Abstract
The alternation hierarchy for Turing machines with a space bound between loglog and log is infinite. That applies to all common concepts, especially a) to two-way machines with weak space-bounds, b) to two-way machines with strong space-bounds, and c) to one-way machines with weak space-bounds. In all of these cases the ∑ k -and II k -classes are not comparable fork>-2. Furthermore the ∑ k -classes are not closed under intersection and the II k -classes are not closed under union. Thus these classes are not closed under complementation. The hierarchy results also apply to classes determined by an alternation depth which is a function depending on the input rather than on a constant.
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