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On Probabilistic Kleene Algebras, Automata and Simulations

  • Conference paper
Relational and Algebraic Methods in Computer Science (RAMICS 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6663))

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Abstract

We show that a class of automata modulo simulation equivalence forms a model of probabilistic Kleene algebra. We prove completeness of this model with respect to continuous probabilistic Kleene algebras. Hence an identity is valid in continuous probabilistic Kleene algebras if and only if the associated automata are simulation equivalent.

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References

  1. Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.): Handbook of Process Algebra. Elsevier, Amsterdam (2001)

    MATH  Google Scholar 

  2. Braibant, T., Pous, D.: An efficient coq tactic for deciding kleene algebras. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 163–178. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  3. Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bustan, D., Grumberg, O.: Simulation-based minimization. ACM Trans. Comput. Logic 4(2), 181–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen, E.: Weak Kleene algebra is sound and (possibly) complete for simulation. CoRR abs/0910.1028 (2009)

    Google Scholar 

  6. Fokkink, W., Zantema, H.: Basic process algebra with iteration: Completeness of its equational axioms. Comput. J. 37(4), 259–268 (1994)

    Article  Google Scholar 

  7. Furusawa, H., Nishizawa, K.: Relational and multirelational representation theorems for complete idempotent left semirings. In: de Swart, H. (ed.) RAMICS 2011. LNCS, vol. 6663, Springer, Heidelberg (2011)

    Google Scholar 

  8. Furusawa, H., Tsumagari, N., Nishizawa, K.: A non-probabilistic relational model of probabilistic kleene algebras. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 110–122. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  9. van Glabbeek, R.G.: The linear time-branching time spectrum (extended abstract). In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 278–297. Springer, Heidelberg (1990)

    Google Scholar 

  10. Hoare, C.A.R.T., Möller, B., Struth, G., Wehrman, I.: Concurrent kleene algebra. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 399–414. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  11. Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kozen, D.: On Kleene algebras and closed semirings. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 26–47. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  13. Krauss, A., Nipkow, T.: Proof pearl: Regular expression equivalence and relation algebra. Journal of Automated Reasoning (to appear, 2011)

    Google Scholar 

  14. McIver, A.K., Cohen, E., Morgan, C.C.: Using probabilistic kleene algebra for protocol verification. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 296–310. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. McIver, A.K., Morgan, C.C.: Abstraction, Refinement and Proof for Probabilistic Systems. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  16. McIver, A.K., Weber, T.: Towards automated proof support for probabilistic distributed systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 534–548. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Milner, R.: An algebraic definition of simulation between programs. Tech. rep., Stanford University, Stanford, CA, USA (1971)

    Google Scholar 

  18. Milner, R.: A complete inference system for a class of regular behaviours. J. Comput. Syst. Sci. 28(3), 439–466 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rutten, J.J.M.M.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  20. Takai, T., Furusawa, H.: Monodic tree kleene algebra. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 402–416. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  21. von Wright, J.: Towards a refinement algebra. Sci. Comput. Program. 51(1-2), 23–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computing, Macquarie University, Australia

    Annabelle McIver

  2. Department of Computer Science, The University of Sheffield, UK

    Tahiry M. Rabehaja & Georg Struth

Authors
  1. Annabelle McIver
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  2. Tahiry M. Rabehaja
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  3. Georg Struth
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Editor information

Editors and Affiliations

  1. Faculty of Philosophy, Tilburg University, P.O. Box 90153, 5000, Tilburg, LE, The Netherlands

    Harrie de Swart

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McIver, A., Rabehaja, T.M., Struth, G. (2011). On Probabilistic Kleene Algebras, Automata and Simulations. In: de Swart, H. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2011. Lecture Notes in Computer Science, vol 6663. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21070-9_20

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  • DOI: https://doi.org/10.1007/978-3-642-21070-9_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21069-3

  • Online ISBN: 978-3-642-21070-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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