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Convex Lattice Equation Systems

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Principles of Systems Design

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13660))

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Abstract

In this paper we revisit the paradigm shift “From Boolean to Quantitative Notions of Correctness” proposed by Henzinger more than 10 years ago. In particular, we present the notion of Convex Lattice Equation Systems as a universal framework for encoding and inferring behavioural metrics between quantitative system behaviours. We demonstrate how the framework may be applied to infer bounds on values of stochastic games and distances between timed systems.

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Notes

  1. 1.

    Note that convex combinations are treated as described in Remark 1.

References

  1. Algayres, B., Coelho, V., Doldi, L., Garavel, H., Lejeune, Y., Rodríguez, C.: VESAR: a pragmatic approach to formal specification and verification. Comput. Netw. ISDN Syst. 25(7), 779–790 (1993)

    Article  Google Scholar 

  2. Andersen, H.R.: Model checking and Boolean graphs. In: Krieg-Brückner, B. (ed.) ESOP 1992. LNCS, vol. 582, pp. 1–19. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55253-7_1

    Chapter  Google Scholar 

  3. Andersen, J.R., et al.: CAAL: concurrency workbench, Aalborg edition. In: Leucker, M., Rueda, C., Valencia, F.D. (eds.) ICTAC 2015. LNCS, vol. 9399, pp. 573–582. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25150-9_33

    Chapter  Google Scholar 

  4. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: Computing behavioral distances, compositionally. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 74–85. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40313-2_9

    Chapter  MATH  Google Scholar 

  5. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: On-the-fly exact computation of bisimilarity distances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 1–15. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36742-7_1

    Chapter  MATH  Google Scholar 

  6. 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, Québec City, Canada, 23–26 August 2016, volume 59 of LIPIcs, pp. 21:1–21:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

    Google Scholar 

  7. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: Complete axiomatization for the total variation distance of Markov chains. In: Staton, S. (ed.) Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, MFPS 2018, Dalhousie University, Halifax, Canada, 6–9 June 2018, volume 341 of Electronic Notes in Theoretical Computer Science, pp. 27–39. Elsevier (2018)

    Google Scholar 

  8. Bacci, G., Bacci, G., Larsen, K.G., Mardare, R., Tang, Q., van Breugel, F.: Computing probabilistic bisimilarity distances for probabilistic automata. Log. Methods Comput. Sci. 17(1) (2021)

    Google Scholar 

  9. Čerāns, K., Godskesen, J.C., Larsen, K.G.: Timed modal specification—Theory and tools. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 253–267. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56922-7_21

    Chapter  Google Scholar 

  10. Chen, D., van Breugel, F., Worrell, J.: On the complexity of computing probabilistic bisimilarity. In: Birkedal, L. (ed.) FoSSaCS 2012. LNCS, vol. 7213, pp. 437–451. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28729-9_29

    Chapter  MATH  Google Scholar 

  11. Cleaveland, R., Parrow, J., Steffen, B.: The concurrency workbench. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 24–37. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52148-8_3

    Chapter  Google Scholar 

  12. Condon, A.: On algorithms for simple stochastic games. In: Advances in Computational Complexity Theory, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 51–72. DIMACS/AMS (1990)

    Google Scholar 

  13. Condon, A.: The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Desharnais, J., Laviolette, F., Tracol, M.: Approximate analysis of probabilistic processes: logic, simulation and games. In: Fifth International Conference on the Quantitative Evaluaiton of Systems (QEST 2008), Saint-Malo, France, 14–17 September 2008, pp. 264–273. IEEE Computer Society (2008)

    Google Scholar 

  16. Easterbrook, S.M., Chechik, M.: A framework for multi-valued reasoning over inconsistent viewpoints. In: Müller, H.A., Harrold, M.J., Schäfer, W. (eds.) Proceedings of the 23rd International Conference on Software Engineering, ICSE 2001, Toronto, Ontario, Canada, 12–19 May 2001, pp. 411–420. IEEE Computer Society (2001)

    Google Scholar 

  17. Fahrenberg, U., Thrane, C.R., Larsen, K.G.: Distances for weighted transition systems: games and properties. In: Massink, M., Norman, G. (eds.) Proceedings Ninth Workshop on Quantitative Aspects of Programming Languages, QAPL 2011, Saarbrücken, Germany, 1–3 April 2011, volume 57 of EPTCS, pp. 134–147 (2011)

    Google Scholar 

  18. Garavel, H., Lang, F., Mateescu, R., Serwe, W.: CADP 2010: a toolbox for the construction and analysis of distributed processes. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 372–387. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19835-9_33

    Chapter  MATH  Google Scholar 

  19. Goubault-Larrecq, J.: Prevision domains and convex powercones. In: Amadio, R. (ed.) FoSSaCS 2008. LNCS, vol. 4962, pp. 318–333. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78499-9_23

