Computing the order of a locally testable automaton

  • Sam Kim
  • Robert McNaughton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)


A locally testable language is a language with the property that, for some nonnegative integer j, whether or not a word x is in the language depends on (1) the prefix and suffix of x of length j, and (2) the set of substrings of x of length j+1, without regard to the order in which these substrings occur or the number of times each substring occurs. This paper shows that computing the smallest j of a given locally testable deterministic automaton is NP-hard, and presents a polynomial-time ε- approximation algorithm for computing such j. It turns out that, for a fixed j, there is a polynomial time algorithm to decide whether a given automaton satisfies the above condition. In addition, we have obtained an upper bound of 2n2+1 on the smallest such j for a locally testable automaton of n states.


Terminal Node Longe Path Finite Automaton Outgoing Edge Testable Language 
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. [1]
    A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
  2. [2]
    J. Brzozowski and I. Simon, Characterizations of locally testable events, Discrete Mathematics, 4 (1973), pp. 243–271.CrossRefGoogle Scholar
  3. [3]
    M. R. Garey and D. S. Johnson, Computers and Intractability, Freeman, 1979.Google Scholar
  4. [4]
    M. Harrison, Introduction to Formal Language Theory, Addison-Wesley, 1978.Google Scholar
  5. [5]
    J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 1979.Google Scholar
  6. [6]
    E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, Computer Science Press, 1984.Google Scholar
  7. [7]
    S. M. Kim, R. McNaughton, and R. McCloskey, A polynomial time algorithm for the local testability problem of deterministic finite automata (to appear in IEEE Trans. Computers, October, 1991).Google Scholar
  8. [8]
    S. M. Kim and R. McNaughton, Computing the order of a locally testable automaton, Tech. Report 91-24, Department of Computer Science, Rensselaer Polytechnic InstituteGoogle Scholar
  9. [9]
    M. Minsky and S. Papert, Perceptions, M.I.T. Press, 1969.Google Scholar
  10. [10]
    R. McNaughton and S. Papert, Counter-free Automata, M.I.T. Press, 1971.Google Scholar
  11. [11]
    R. McNaughton, Algebraic decision procedures for local testability, Mathematical Systems Theory, 8 (1974), pp. 60–76.CrossRefGoogle Scholar
  12. [12]
    J. Stern, Complexity of some problems from the theory of automata, Information and Control, 66(185), pp. 163–176.Google Scholar
  13. [13]
    Y. Zalcstein, Locally testable languages, Journal of Computer and System Sciences, 6 (1972), pp. 151–167.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Sam Kim
    • 1
  • Robert McNaughton
    • 1
  1. 1.Computer Science DepartmentRensselaer Polytechnic InstituteTroy

Personalised recommendations