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Brzozowski’s Algorithm (Co)Algebraically

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Part of the Lecture Notes in Computer Science book series (LNPSE,volume 7230)

Abstract

We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on the duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations.

Keywords

  • Regular Expression
  • Universal Algebra
  • Deterministic Automaton
  • Probabilistic Automaton
  • Nondeterministic Automaton

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.

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References

  1. Adámek, J., Milius, S., Moss, L.S., Sousa, L.: Well-pointed Coalgebras (Unpublished note)

    Google Scholar 

  2. Arbib, M.A., Manes, E.G.: Adjoint machines, state-behaviour machines, and duality. Journal of Pure and Applied Algebra 6, 313–344 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Arbib, M.A., Manes, E.G.: Machines in a category. Journal of Pure and Applied Algebra 19, 9–20 (1980)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Bidoit, M., Hennicker, R., Kurz, A.: On the Duality between Observability and Reachability. In: Honsell, F., Miculan, M. (eds.) FOSSACS 2001. LNCS, vol. 2030, pp. 72–87. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  5. Bezhanishvili, N., Panangaden, P., Kupke, C.: Minimization via duality (Unpublished note)

    Google Scholar 

  6. Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. Mathematical Theory of Automata 12(6), 529–561 (1962)

    Google Scholar 

  7. Castiglione, G., Restivo, A., Sciortino, M.: Nondeterministic Moore Automata and Brzozowski’s Algorithm. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 88–99. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  8. Kozen, D.: Automata and Computability. Springer, Heidelberg (1997)

    CrossRef  MATH  Google Scholar 

  9. Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19, 427–443 (1997)

    CrossRef  Google Scholar 

  10. Hundt, C., Panangaden, P., Pineau, J., Precup, D., Dinculescu, M.: The duality of state and observations (Unpublished note)

    Google Scholar 

  11. Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)

    CrossRef  Google Scholar 

  12. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000); Fundamental Study

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press (2009)

    Google Scholar 

  14. Schützenberger, M.P.: On the definition of a family of automata. Information and Control 4(2-3), 245–270 (1961)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Silva, A., Bonchi, F., Bonsangue, M.M., Rutten, J.J.M.M.: Generalizing the powerset construction, coalgebraically. In: Proc. of FSTTCS 2010. Leibniz International Proceedings in Informatics (LIPIcs) Series, vol. 8, pp. 272–283 (2010)

    Google Scholar 

  16. Watson, B.W.: Taxonomies and Toolkits of Regular Language Algorithms. Ph.D thesis, Eindhoven University of Technology, The Netherlands (1995)

    Google Scholar 

  17. Watson, B.W.: Directly Constructing Minimal DFAs: Combining Two Algorithms by Brzozowski. In: Yu, S., Păun, A. (eds.) CIAA 2000. LNCS, vol. 2088, pp. 311–317. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

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Bonchi, F., Bonsangue, M.M., Rutten, J.J.M.M., Silva, A. (2012). Brzozowski’s Algorithm (Co)Algebraically. In: Constable, R.L., Silva, A. (eds) Logic and Program Semantics. Lecture Notes in Computer Science, vol 7230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29485-3_2

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  • DOI: https://doi.org/10.1007/978-3-642-29485-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29484-6

  • Online ISBN: 978-3-642-29485-3

  • eBook Packages: Computer ScienceComputer Science (R0)