computational complexity

, Volume 3, Issue 3, pp 207–230 | Cite as

The alternation hierarchy for sublogarithmic space is infinite

  • Burchard von Braunmühl
  • Romain Gengler
  • Robert Rettinger


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.

Key words

Alternating Turing machines alternation hierarchy sublogarithmic space 

Subject classifications

68Q10 68Q15 


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Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Burchard von Braunmühl
    • 1
  • Romain Gengler
    • 1
  • Robert Rettinger
    • 1
  1. 1.Institut für Informatik IUniversität BonnBonnGermany

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