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 σ k− and IIk−classes are not comparable for k ≥ 2. Furthermore the σ k−classes are not closed under intersection and the IIkclasses 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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References
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© 1994 Springer-Verlag Berlin Heidelberg
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von Braunmühl, B., Gengler, R., Rettinger, R. (1994). The alternation hierarchy for machines with sublogarithmic space is infinite. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_133
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DOI: https://doi.org/10.1007/3-540-57785-8_133
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