Downward translations of equality

AbstractIn this paper we construct oracles relative to which DTIME(T(n)) equals NTIME(T(n)) and DTIME(t(n)) does not equal NTIME (t(n)), for t(n) sufficiently smaller than T(n). A stronger result than this is also obtained, though for fewer T(n), expressedin two parts. For T(n)⩽2n, there is an oracle relative to which DTIME(T(n)) equals NTIME(T(n)) and NTIME(2n) contains a set not in DTIME(t(n)) for any t(n) growing more slowly than T(n). For T(n)⩽22n+0(1), there is an oracle relative to which DTIME(T(n)) equals NTIME(T(n)) and NTIME(log T(n)) contains a set not in DTIME(t(n)) for t(n) growing more slowly than T(n). These results expand on those obtained by Dekhtyar (1976), Book et al. (1982), and Allender (1989)