    Chapter  MATH  Google Scholar 

  20. Henzinger, T.A.: From Boolean to quantitative notions of correctness. In: Hermenegildo, M.V., Palsberg, J. (eds.) Proceedings of the 37th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2010, Madrid, Spain, 17–23 January 2010, pp. 157–158. ACM (2010)

    Google Scholar 

  21. Henzinger, T.A., Majumdar, R., Prabhu, V.S.: Quantifying similarities between timed systems. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 226–241. Springer, Heidelberg (2005). https://doi.org/10.1007/11603009_18

    Chapter  MATH  Google Scholar 

  22. Henzinger, T.A., Sifakis, J.: The embedded systems design challenge. In: Misra, J., Nipkow, T., Sekerinski, E. (eds.) FM 2006. LNCS, vol. 4085, pp. 1–15. Springer, Heidelberg (2006). https://doi.org/10.1007/11813040_1

    Chapter  Google Scholar 

  23. Juba, B.: On the hardness of simple stochastic games. Master’s thesis, Carnegie Mellon University, Pittsburgh, PA, USA, May 2005

    Google Scholar 

  24. Kupferman, O., Lustig, Y.: Latticed simulation relations and games. Int. J. Found. Comput. Sci. 21(2), 167–189 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Larsen, K.G., Fahrenberg, U., Thrane, C.R.: Metrics for weighted transition systems: axiomatization and complexity. Theoret. Comput. Sci. 412(28), 3358–3369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Larsen, K.G.: Efficient local correctness checking. In: von Bochmann, G., Probst, D.K. (eds.) CAV 1992. LNCS, vol. 663, pp. 30–43. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56496-9_4

    Chapter  Google Scholar 

  27. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. In: Conference Record of the Sixteenth Annual ACM Symposium on Principles of Programming Languages, Austin, Texas, USA, 11–13 January 1989, pp. 344–352. ACM Press (1989)

    Google Scholar 

  28. Liggett, T., Lippman, S.A.: Stochastic games with perfect information and time average payoff. SIAM Rev. 11(4), 604–607 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, X., Smolka, S.A.: Simple linear-time algorithms for minimal fixed points. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 53–66. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055040

    Chapter  Google Scholar 

  30. Man, K.L.: \(\upmu \)CRL: a computer science based approach for specification and verification of hardware circuits, vol. 01, pp. I-387–I-390 (2008)

    Google Scholar 

  31. Mislove, M.: Nondeterminism and probabilistic choice: obeying the laws. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 350–365. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44618-4_26

    Chapter  Google Scholar 

  32. Rosenmann, A.: On the distance between timed automata. In: André, É., Stoelinga, M. (eds.) FORMATS 2019. LNCS, vol. 11750, pp. 199–215. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29662-9_12

    Chapter  MATH  Google Scholar 

  33. Tang, Q., van Breugel, F.: Computing probabilistic bisimilarity distances via policy iteration. In: CONCUR, volume 59 of LIPIcs, pp. 22:1–22:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

    Google Scholar 

  34. Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5(2), 285–309 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thrane, C.R., Fahrenberg, U., Larsen, K.G.: Quantitative analysis of weighted transition systems. J. Log. Algebr. Methods Program. 79(7), 689–703 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tix, R., Keimel, K., Plotkin, G.D.: Semantic domains for combining probability and non-determinism. Electron. Notes Theoret. Comput. Sci. 222, 3–99 (2009)

    Article  MATH  Google Scholar 

  37. Varacca, D., Winskel, G.: Distributing probability over non-determinism. Math. Struct. Comput. Sci. 16(1), 87–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, Y.: Real-time behaviour of asynchronous agents. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 502–520. Springer, Heidelberg (1990). https://doi.org/10.1007/BFb0039080

    Chapter  Google Scholar 

  39. Yi, W.: CCS + time = an interleaving model for real time systems. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds.) ICALP 1991. LNCS, vol. 510, pp. 217–228. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54233-7_136

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Aalborg University, Aalborg, Denmark

    Giorgio Bacci, Giovanni Bacci, Mathias Claus Jensen & Kim G. Larsen

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  1. Giorgio Bacci
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  2. Giovanni Bacci
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  3. Mathias Claus Jensen
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  4. Kim G. Larsen
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Correspondence to Giorgio Bacci .

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Editors and Affiliations

  1. Université Libre de Bruxelles, Brussels, Belgium

    Jean-François Raskin

  2. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Krishnendu Chatterjee

  3. École Normale Supérieure Paris-Saclay, Gif-sur-Yvette, France

    Laurent Doyen

  4. MPI for Software Systems, Kaiserslautern, Germany

    Rupak Majumdar

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Bacci, G., Bacci, G., Jensen, M.C., Larsen, K.G. (2022). Convex Lattice Equation Systems. In: Raskin, JF., Chatterjee, K., Doyen, L., Majumdar, R. (eds) Principles of Systems Design. Lecture Notes in Computer Science, vol 13660. Springer, Cham. https://doi.org/10.1007/978-3-031-22337-2_21

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  • DOI: https://doi.org/10.1007/978-3-031-22337-2_21

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