A Logical Characterisation of Linear Time on Nondeterministic Turing Machines

  • Clemens Lautemann
  • Nicole Schweikardt
  • Thomas Schwentick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1563)


The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form ∃f1··∃fkR1··∃Rmxϕ that is true exactly for all strings in L. In this formula the fi are unary function symbols, the Ri are unary relation symbols and ϕ is a quantifier-free formula. Furthermore, the quantification of functions is restricted to non-crossing, decreasing functions and in ϕ no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these results we derive an Ehrenfeucht game characterisation of NTIME(n).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Ajtai and R. Fagin. Reachability is harder for directed than for undirected finite graphs. Journal of Symbolic Logic, 55(1):113–150, 1990.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Autebert, J. Berstel, and L. Boasson. Context-free languages and pushdown automata. Handbook of Formal Languages, 2:111–174, 1997.MathSciNetGoogle Scholar
  3. [3]
    R. Book and S. Greibach. Quasi-realtime languages. Math. Syst. Theory, 4:97–111, 1970.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J.R. Büchi. Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 6:66–92, 1960.MATHCrossRefGoogle Scholar
  5. [5]
    H.D. Ebbinghaus and J. Flum. Finite Model Theory. Springer-Verlag, New York, 1995.MATHGoogle Scholar
  6. [6]
    R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In R.M. Karp, editor, Complexity of Computation, volume 7 of SIAM-Proceedings, pages 43–73. AMS, 1974.Google Scholar
  7. [7]
    E. Grädel. Capturing complexity classes by fragments of second order logic. Theoretical Computer Science, 101:35–57, 1992.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    E. Grandjean. Invariance properties of RAMs and linear time. Computational Complexity, 4:62–106, 1994.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    E. Grandjean. Linear time algorithms and NP-complete problems. SIAM Journal of Computing, 23:573–597, 1994.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    E. Grandjean. Sorting, linear time and the satisfiability problem. Annals of Mathematics and Artificial Intelligence, 16:183–236, 1996.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Company, 1979.Google Scholar
  12. [12]
    N. Immerman. Descriptive and Computational Complexity. Springer-Verlag, New York, 1998.Google Scholar
  13. [13]
    C. Lautemann, T. Schwentick, and D. Thérien. Logics for context-free languages. In Proceedings of the Annual Conference of the EACSL, volume 933 of Lecture Notes in Computer Science, pages 205–216, 1994.Google Scholar
  14. [14]
    J. F. Lynch. The quantifier structure of sentences that characterize nondeterministic time complexity. Computational Complexity, 2:40–66, 1992.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W. Maass, G. Schnitger, E. Szemerédi, and G. Turán. Two tapes versus one for off-line turing machines. Computational Complexity, 3:392–401, 1993.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    W. Paul, N. Pippenger, E. Szemerédi, and W. Trotter. On determinism versus nondeterminism and related problems. In Proc. 24th Ann. Symp. Found. Comput. Sci., pages 429–438, 1983.Google Scholar
  17. [17]
    T. Schwentick. Padding and the expressive power of existential second-order logics. In 11th Annual Conference of the EACSL, CSL’ 97, pages 461–477, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Clemens Lautemann
    • 1
  • Nicole Schweikardt
    • 1
  • Thomas Schwentick
    • 1
  1. 1.Institut für Informatik / FB 17Johannes Gutenberg-UniversitätMainz

Personalised recommendations