Timothy J. Long

Quantitative relativizations of complexity classes

Consider the following open problems: (i)

Strong nondeterministic polynomial-time reducibilities

{\text{P}} = ? {\text{ NP}}

Relativizing complexity classes with sparse oracles

(ii)

Reasoning about Software-Component Behavior

{\text{NP}} = ? {\text{ co-NP}}

On one-way functions and polynomial-time isomorphisms

; (iii)

Qualitative relativizations of complexity classes

{\text{P}} = ? {\text{ PSPACE}}

Erratum: On restricting the size of Oracles compared with restricting access to Oracles

; (iv)

Part II: specifying components in RESOLVE

{\text{NP}} = ? {\text{ PSPACE}}

A note on sparse oracles for NP

. In this paper we study these four problems from a particular point of view. To illustrate our approach, consider the first problem. It is known that there exist recursive sets A and B such that

A REFINEMENT OF THE LOW AND HIGH HIERARCHIES

{\text{P}}(A) = {\text{NP}}(A)