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 ∑ 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.
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References
L. Babai andS. Moran, Arthur-Merlin games: a randomized proof-system, and a hierarchy of complexity classes.J. Comput. System. Sci. 36 (1988), 254–276.
J. L. Balcázar, J. Díaz andJ. Gabarró,Structural Complexity I. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.
E. Börger,Computability, Complexity, Logic. North-Holland, Amsterdam, New York, Oxford, Tokyo, 1989.
B. von Braunmühl, Alternationshierarchien von Turingmaschinen mit kleinem Speicher. Informatik Berichte 83, Institut für Informatik, Universität Bonn, 1991.
J. H. Chang, O. H. Ibarra, B. Ravikumar, andL. Berman Some observations concerning Turing machines using small space.Inform. Process. Lett. 25 (1987), 1–9. Erratum,Inform. Process. Lett. 25 (1988) 53.
P. van Emde Boas, Machine models and simulations. InHandbook of Theoretical Computer Science, Volume A, Algorithms and Complexity ed.J. van Leeuwen, 1–66. Elsevier, Amsterdam, New York, Oxford, Tokyo, 1990.
R. Freivalds, On time complexity of deterministic and nondeterministic Turing machines.Latvijski Mathematičeskij Eshegodnik 23 (1979), 158–165. In Russian.
M. R. Garey andD. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Fransisco, 1979.
V. Geffert, Nondeterministic computations in sublogarithmic space and space constructibility. InProc. 17 ICALP, Lecture Notes in Computer Science 443, 1990 111–124.
V. Geffert, Tally versions of the Savitch and the Immerman-Szelepcsényi theorems for sublogarithmic space.SIAM J. Comput. 22 (1993), 102–113.
L. A. Hemachandra, The strong exponential hierarchy collapses. InProc. Nineteenth Ann. ACM Symp. Theor. Comput, 1987, 110–122.
N. Immerman, NSPACE is closed under complement.SIAM J. Comput. 17 (1988), 935–938.
A. Ito, K. Inoue, andI. Takanami, A note on alternating Turing machines using small space.The Transactions of the IEICE E 70 no. 10 (1987), 990–996.
K. Iwama, ASPACE(o(log log)) is regular. Research report KSU/ICS Kyoto Sangyo University, Kyoto, 603, Japan, 1986. See alsoSIAM J. Comput 22 (1993), 136–146.
M. Kutylowski, M. Liśkiewicz, andK. Loryś, Reversal complexity classes for alternating Turing machines.SIAM J. Comput. 19 (1990), 207–221.
P. M. Lewis, R. E. Stearns, and J. Hartmanis, Memory bounds for recognition of context-free and context-sensitive languages. InIEEE. Conf. Switch. Circuit Theory and Logic Design, 1965, 191–202.
M. Liśkiewicz and R. Reischuk, Separating the lower levels of the sublogarithmic space hierarchy. InProc. 10.STACS, Lecture Notes in Computer Science 665, 1993a, 16–28.
M. Liśkiewicz and R. Reischuk, The sublogarithmic space hierarchy is infinite. Technical report, Institut für theoretische Informatik, TH Darmstadt, 1993b.
M. Liśkiewicz, K. Loryś, andM. Piotrów On reversal bounded alternating Turing machines.Theoret. Comput. Sci. 54 (1987), 331–339.
W. J. Paul,Komplexitätstheorie. Teubner Studienbücher, Informatik, Stuttgart, 1978.
U. Schöning and K. W. Wagner, Collapsing oracle hierarchies, census functions and logarithmically many queries. InProc. 5th. STACS 88, Lecture Notes in Computer Science 294, 1988, 91–97.
M. Sipser, Halting bounded computations.,Theoret. Comput. Sci. 10 (1980), 335–338.
M. Sipser, Borel sets and circuit complexity. InProc. Fifteenth Ann. ACM Symp. Theor. Comput., 1983, 330–335.
R. Szelepcsényi, The method of forced enumeration for nondeterministic automata.Acta Infor. 26 (1988), 279–284.
K. W. Wagner, The alternation hierarchy for sublogarithmic space: an exciting race to STACS'93 (Editorial note). InProc. 10 STACS, Lecture Notes in Computer Science 665, 1993, 2–4.
K. Wagner andG. Wechsung,Computational Complexity. D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1986.
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von Braunmühl, B., Gengler, R. & Rettinger, R. The alternation hierarchy for sublogarithmic space is infinite. Comput Complexity 3, 207–230 (1993). https://doi.org/10.1007/BF01271368
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DOI: https://doi.org/10.1007/BF01271368