An upper bound on the order of locally testable deterministic finite automata

  • Sam Kim
  • Robert McNaughton
  • Robert McCloskey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 401)


A locally testable language is a language with the property that for some nonnegative integer k, called the order of locality, whether or not a word w is in the language depends on (1) the prefix and suffix of w of length k, and (2) the set of intermediate substrings of w of length k + 1, without regard to the order in which these substrings occur. The local testability problem is, given a deterministic finite automaton, to decide whether it accepts a locally testable language or not. Recently, we introduced the first polynomial time algorithm for the local testability problem based on a simple characterization of locally testable deterministic automata. This paper investigates the upper bound on the order of locally testable automata. It shows that the order of a locally testable deterministic automaton is at most n4 + 1, where n is the number of states of the automaton.


Polynomial Time Algorithm Finite Automaton Transition Graph Closed Path Testable Language 
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  1. (1).
    Aho, A., Hopcroft, J., and Ullman, J., The Design and Analysis of Computer Algorithms, Addison-Wesley (1974).Google Scholar
  2. (2).
    Brzozowski, J. and Simon, I., Characterizations of locally testable events, Discrete Mathematics 4 (1973), pp. 243–271.CrossRefGoogle Scholar
  3. (3).
    Harrison, M., Introduction to Formal Language Theory, Addison Wesley (1978).Google Scholar
  4. (4).
    Hopcroft, J., and Ullman, J., Introduction to Automata Theory, Languages, and Computation, Addison Weslely (1979).Google Scholar
  5. (5).
    Hunt, H. and Rosenkrantz, D., Computational parallels between the regular and context-free languages, SIAM J. COMPUT., 7 (1978), pp. 99–114.CrossRefGoogle Scholar
  6. (6).
    Kim, S., McNaughton, R., and McCloskey, R., A polynomial time algorithm for the local testability problem of deterministic finite automata, Workshop on Algorithms and Data Structures, (1989).Google Scholar
  7. (7).
    Martin, R., Studies in Feedback-Shift-Register Synthesis of Sequential Machines, M.I.T. Press, (1969).Google Scholar
  8. (8).
    McNaughton, R., Algebraic decision procedures for local testability, Mathematical Systems Theory, Vol.8 (1974), pp. 60–76.CrossRefGoogle Scholar
  9. (9).
    McNaughton, R. and Papert, S., Counter-free Automata, M.I.T. Press, (1971)Google Scholar
  10. (10).
    Menon, P., and Friedman, A., Fault detection in iterative logic arrays, IEEE Trans. on Computers, C-20 (1971), pp. 524–535.Google Scholar
  11. (11).
    Zalcstein, Y., Locally testable languages, Journal of Computer and System Sciences, 6 (1972), pp. 151–167.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

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

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