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Concurrent Kleene Algebra

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CONCUR 2009 - Concurrency Theory (CONCUR 2009)

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A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partially-ordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Jones’s rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.

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  1. Bergstra, J.A., Bethke, I., Ponse, A.: Process algebra with iteration and nesting. The Computer Journal 37(4), 243–258 (1994)

    Article  MATH  Google Scholar 

  2. Birkhoff, G.: Lattice Theory, 3rd edn. Amer. Math. Soc. (1967)

    Google Scholar 

  3. Bistarelli, S., Montanari, U., Rossi, F.: Semiring-based constraint satisfaction and optimization. J. ACM 44(2), 201–236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boudol, G., Castellani, I.: On the semantics of concurrency: partial orders and transition systems. In: Ehrig, H., Levi, G., Montanari, U. (eds.) CAAP 1987 and TAPSOFT 1987. LNCS, vol. 249, pp. 123–137. Springer, Heidelberg (1987)

    Google Scholar 

  5. Chothia, T., Kleijn, J.: Q-Automata: modelling the resource usage of concurrent components. Electr. Notes Theor. Comput. Sci. 175(2), 153–167 (2007)

    Article  Google Scholar 

  6. Cohen, E.: Separation and reduction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  7. Conway, J.: Regular Algebra and Finite Machines. Chapman & Hall, Sydney (1971)

    MATH  Google Scholar 

  8. Desharnais, J., Möller, B., Struth, G.: Kleene Algebra with domain. Trans. Computational Logic 7, 798–833 (2006)

    Article  MathSciNet  Google Scholar 

  9. Gischer, J.: Partial orders and the axiomatic theory of shuffle. PhD thesis, Stanford University (1984)

    Google Scholar 

  10. Grabowski, J.: On partial languages. Fundamenta Informaticae 4(1), 427–498 (1981)

    MathSciNet  MATH  Google Scholar 

  11. Groote, J., Mathijssen, A., van Weerdenburg, M., Usenko, Y.: From μCRL to mCRL2: motivation and outline. In: Proc. Workshop Essays on Algebraic Process Calculi (APC 25). ENTCS, vol. 162, pp. 191–196 (2006)

    Google Scholar 

  12. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12, 576–585 (1969)

    Article  MATH  Google Scholar 

  13. Hoare, C.A.R.: Communicating sequential processes. Prentice Hall, Englewood Cliffs (1985)

    MATH  Google Scholar 

  14. Hoare, C.A.R., Möller, B., Struth, G., Wehrman, I.: Concurrent Kleene Algebra. Institut für Informatik, Universität Augsburg, Technical Report 2009-04 (April 2009)

    Google Scholar 

  15. Hoare, C.A.R., Möller, B., Struth, G., Wehrman, I.: Foundations of Concurrent Kleene Algebra. In: Berghmmar, R., Jaoua, A., Möller, B. (eds.) Relations and kleene Algebra in Computer Science. Proc. 11th International Conference on Relational Methods in Computer Science (RelMiCS 11) and 6th International Conference on Applications of Kleene Algebra (AKA 6), Doha, Qatar, November 1–5. LNCS, vol. 5827. Springer, Heidelberg (2009) (forthcoming)

    Google Scholar 

  16. Hoare, C.A.R., Wehrman, I., O’Hearn, P.: Graphical models of separation logic. In: Proc. Marktoberdorf Summer School (forthcoming, 2008)

    Google Scholar 

  17. Jones, C.: Development methods for computer programs including a notion of interference. PhD Thesis, University of Oxford. Programming Research Group, Technical Monograph 25 (1981)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Kozen, D.: Kleene algebra with tests. Trans. Programming Languages and Systems 19, 427–443 (1997)

    Article  MATH  Google Scholar 

  20. Mac Lane, S.: Categories for the working mathematician, 2nd edn. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  21. Milner, R.: A Calculus of Communication Systems. LNCS, vol. 92. Springer, Heidelberg (1980)

    Book  MATH  Google Scholar 

  22. Misra, J.: Axioms for memory access in asynchronous hardware systems. ACM Trans. Program. Lang. Syst. 8, 142–153 (1986)

    Article  MATH  Google Scholar 

  23. Morgan, C.: Programming from Specifications. Prentice Hall, Englewood Cliffs (1990)

    MATH  Google Scholar 

  24. Mulvey, C.: Rendiconti del Circolo Matematico di Palermo 12, 99–104 (1986)

    Google Scholar 

  25. O’Hearn, P.: Resources, concurrency, and local reasoning. Theor. Comput. Sci. 375, 271–307 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pratt, V.R.: Modelling concurrency with partial orders. Journal of Parallel Programming 15(1) (1986)

    Google Scholar 

  27. Prisacariu, C.: Extending Kleene lgebra with synchrony — technicalities. University of Oslo, Department of Informatics, Research Report No. 376 (October 2008)

    Google Scholar 

  28. McCune, W.: Prover9 and Mace4, (accessed March 1, 2009)

  29. Rosenthal, K.: Quantales and their applications. Pitman Research Notes in Math. No. 234. Longman Scientific and Technical (1990)

    Google Scholar 

  30. Sangiorgi, D., Walker, D.: The π-calculus — A theory of mobile processes. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  31. Winskel, G.: Event structures. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) APN 1986. LNCS, vol. 255, pp. 325–392. Springer, Heidelberg (1987)

    Google Scholar 

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Hoare, C.A.R.T., Möller, B., Struth, G., Wehrman, I. (2009). Concurrent Kleene Algebra. In: Bravetti, M., Zavattaro, G. (eds) CONCUR 2009 - Concurrency Theory. CONCUR 2009. Lecture Notes in Computer Science, vol 5710. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04080-1

  • Online ISBN: 978-3-642-04081-8

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