The alternation hierarchy for sublogarithmic space is infinite. (English) Zbl 0796.68099

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 \(\Sigma_ k\)- and \(\Pi_ k\)- classes are not comparable for \(k\geq 2\). Furthermore, the \(\Sigma_ k\)- classes are not closed under intersection and the \(\Pi_ k\)-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.


68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
03D10 Turing machines and related notions
03D15 Complexity of computation (including implicit computational complexity)
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