On the Tape-Number Problem for Deterministic Time Classes

TL;DR

This paper investigates the hierarchy conjecture related to deterministic Turing machines, showing how certain properties imply separations between deterministic and nondeterministic time complexity classes, with all results relativizing.

Contribution

It establishes that the hierarchy conjecture H(f) implies separations between deterministic and nondeterministic time classes, and connects it to the gap property G(f), providing new insights into time class separations.

Findings

H(f) implies deterministic-nondeterministic class separations

G(f) implies further class separations

All relationships relativize

Abstract

For any time bound f, let H(f) denote the hierarchy conjecture which means that the restriction of the numbers of work tapes of deterministic Turing machines to some b generates an infinite hierarchy of proper subclasses DTIME_b(f) \subset \DTIME(f). We show that H(f) implies separations of deterministic from nondeterministic time classes. H(f) follows from the gap property, G(f), which says that there is a time-constructible bound f_2 such that f \in o(f_2) and DTIME(f)=DTIME(f_2). G(f) implies further separations. All these relationships relativize.