Minimization via Duality

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


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.


Boolean Algebra Left Adjoint Forgetful Functor Minimal Realization Contravariant Functor 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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

Personalised recommendations