Some results on relativized time-bounded complexity classes are presented. There can be many kinds of hierarchies of complexity subclasses of relativized NP. For brevity, let P(
A,
k) [NP(
A,
k)] be the relativized complexity class DTIME
A(
nk) [resp. NTIME
A(
nk)] with respect to oracle set
A. (For
k=1, replace
nk by 2
n). Then for example: 1). There is an oracle set
A such that for all
k>0 P(
A,
k) is properly contained in NP(
A,
k) and NP(
A,
k) is properly contained in P(
A,
k+1). 2) For each
k>0, there is an oracle set
D (depending on
k) such that for any
i≤
k P(
D,
i)≠NP(
D,
i), but for all
j>
k P(
D,
j)=NP(
D,
j). Besides, we show a theorem which is a higher level analog to a theorem of Book, Wilson and Mei-Rui [3].
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