On the Tape-Number Problem for Deterministic Time Classes

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.