Advertisement

Surminimisation of Automata

  • Victor MarsaultEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)

Abstract

We introduce the notion of surminimisation of a finite deterministic automaton; it consists in performing a transition relabelling before executing the minimisation and it produces an automaton smaller than a sole minimisation would. While the classical minimisation process preserves the accepted language, the surminimisation process preserves its underlying ordered tree only. Surminimisation induces on languages and on Abstract Rational Numeration Systems (ARNS) a transformation that we call label reduction. We prove that all positional numeration systems are label-irreducible and that an ARNS and its label reduction are very close, in the sense that converting the integer representations from one system into the other is done by a simple Mealy machine.

Keywords

Sole Minimisation Regular Language Numeration System Label Tree Outgoing Transition 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berthé, V., Rigo, M.: Odometers on regular languages. Theory Comput. Syst. 40(1), 1–31 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cobham, A.: Uniform tag sequences. Math. Systems Theory 6, 164–192 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dumont, J.-M., Thomas, A.: Digital sum problems and substitutions on a finite alphabet. Journal of Number Theory 39(3), 351–366 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Frougny, C., Klouda, K.: Rational base number systems for p-adic numbers. RAIRO - Theor. Inf. and Applic. 46(1), 87–106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Frougny, C., Sakarovitch, J.: Number representation and finite automata. In: Berthé, V., Rigo, M. (eds) Combinatorics, Automata and Number Theory. Encyclopedia of Mathematics and its Applications 135, pp. 34–107. Cambridge Univ. Press (2010)Google Scholar
  6. 6.
    Mealy, G.H.: A method for synthesizing sequential circuits. Bell Syst. Tech. J. 34, 1045–1079 (1955)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)Google Scholar
  8. 8.
    Lecomte, P., Rigo, M.: Abstract numeration systems. In: Berthé, V., Rigo, M. (eds.) Combinatorics, Automata and Number Theory. Encyclopedia of Mathematics and its Applications 135, pp. 108–162. Cambridge Univ. Press (2010)Google Scholar
  9. 9.
    Lecomte, P., Rigo, M.: Numeration systems on a regular language. Theory Comput. Syst. 34, 27–44 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press (2002)Google Scholar
  11. 11.
    Rigo, M., Maes, A.: More on generalized automatic sequences. Journal of Automata, Languages and Combinatorics 7(3), 351–376 (2002)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sakarovitch, J.: Eléments de théorie des automates. Vuibert, 2003. Corrected English translation: Elements of Automata Theory. Cambridge University Press (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Telecom-ParisTechParisFrance

Personalised recommendations