The Sublogarithmic Alternating Space World

{ the most stringent form of a bounded language { and still separate 2 Space(S) from 2 Space(S). Thus a separation of the rst level would require a syntatically more complex languages than the second level. For k > 2 the languages L k and L k used in this paper to establish the separation are no longer bounded. But by Proposition 4.1 the third level can also be separated using simple bounded languages A 2 \ B 2 and A 2 B 2 that both are subsets of f1g f0gf1g. Nothing seems to be known for level 4 and higher. Thus, the sublogartihmic space hierarchy for bounded languages may be even more complex. We have made some observations leading to the conjecture that for bounded languages this hierarchy might indeed consist of only a nite number of distinct levels. Finally, it would be nice to characterize the exact relationship between classes co-k Space(S) and k Space(S) for sublogarithmic space bounds S and the class of arbitrary languages.