The alternation hierarchy for machines with sublogarithmic space is infinite

  • Burchard von Braunmühl
  • Romain Gengler
  • Robert Rettinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)


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 IIk−classes are not comparable for k ≥ 2. Furthermore the σk−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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BGR93]
    B. von Braunmühl, R. Gengler, and R. Rettinger. The alternation hierarchy with sublogarithmic space is infinite. Computational Complexity, 3/3:207–230, 1993. To appear.Google Scholar
  2. [BM88]
    L. Babai and S. Moran. Arthur-Merlin games: a randomized proof-system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36:254–276, 1988.Google Scholar
  3. [Bra91]
    B. v. Braunmühl. Alternationshierarchien von Turingmaschinen mit kleinem Speicher. Informatik Berichte 83, Inst. f. Informatik, Universität Bonn, 1991.Google Scholar
  4. [CIRB87]
    J. H. Chang, O. H. Ibarra, B. Ravikumar, and L. Berman. Some observations concerning Turing machines using small space. Information Processing Letters, 25:1–9, 1987. Erratum, Information Processing Letters, 25:53, 1988.Google Scholar
  5. [Fre79]
    R. Freivalds. On time complexity of deterministic and nondeterministic Turing machines. Latvijski Mathematičeskij Eshegodnik, 23:158–165, 1979. In Russian.Google Scholar
  6. [Gef93]
    V. Geffert. A hierarchy that does not collapse: Alternations in low level space. Research report, šafárik University, Košice, 1993.Google Scholar
  7. [Hem87]
    L. A. Hemachandra. The strong exponential hierarchy collapses. In Proc. 19th. STOC Conference, pages 110–122, 1987.Google Scholar
  8. [IIT87]
    A. Ito, K. Inoue, and I. Takanami. A note on alternating Turing machines using small space. The Trans. of the IEICE, E 70 no. 10:990–996, 1987.Google Scholar
  9. [Imm88]
    N. Immerman. NSPACE is closed under complement. SIAM J. Comput., 17:935–938, 1988.Google Scholar
  10. [Iwa86]
    K. Iwama. ASPACE(o(log log)) is regular. Research report, KSU/ICS Kyoto Sangyo University, Kyoto, 603, Japan, March 1986. See also SIAM J. Comput. 22:136–146, 1993.Google Scholar
  11. [LL89]
    M. LiŚkiewicz and K. LoryŚ. On reversal complexity for alternating Turing machines. In Proc. 30st FOCS, pages 618–623, 1989.Google Scholar
  12. [LR93a]
    M. LiŚkiewicz and R. Reischuk. Separating the lower levels of the sublogarithmic space hierarchy. In Proc. 10. STACS, LNCS 665, pages 16–28, 1993.Google Scholar
  13. [LR93b]
    M. Liskiewicz and R. Reischuk. The sublogarithmic space world. Technical report, Institut für Theoretische Informatik, TH Darmstadt, 1993.Google Scholar
  14. [Sip83]
    M. Sipser. Borel sets and circuit complexity. In Proc 15. Ann. ACM Symp. on Theory of Computing, pages 330–335, 1983.Google Scholar
  15. [Sze88]
    R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Informatica, 26:279–284, 1988.Google Scholar
  16. [Wag93]
    K. W. Wagner. The alternation hierarchy for sublogarithmic space: an exciting race to STACS'93 (Editorial note). In Proc. 10. STAGS, LNCS 665, pages 2–4, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Burchard von Braunmühl
    • 1
  • Romain Gengler
    • 1
  • Robert Rettinger
    • 1
  1. 1.Institut für Informatik IUniversität Bonn Römerstraße 164BonnGermany

Personalised recommendations