorphic under first-order computable isomorphisms. 1

We show that all sets complete for NC¹ under AC 0 reductions are isomorphic under AC 0 -computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC¹-computable manyone reductions, the sets complete for C under NC 0 reductions are all isomorphic under AC 0 -computable isomorphisms. Our result showing that the complete degree for NC¹ collapses to an isomorphism type follows from a theorem showing that in NC¹, the complete degrees for AC 0 and NC 0 reducibility coincide.