The logarithmic alternation hierarchy collapses: \(A\Sigma _2^\mathcal{L} = A\Pi _2^\mathcal{L}\)

  • Klaus-Jörn Lange
  • Birgit Jenner
  • Bernd Kirsig
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)


We show that \(A\Sigma _2^\mathcal{L}\) coincides with \(A\Pi _2^\mathcal{L}\). Essentially this is done by Hausdorff reducing the \(A\Sigma _2^\mathcal{L}\)-complete set (GAP¢Co-GAP)(∃) to the question whether of two vectors A and B of n components, A contains more “solvable” components, i.e. components which are contained in GAP, than B. Moreover, we show \(A\Sigma _2^\mathcal{L} = \mathcal{L}_{hd} (N\mathcal{L})\).


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  1. [Cha Ko Sto]
    A. Chandra, D. Kozen, L. Stockmeyer: Alternation, J. Assoc. Comput. Mach. 28 (1981), 114–133.Google Scholar
  2. [Gol]
    L. Goldschlager: The monotone and planar circuit value problems are LOG space complete for P, SIGACT NEWS 9#35 (1977), 25–29.Google Scholar
  3. [HU]
    J. Hopcroft, J. Ullman: Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading Mass., 1979.Google Scholar
  4. [Jo]
    N.D. Jones: Reducibility among combinatorial problems in log n space, Proc. of the 7th Annual Princeton Conf. on Information Sciences and Systems, 1973, 547–551.Google Scholar
  5. [Kir]
    B. Kirsig: Logarithmic Hausdorff reductions, submitted for publication, 1986.Google Scholar
  6. [Kir Lan]
    B. Kirsig, K.-J. Lange: Separation with the Ruzzo, Simon, and Tompa relativization implies DSPACE(log n) ≠ NSPACE(log n), to be published in Inform. Process. Letters, 1986/87.Google Scholar
  7. [Kö Schö Wa]
    J. Köbler, U. Schöning, K.W. Wagner: The difference and truth-table hierarchies for NP, manuscript, 1985.Google Scholar
  8. [La Ly]
    R. Ladner, N. Lynch: Relativization of questions about log space computability, Math. Systems Theory 10 (1976), 19–32.Google Scholar
  9. [Lan]
    K.-J. Lange: Two characterizations of the logarithmic alternation hierarchy, Proc. of the 12th Symp. of Math. Foundations of Comput. Science, 1986, Springer LNCS 233, 518–526.Google Scholar
  10. [Me Sto]
    A. Meyer, L. Stockmeyer: The equivalence problem for regular expressions with squaring requires exponential space, Proc. of the 13th Annual IEEE Symp. on Switching and Automata Theory 1972, 125–129.Google Scholar
  11. [Ro Ye]
    L.E. Rosier, H. Yen: Logspace Hierarchies, Polynomial Time and the Complexity of Fairness Problems concerning ω-Machines, Proc. of the STACS, 1986, Springer LNCS 210, 306–320.Google Scholar
  12. [Ru Si To]
    W. Ruzzo, J. Simon, M. Tompa: Space-bounded hierarchies and probabilistic computations, J. Comput. System Sci. 28 (1984), 216–230.CrossRefGoogle Scholar
  13. [Sa]
    W. Savitch: Relationships between nondeterministic and deterministic tape complexities, J. Comput. System Sci. 4 (1970), 177–192.Google Scholar
  14. [Sto]
    L. Stockmeyer: The polynomial-time hierarchy, Theoret. Comput. Sci. 3 (1976), 1–22.CrossRefGoogle Scholar
  15. [Wa 86a]
    K.W. Wagner: More complicated questions about maxima and minima, and some closures of NP, Report, University of Passau, 1986.Google Scholar
  16. [Wa 86b]
    K.W. Wagner: The complexity of combinatorial problems with succint input representation, Acta Informatica 23, 3 (1986), 325–356.CrossRefGoogle Scholar
  17. [We]
    G. Wechsung: On the Boolean closure of NP, Proc. FCT Conf., 1985, Springer LNCS 199, 485–493.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Klaus-Jörn Lange
    • 1
  • Birgit Jenner
    • 2
  • Bernd Kirsig
    • 2
  1. 1.TU München, Inst. für InformatikMünchen 2
  2. 2.Fachbereich InformatikUniv. HamburgHamburg 13

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