The alternation hierarchy for machines with sublogarithmic space is infinite

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 Σk− and Πk−classes are not comparable for k ≥ 2. Furthermore the Σk−classes are not closed under intersection and the Πk-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.