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Mathematical systems theory

, Volume 24, Issue 1, pp 117–146 | Cite as

A geometrical view of the determinization and minimization of finite-state automata

  • Bruno Courcelle
  • Damian Niwinski
  • Andreas Podelski
Article

Abstract

With every finite-state word or tree automaton, we associate a binary relation on words or trees. We then consider the “rectangular decompositions” of this relation, i.e., the various ways to express it as a finite union of Cartesian products of sets of words or trees, respectively. We show that the determinization and the minimization of these automata correspond to simple geometrical reorganizations of the rectangular decompositions of the associated relations.

Keywords

Computational Mathematic Binary Relation Finite Union Tree Automaton Geometrical View 
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References

  1. [A]
    Arnold, A., A syntactic congruence for rational ω-languages,Theoret. Comput. Sci.,39 (1985), 333–335.Google Scholar
  2. [B]
    Brzozowski, J., Canonical regular expressions and minimal state graphs for definite events, inMathematical Theory of Automata, Vol. 12, MRI Symposium Series, Polytechnic Press of the Polytechnic Institute of Brooklyn, 1963, pp. 529–561.Google Scholar
  3. [C1]
    Courcelle, B., On recognizable sets and tree automata, inResolution of Equations in Algebraic Structures, Vol. 1 (H. Aït-Kaci and M. Nivat, eds.), Academic Press, New York, 1989, pp. 93–126.Google Scholar
  4. [C2]
    Courcelle, B., The monadic second-order logic of graphs, I: Recognizable sets of finite graphs,Inform. and Comput.,85 (1990), 12–75.Google Scholar
  5. [E]
    Eilenberg, S.,Automata, Languages, and Machines, Vol. A, Academic Press, New York, 1974.Google Scholar
  6. [GS]
    Gecseg, F., and Steinby, M.,Tree Automata, Akademiai Kiado, Budapest, 1984.Google Scholar
  7. [MW]
    Mezei, J., and Wright, J., Algebraic automata and context-free sets,Informat. and Control,11 (1967), 3–29.Google Scholar
  8. [N]
    Nerode, A., Linear automata transformations,Proc. Amer. Math. Soc.,9 (1958), 541–544.Google Scholar
  9. [NP]
    Nivat, M., and Podelski, A., Tree monoids and recognizable sets of finite trees, inResolution of Equations in Algebraic Structures, Vol. 1 (H. Aït-Kaci and M. Nivat, eds.), Academic Press, New York, 1989, pp. 351–367.Google Scholar
  10. [P]
    Podelski, A., Monoïdes d'arbres et automates d'arbres, Thèse, Université Paris-7, 1989.Google Scholar
  11. [RS]
    Rabin, M., and Scott, D., Finite automata and their decision problems,IBM J. Res. Develop.,3 (1959), 114–125. Reprinted inSequential Machines (E. Moore, ed.), Addison-Wesley, Reading, MA, 1964.Google Scholar
  12. [S]
    Staiger, L., Finite-state ω-languages,J. Comput. System Sci.,27 (1983), 434–448.Google Scholar
  13. [T]
    Trakhtenbrot, B., Finite automata and the logic of 1-place predicates,Siberian Math. J.,3 (1962), 103–131 (in Russian).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Bruno Courcelle
    • 1
  • Damian Niwinski
    • 2
  • Andreas Podelski
    • 3
  1. 1.Laboratoire d'InformatiqueUniversité Bordeaux-1Talence cedexFrance
  2. 2.Institute of MathematicsWarsaw University, PKiN IXWarszawaPoland
  3. 3.LITP, Université Paris-7Paris Cedex 05France

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