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A complete axiomatization of weighted branching bisimulation

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Abstract

We propose an axiomatization for weighted branching bisimulation over a weighted process algebra with positive rational weights including zero and show that this axiomatization is both sound and complete. Our proof of soundness and completeness are inspired by similar results by Milner for strong and weak bisimulation and by van Glabbeek for branching bisimulation. We also show that the claim that weighted branching bisimilarity is an equivalence relation indeed holds true. As auxiliary results, we give two alternative characterizations of weighted branching bisimulation, one in terms of weighted stuttering transitions and another in terms of a relative branching base which can be seen as a linear basis from which we can construct all weighted stuttering transitions.

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Notes

  1. You can just map actions to distinct weights and then strongly bisimilar CCS processes would be weighted (strongly) bisimilar; and vice versa.

  2. Another plausible approach would have been to use some rooted variant of the equivalence, as was done for (regular) branching bisimulation [17]. This would however leads to a less intuitive axiomatization.

  3. We try to mimic the prefix and non-deterministic choice operations of CCS, as they in our case have been merged.

  4. Recall that and are shorthands for and respectively.

  5. That is, \({\mathcal {R}}\circ {\mathcal {R}}= \{\langle E, G \rangle \mathrel {\big |}\exists F \in {\mathcal {E}}: E {\mathcal {R}}F \text { and } F {\mathcal {R}}G\}\).

  6. Usually given as \(E + F \equiv F + E\), \(E + (F + G) \equiv (E + F) + G\), and \(E + E \equiv E\) respectively.

  7. Not counting the implicit uses of A1, A2, and A3.

  8. It should be noted that we here do not address the issue that substitution might rename bound variables. Instead, we use the convention that if \(E =_\alpha F\), then \(E = F\), where \(=_\alpha \) is the relation given by \(\alpha \)-equivalence. This decision is justified by Lemma 3.

References

  1. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: Complete axiomatization for the bisimilarity distance on Markov chains. In: Desharnais, J., Jagadeesan, R. (eds.) 27th International Conference on Concurrency Theory, CONCUR 2016, 23–26 Aug 2016, Québec City, Canada, LIPIcs, vol. 59, pp. 21:1–21:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016). https://doi.org/10.4230/LIPIcs.CONCUR.2016.21

  2. Basten, T.: Branching bisimilarity is an equivalence indeed!. Inf. Process. Lett. 58(3), 141–147 (1996). https://doi.org/10.1016/0020-0190(96)00034-8

    Article  MathSciNet  MATH  Google Scholar 

  3. Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Kozen, D. (ed.) Logics of Programs, Workshop, Yorktown Heights, New York, USA, May 1981, Lecture Notes in Computer Science, vol. 131, pp. 52–71. Springer (1981). https://doi.org/10.1007/BFb0025774

  4. De Nicola, R., Vaandrager, F.W.: Three logics for branching bisimulation. J. ACM 42(2), 458–487 (1995). https://doi.org/10.1145/201019.201032

    Article  MathSciNet  MATH  Google Scholar 

  5. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labeled markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR ’99: Concurrency Theory, 10th International Conference, Eindhoven, The Netherlands, 24–27 Aug 1999, Proceedings, Lecture Notes in Computer Science, vol. 1664, pp. 258–273. Springer (1999). https://doi.org/10.1007/3-540-48320-9_19

  6. Foshammer, L., Larsen, K.G., Mardare, R., Xue, B.: Logical characterization and complexity of weighted branching preorders and distances. Unpublished draft (2017)

  7. Foshammer, L., Larsen, K.G., Mariegaard, A.: Weighted branching simulation distance for parametric weighted kripke structures. In: Brihaye, T., Delahaye, B., Jezequel, L., Markey, N., Srba, J. (eds.) Proceedings Cassting Workshop on Games for the Synthesis of Complex Systems and 3rd International Workshop on Synthesis of Complex Parameters and 3rd International Workshop on Synthesis of Complex Parameters, Cassting/SynCoP 2016, Eindhoven, The Netherlands, 2–3 Apr 2016. EPTCS, vol. 220, pp. 63–75 (2016). https://doi.org/10.4204/EPTCS.220.6

