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Sublogarithmic-space turing machines, nonuniform space complexity, and closure properties

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Abstract

We show that some of the fundamental closure properties (such as concatenation) that hold for Turing machines (TMs) operating in space above logn, do not hold for TMs operating in space below logn. We also compare the powers of TMs andsweeping TMs operating in space below logn. While the proof that the powers of TMs and sweeping TMs are the same is trivial for space greater than or equal to logn, it is not obvious when the space is sublogarithmic. To explore the nature of sublogarithmic space computations further, we introduce a nonuniform space complexity measure and study some of its fundamental properties (such as closure, hierarchy, and gap) in the sublogarithmic range.

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References

  1. Alberts, M., Space complexity of alternating Turing machines,Fundamentals of Computation Theory, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1986, pp. 1–7.

    Google Scholar 

  2. Alt, H. and K. Mehlhorn, A language over a one-symbol alphabet requiring onlyO(log logn) space,SIGACT Newsletter, Nov.–Dec. 1975.

  3. Alt, H., Lower bounds on space complexity for context-free recognition,Acta Informatica,12 (1979), 33–61.

    Article  MATH  MathSciNet  Google Scholar 

  4. Chang, J., O. Ibarra, M. Palis, and B. Ravikumar, On pebble automata,Theoretical Computer Science,44 (1986), 111–121.

    Article  MATH  MathSciNet  Google Scholar 

  5. Chang, J., O. Ibarra, T. Jiang, and B. Ravikumar, Some languages in NC1 (submitted for publication).

  6. Chang, J., O. Ibarra, B. Ravikumar, and L. Berman, Some observations concerning alternating Turing machines using small space,Information Processing Letters,25 (1987), 1–9.

    Article  MATH  MathSciNet  Google Scholar 

  7. Chang, J., O. Ibarra, and A. Vergis, On the power of one-way communication, to appear inJournal of the Association for Computing Machinery. (Preliminary version appeared in theProc. of the 27th Symp. on Foundations of Computer Science, pp. 455–464.)

  8. Freedman, E. and R. Ladner, Space bounds for processing contentless inputs,Journal of Computer and System Sciences,11 (1975), 118–128.

    MATH  MathSciNet  Google Scholar 

  9. Greibach, S., Remarks on the complexity of nondeterministic counter languages,Theoretical Computer Science,1 (1976), 287.

    Article  Google Scholar 

  10. Hartmanis, J. and L. Berman, Tape bounds for processing languages over unary alphabet,Theoretical Computer Science,3 (1976), 213–224.

    Article  MathSciNet  Google Scholar 

  11. Harrison, M.,Introduction to Formal Language Theory, Addison-Wesley, Reading, MA, 1978.

    MATH  Google Scholar 

  12. Hopcroft, J. and J. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979.

    MATH  Google Scholar 

  13. Kannan, R., Circuit-size lower bounds and nonreducibility to sparse sets,Information and Control,55 (1982), 40–56.

    Article  MATH  MathSciNet  Google Scholar 

  14. Litow, B., On efficient deterministic simulation of Turing machine computations below logspace,Mathematical Systems Theory,18 (1985), 11–18.

    Article  MATH  MathSciNet  Google Scholar 

  15. Sakoda, W. and M. Sipser, Nondeterminism and the size of two-way finite automata,Proc. of the Tenth ACM Symp. on Theory of Computing, 1978, pp. 275–286.

  16. Savitch, W., Relationships between nondeterministic and deterministic tape complexities,Journal of Computer and System Sciences,4 (1970), 177–192.

    MATH  MathSciNet  Google Scholar 

  17. Savitch, W., A note on multihead automata and context-sensitive languages,Acta Informatica,2 (1973), 249–252.

    Article  MATH  MathSciNet  Google Scholar 

  18. Schoning, U.,Complexity and Structure, Lecture Notes in Computer Science, Vol. 211, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  19. Sipser, M., Halting space-bounded computations,Theoretical Computer Science,10 (1980), 335–338.

    Article  MATH  MathSciNet  Google Scholar 

  20. Stearns, R., J. Hartmanis, and P. Lewis, Hierarchies of memory limited computations,IEEE Conference Record on Switching Circuit Theory and Logical Design, IEEE Pub. 16C13, 1965, pp. 179–190.

  21. Szepietowski, A., There are no fully space constructible functions between log logn and logn, Information Processing Letters,24 (1987), 361–362.

    Article  MATH  MathSciNet  Google Scholar 

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This research was supported in part by NSF Grant DCR-8604603.

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Ibarra, O.H., Ravikumar, B. Sublogarithmic-space turing machines, nonuniform space complexity, and closure properties. Math. Systems Theory 21, 1–17 (1988). https://doi.org/10.1007/BF02088003

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  • DOI: https://doi.org/10.1007/BF02088003

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