Minimization via Duality

  • Nick Bezhanishvili
  • Clemens Kupke
  • Prakash Panangaden
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7456)

Abstract

We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object.

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References

  1. 1.
    Adámek, J., Bonchi, F., Hülsbusch, M., König, B., Milius, S., Silva, A.: A Coalgebraic Perspective on Minimization and Determinization. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 58–73. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Dover (1990)Google Scholar
  3. 3.
    Arbib, M., Manes, E.: Adjoint machines, state behavior machines and duality. J. Pure Appl. Algebra 6, 313–343 (1975)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Beimel, A., Bergadano, F., Bshouty, N.H., Kushelevitz, E., Varricchio, S.: Learning functions represented as multiplicity automata. Journal of the ACM 47(5), 506–530 (2000)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bonchi, F., Bonsangue, M., Boreale, M., Rutten, J., Silva, A.: A coalgebraic perspective on linear weighted automata. Information and Computation 211, 77–105 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bonchi, F., Bonsangue, M.M., Rutten, J.J.M.M., Silva, A.: Brzozowski’s Algorithm (Co)Algebraically. In: Constable, R.L., Silva, A. (eds.) Logic and Program Semantics, Kozen Festschrift. LNCS, vol. 7230, pp. 12–23. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Fox, J. (ed.) Proceedings of the Symposium on Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press of the Polytechnic Institute of Brooklyn (April 1962); book appeared in 1963Google Scholar
  8. 8.
    Givant, S., Halmos, P.: Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer (2009)Google Scholar
  9. 9.
    Hundt, C., Panangaden, P., Pineau, J., Precup, D.: Representing systems with hidden state. In: The Twenty-First National Conference on Artificial Intelligence, AAAI (2006)Google Scholar
  10. 10.
    Johnstone, P.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press (1982)Google Scholar
  11. 11.
    Kaelbling, L.P., Littman, M.L., Cassandra, A.R.: Planning and acting in partially observable stochastic domains. Artificial Intelligence 101 (1998)Google Scholar
  12. 12.
    Kiefer, S.: Minimization of weighted automata (2011) (unpublished private communication)Google Scholar
  13. 13.
    Mislove, M., Ouaknine, J., Pavlovic, D., Worrell, J.B.: Duality for Labelled Markov Processes. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 393–407. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Negrepontis, J.W.: Duality in analysis from the point of view of triples. Journal of Algebra 19, 228–253 (1971)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)Google Scholar
  16. 16.
    Smallwood, R.D., Sondik, E.J.: The optimal control of partially observable Markov decision processes over a finite horizon. Operations Research 21(5), 1071–1088 (1973)MATHCrossRefGoogle Scholar
  17. 17.
    Sondik, E.J.: The optimal control of partially observable Markov processes. Ph.D. thesis, Stanford University (1971)Google Scholar
  18. 18.
    Stone, M.H.: A general theory of spectra I. Proc. Nat. Acad. Sci. USA 26, 280–283 (1940)CrossRefGoogle Scholar
  19. 19.
    Stone, M.H.: A general theory of spectra II. Proc. Nat. Acad. Sci. USA 27, 83–87 (1941)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nick Bezhanishvili
    • 1
  • Clemens Kupke
  • Prakash Panangaden
  1. 1.Department of ComputingImperial College LondonUK

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