Stephen A. Fenner

On oracle builder's toolkit

It is shown how to use various notions of genericity as a tool in oracle creation. A general framework for defining different types of generic sets in terms of arithmetic forcing is given. A number of basic facts about Cohen generic sets, many of which are generalizations of known results, are systematically assembled. We define sp-generic sets and extend some previous results.<<ETX>>

The isomorphism conjecture holds relative to an oracle

The authors introduce symmetric perfect generic sets. these sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. They then show that the Berman-Hartmanis (1977) isomorphism conjecture holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP/sup A/-complete sets are polynomial-time isomorphic relative to A. As part of the proof that the isomorphism conjecture holds relative to symmetric perfect generic sets they also show that P/sup A/=FewP/sup A/ for any symmetric perfect generic/sup /A.<<ETX>>

Notions of resource-bounded category and genericity

The author investigates the strength of resource-bounded generic sets for deciding results in relativized complexity. He makes technical improvements to J.H. Lutzs notion of resource-bounded Baire category (1987, 1989) to show that almost every exponential-time set (in the authors sense of category) separate P from NP. It is shown that the authors improved notion of category, while strictly more powerful, still has all the other desirable properties of Lutzs characterization of resource-bounded category in terms of Banach-Mazur games. He then considers the amount of genericity needed to prove result of M. Blum and R. Impagliazzo (1987) regarding NP intersection co-NP and one-way functions. It is found that although these results hold for 1-generic sets, they cannot be guaranteed even by extremely powerful but slightly weaker generics. A crucial difference between 1-genericity and weaker notions is thus isolated. The author studies this weaker notion of genericity and shows that it has recursion-theoretic properties radically different from 1-genericity.<<ETX>>

Inverting onto functions

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: • In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*. • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. • 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.

Bipartite perfect matching is in quasi-NC

We show that the bipartite perfect matching problem is in quasi- NC2. That is, it has uniform circuits of quasi-polynomial size nO(logn), and O(log2 n) depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

Oracles That Compute Values

This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that a multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2k-1 nonadaptive queries to NPMV; a characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP; unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV are different than k+1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness, and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses.

The Isomorphism Conjecture Holds Relative toan Oracle

The authors 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 holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set

PP-Lowness and a Simple Definition of AWPP

A

An oracle builder's toolkit

, all

On Using Oracles That Compute Values

{\bf NP}^A