We are interested in separating classes in the exponential time hierarchy, EXPH, from classes in the polynomial time hierarchy, PH. We show that for any fixed integer c, P/sup NP[O(n(c/)])/spl sube/NEXP. This improves a previous result by B. Fu, H. Li and Y. Zhong (1992). Further, we generalize this separation to related levels of PH and EXPH, showing that for any fixed integer c and i/spl ges/1, /spl Deltasub isup P[O(n(c/)])/spl subespl Sigmasub i-1sup EXP/. This improves the long standing separations which result from the relativization of the time hierarchy theorem. Related results for function classes are also considered.
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