  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.
