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.

Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman--Hartmanis and Joseph--Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...

Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.

We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomial-time computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. ffl In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1 Introduction Understanding the power of nondeterminism has been one of the pri...

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...

In this report we present a new method of diagonalization that is a refinement of the well-known finite injury priority method discovered independently by Friedberg and Muchnik in 1957. In the resource bounded injury method , it is necessary in addition to proving that the number injuries for a given requirement is finite to carefully count these injuries and prove that this number does not exceed a bound given by the index of the requirement. The method is used to construct an oracle relative to which the polynomial time hierarchy collapses to an extent that the second level of this hierarchy (P NP A ) captures nondeterministic exponential time. This oracle is an answer to an open problem posed by Heller in 1984 that has thus far resisted existing methods and that has recently regained interest by work of Fu et. al. and by work of Homer and Mocas. Moreover, our oracle provides a constructive counterexample to Sewelson's conjecture that does not make use of information theoretical l...

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.

In this paper we demonstrate an oracle relative to which there are one-way functions but every paddable 1-li-degree collapses to an isomorphism type, thus yielding a relativized failure of the Joseph-Young Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but one-way functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of one-way functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomial-time many-to-one reductions and if they are both paddable then they are polynomial-time isomorphic. After surveying all of the then-known NP-complete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NP-complete lan...

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