Theory of Átomata

  • Janusz Brzozowski
  • Hellis Tamm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call “átomaton”, whose states are the “atoms” of the language, that is, non-empty intersections of complemented or uncomplemented left quotients of the language. We describe methods of constructing the átomaton, and prove that it is isomorphic to the normal automaton of Sengoku, and to an automaton of Matz and Potthoff. We study “atomic” NFA’s in which the right language of every state is a union of atoms. We generalize Brzozowski’s double-reversal method for minimizing a deterministic finite automaton (DFA), showing that the result of applying the subset construction to an NFA is a minimal DFA if and only if the reverse of the NFA is atomic.

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References

  1. 1.
    Arnold, A., Dicky, A., Nivat, M.: A note about minimal non-deterministic automata. Bull. EATCS 47, 166–169 (1992)MATHGoogle Scholar
  2. 2.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Proceedings of the Symposium on Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn, N.Y (1963)Google Scholar
  3. 3.
    Carrez, C.: On the minimalization of non-deterministic automaton. Technical report, Lille University, Lille, France (1970)Google Scholar
  4. 4.
    Conway, J.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)MATHGoogle Scholar
  5. 5.
    Denis, F., Lemay, A., Terlutte, A.: Residual finite state automata. Fund. Inform. 51, 339–368 (2002)MATHMathSciNetGoogle Scholar
  6. 6.
    Kameda, T., Weiner, P.: On the state minimization of nondeterministic automata. IEEE Trans. Comput. C-19(7), 617–627 (1970)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Lombardy, S., Sakarovitch, J.: The universal automaton. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives. Texts in Logic and Games, vol. 2, pp. 457–504. Amsterdam University Press, Amsterdam (2007)Google Scholar
  8. 8.
    Matz, O., Potthoff, A.: Computing small finite nondeterministic automata. In: Engberg, U.H., Larsen, K.G., Skou, A. (eds.) Proceedings of the Workshop on Tools and Algorithms for Construction and Analysis of Systems. BRICS Note Series, pp. 74–88. BRICS, Aarhus (1995)Google Scholar
  9. 9.
    Rabin, M., Scott, D.: Finite automata and their decision problems. IBM J. Res. and Dev. 3, 114–129 (1959)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Sengoku, H.: Minimization of nondeterministic finite automata. Master’s thesis, Kyoto University, Department of Information Science, Kyoto University, Kyoto, Japan (1992)Google Scholar
  11. 11.
    Tabakov, D., Vardi, M.: Experimental evaluation of classical automata constructions. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 396–411. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Watson, B.W.: Taxonomies and toolkits of regular language algorithms. PhD thesis, Faculty of Mathematics and Computing Science. Eindhoven University of Technology, Eindhoven, The Netherlands (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Hellis Tamm
    • 2
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

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