  8. Groote, J.F., Vaandrager, F.W.: An efficient algorithm for branching bisimulation and stuttering equivalence. In: M. Paterson (ed.) Automata, Languages and Programming, 17th International Colloquium, ICALP90, Warwick University, England, UK, 16–20 July 1990, Proceedings, Lecture Notes in Computer Science, vol. 443, pp. 626–638. Springer (1990). https://doi.org/10.1007/BFb0032063

  9. Hansen, M., Larsen, K.G., Mardare, R., Pedersen, M.R.: Reasoning about bounds in weighted transition systems. Log. Methods Comput. Sci. 14(4) (2018). https://doi.org/10.23638/LMCS-14(4:19)2018. https://lmcs.episciences.org/4992

  10. Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985). https://doi.org/10.1145/2455.2460

    Article  MathSciNet  MATH  Google Scholar 

  11. Jensen, M.C., Larsen, K.G., Mardare, R.: Weighted branching systems: Behavioural equivalence, behavioural distance, and their logical characterisations. In: Jansen, D.N., Prabhakar, P. (eds.) Formal Modeling and Analysis of Timed Systems—16th International Conference, FORMATS 2018, Beijing, China, 4–6 Sept 2018, Proceedings, Lecture Notes in Computer Science, vol. 11022, pp. 145–161. Springer (2018). https://doi.org/10.1007/978-3-030-00151-3_9

  12. Milner, R.: A Calculus of Communicating Systems, Lecture Notes in Computer Science, vol. 92. Springer, Berlin (1980). https://doi.org/10.1007/3-540-10235-3

  13. Milner, R.: A complete inference system for a class of regular behaviours. J. Comput. Syst. Sci. 28(3), 439–466 (1984). https://doi.org/10.1016/0022-0000(84)90023-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Milner, R.: A complete axiomatisation for observational congruence of finite-state behaviors. Inf. Comput. 81(2), 227–247 (1989). https://doi.org/10.1016/0890-5401(89)90070-9

    Article  MATH  Google Scholar 

  15. Milner, R.: CHAPTER 19 - Operational and algebraic semantics of concurrent processes. In: Leeuwen, J.V. (ed.) Formal Models and Semantics. Handbook of Theoretical Computer Science, pp. 1201–1242. Elsevier, Amsterdam (1990). ISBN: 978-0-444-88074-1. https://doi.org/10.1016/B978-0-444-88074-1.50024-X

  16. Nykänen, M., Ukkonen, E.: The exact path length problem. J. Algorithms 42(1), 41–53 (2002). https://doi.org/10.1006/jagm.2001.1201

    Article  MathSciNet  MATH  Google Scholar 

  17. van Glabbeek, R.J.: A complete axiomatization for branching bisimulation congruence of finite-state behaviours. In: Borzyszkowski A.M., Sokolowski S. (eds.) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg (1993). https://doi.org/10.1007/3-540-57182-5_39

  18. van Glabbeek, R.J.: The linear time – Branching time spectrum II. In: Best E. (ed.) CONCUR’93. Lecture Notes in Computer Science, vol 715. Springer, Berlin, Heidelberg (1993)

  19. van Glabbeek, R.J., Weijland, W.P.: Branching time and abstraction in bisimulation semantics (extended abstract). In: IFIP Congress, pp. 613–618 (1989)

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Acknowledgements

We would like to thank our peers who acted as referees in reviewing this paper. Their well-thought-out suggestions, insight, and criticism have been of great help in completing this work. Of considerable note, their comments on whether weighted branching bisimularity indeed was an equivalence relation or not and on the proof of completeness have been most substantive and of use. Again, thank you.

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  1. Department of Computer Science, Aalborg University, Selma Lagerlöfs Vej 300, 9220, Aalborg, Denmark

    Mathias Claus Jensen & Kim Guldstrand Larsen

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  1. Mathias Claus Jensen
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Correspondence to Mathias Claus Jensen.

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Jensen, M.C., Larsen, K.G. A complete axiomatization of weighted branching bisimulation. Acta Informatica 57, 689–725 (2020). https://doi.org/10.1007/s00236-020-00375-6

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