Skip to main content
SpringerLink
Log in

Minimization via Duality

  • Conference paper
Book cover Logic, Language, Information and Computation (WoLLIC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Chapter  Google Scholar 

  2. Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Dover (1990)

    Google Scholar 

  3. Arbib, M., Manes, E.: Adjoint machines, state behavior machines and duality. J. Pure Appl. Algebra 6, 313–343 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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 1963

    Google Scholar 

  8. Givant, S., Halmos, P.: Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer (2009)

    Google Scholar 

  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. Johnstone, P.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press (1982)

    Google Scholar 

  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. Kiefer, S.: Minimization of weighted automata (2011) (unpublished private communication)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  14. Negrepontis, J.W.: Duality in analysis from the point of view of triples. Journal of Algebra 19, 228–253 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  17. Sondik, E.J.: The optimal control of partially observable Markov processes. Ph.D. thesis, Stanford University (1971)

    Google Scholar 

  18. Stone, M.H.: A general theory of spectra I. Proc. Nat. Acad. Sci. USA 26, 280–283 (1940)

    Article  Google Scholar 

  19. Stone, M.H.: A general theory of spectra II. Proc. Nat. Acad. Sci. USA 27, 83–87 (1941)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing, Imperial College London, UK

    Nick Bezhanishvili

Authors
  1. Nick Bezhanishvili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clemens Kupke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Prakash Panangaden
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Luke Ong

  2. Centro de Informática, Universidade Federal de Pernambuco, Av. jornalista Anibal Fernandes, s/n, 50740-560, Recife, PE, Brazil

    Ruy de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bezhanishvili, N., Kupke, C., Panangaden, P. (2012). Minimization via Duality. In: Ong, L., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2012. Lecture Notes in Computer Science, vol 7456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32621-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-32621-9_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32620-2

  • Online ISBN: 978-3-642-32621-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation