Mathematical systems theory

, Volume 21, Issue 1, pp 1–17 | Cite as

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

  • Oscar H. Ibarra
  • Bala Ravikumar


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.


Space Complexity Turing Machine Binary Representation Finite Automaton Closure Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Alb86]
    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. [Alt76]
    Alt, H. and K. Mehlhorn, A language over a one-symbol alphabet requiring onlyO(log logn) space,SIGACT Newsletter, Nov.–Dec. 1975.Google Scholar
  3. [Alt79]
    Alt, H., Lower bounds on space complexity for context-free recognition,Acta Informatica,12 (1979), 33–61.MATHCrossRefMathSciNetGoogle Scholar
  4. [Cha86a]
    Chang, J., O. Ibarra, M. Palis, and B. Ravikumar, On pebble automata,Theoretical Computer Science,44 (1986), 111–121.MATHCrossRefMathSciNetGoogle Scholar
  5. [Cha86b]
    Chang, J., O. Ibarra, T. Jiang, and B. Ravikumar, Some languages in NC1 (submitted for publication).Google Scholar
  6. [Cha87a]
    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.MATHCrossRefMathSciNetGoogle Scholar
  7. [Cha87b]
    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.)Google Scholar
  8. [Fre75]
    Freedman, E. and R. Ladner, Space bounds for processing contentless inputs,Journal of Computer and System Sciences,11 (1975), 118–128.MATHMathSciNetGoogle Scholar
  9. [Gre76]
    Greibach, S., Remarks on the complexity of nondeterministic counter languages,Theoretical Computer Science,1 (1976), 287.CrossRefGoogle Scholar
  10. [Har76]
    Hartmanis, J. and L. Berman, Tape bounds for processing languages over unary alphabet,Theoretical Computer Science,3 (1976), 213–224.CrossRefMathSciNetGoogle Scholar
  11. [Har78]
    Harrison, M.,Introduction to Formal Language Theory, Addison-Wesley, Reading, MA, 1978.MATHGoogle Scholar
  12. [Hop79]
    Hopcroft, J. and J. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979.MATHGoogle Scholar
  13. [Kan82]
    Kannan, R., Circuit-size lower bounds and nonreducibility to sparse sets,Information and Control,55 (1982), 40–56.MATHCrossRefMathSciNetGoogle Scholar
  14. [Lit85]
    Litow, B., On efficient deterministic simulation of Turing machine computations below logspace,Mathematical Systems Theory,18 (1985), 11–18.MATHCrossRefMathSciNetGoogle Scholar
  15. [Sak78]
    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.Google Scholar
  16. [Sav70]
    Savitch, W., Relationships between nondeterministic and deterministic tape complexities,Journal of Computer and System Sciences,4 (1970), 177–192.MATHMathSciNetGoogle Scholar
  17. [Sav73]
    Savitch, W., A note on multihead automata and context-sensitive languages,Acta Informatica,2 (1973), 249–252.MATHCrossRefMathSciNetGoogle Scholar
  18. [Sch85]
    Schoning, U.,Complexity and Structure, Lecture Notes in Computer Science, Vol. 211, Springer-Verlag, Berlin, 1985.Google Scholar
  19. [Sip80]
    Sipser, M., Halting space-bounded computations,Theoretical Computer Science,10 (1980), 335–338.MATHCrossRefMathSciNetGoogle Scholar
  20. [Ste65]
    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.Google Scholar
  21. [Sze87]
    Szepietowski, A., There are no fully space constructible functions between log logn and logn, Information Processing Letters,24 (1987), 361–362.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1988

Authors and Affiliations

  • Oscar H. Ibarra
    • 1
  • Bala Ravikumar
    • 1
  1. 1.Department of Computer ScienceUniversity of MinnesotaMinneapolisUSA

Personalised recommendations