Advertisement

Quantitative static analysis of communication protocols using abstract Markov chains

  • Abdelraouf OuadjaoutEmail author
  • Antoine Miné
Article
  • 4 Downloads

Abstract

In this paper we present a static analysis of probabilistic programs to quantify their performance properties by taking into account both the stochastic aspects of the language and those related to the execution environment. More particularly, we are interested in the analysis of communication protocols in lossy networks and we aim at inferring statically parametric bounds of some important metrics such as the expectation of the throughput or the energy consumption. Our analysis is formalized within the theory of abstract interpretation and soundly takes all possible executions into account. We model the concrete executions as a set of Markov chains and we introduce a novel notion of abstract Markov chains that provides a finite and symbolic representation to over-approximate the (possibly unbounded) set of concrete behaviors. We show that our proposed formalism is expressive enough to handle both probabilistic and pure non-deterministic choices within the same semantics. Our analysis operates in two steps. The first step is a classic abstract interpretation of the source code, using stock numerical abstract domains and a specific automata domain, in order to extract the abstract Markov chain of the program. The second step extracts from this chain particular invariants about the stationary distribution and computes its symbolic bounds using a parametric Fourier–Motzkin elimination algorithm. We present a prototype implementation of the analysis and we discuss some preliminary experiments on a number of communication protocols. We compare our prototype to the state-of-the-art probabilistic model checker Prism and we highlight the advantages and shortcomings of both approaches.

Keywords

Static analysis Abstract interpretation Probabilistic programs Quantitative analysis Markov chains Performance analysis Communication protocols 

Supplementary material

References

  1. 1.
    Abate A, Katoen J-P, Lygeros J, Prandini M (2010) Approximate model checking of stochastic hybrid systems. Eur J Control 16(6):624–641MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anand S, Păsăreanu, CS, Visser W (2007) JPF-SE: a symbolic execution extension to Java Path Finder. In: TACAS’07, volume 4424 of LNCS. Springer, pp 134–138Google Scholar
  3. 3.
    Barthe G, Espitau T, Ferrer Fioriti L, Hsu J (2016) Synthesizing probabilistic invariants via Doob’s decomposition. In: CAV ’16, volume 9779 of LNCS. Springer, pp 43–61Google Scholar
  4. 4.
    Bouissou O, Goubault E, Goubault-Larrecq J, Putot S (2012) A generalization of p-boxes to affine arithmetic. Computing 94(2):189–201MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bouissou, O, Goubault E, Putot S, Chakarov A, Sankaranarayanan S (2016) Uncertainty propagation using probabilistic affine forms and concentration of measure inequalities. In: TACAS ’16, volume 9636 of LNCS. Springer, pp 225–243Google Scholar
  6. 6.
    Buchholz P (1994) Exact and ordinary lumpability in finite Markov chains. J Appl Probab 31(1):59–75MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chakarov A, Sankaranarayanan S (2013) Probabilistic program analysis with martingales. In: CAV ’13, volume 8044 of LNCS. Springer, pp 511–526Google Scholar
  8. 8.
    Chakarov A, Sankaranarayanan S (2014) Expectation invariants for probabilistic program loops as fixed points. In: SAS ’14, volume 8723 of LNCS. Springer, pp 85–100Google Scholar
  9. 9.
    Cousot P, Cousot R (1977) Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL ’77. ACM, pp 238–252Google Scholar
  10. 10.
    Cousot P, Halbwachs N (1978) Automatic discovery of linear restraints among variables of a program. In: POPL ’78. ACM, pp 84–97Google Scholar
  11. 11.
    Cousot P, Monerau M (2012) Probabilistic abstract interpretation. In: ESOP ’12, volume 7211 of LNCS. Springer, pp 169–193Google Scholar
  12. 12.
    Cousot R (1985) Fondements des méthodes de preuve d’invariance et de fatalité de programmes parallèles. Thèse d’État ès sciences mathématiques, Institut National Polytechnique de Lorraine, Nancy, FranceGoogle Scholar
  13. 13.
    Dattatreya GR (2008) Performance analysis of queuing and computer networks. Chapman and Hall/CRC, Boca RatonCrossRefzbMATHGoogle Scholar
  14. 14.
    Daws C (2004) Symbolic and parametric model checking of discrete-time Markov chains. In: ICTAC ’04, volume 3407 of LNCS. Springer, pp 280–294Google Scholar
  15. 15.
    De Loera JA, Hemmecke R, Tauzer J, Yoshida R (2004) Effective lattice point counting in rational convex polytopes. J Symb Comput 38(4):1273–1302MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dehnert C, Junges S, Jansen N, Corzilius F, Volk M, Bruintjes H, Katoen J.-P, Ábrahám E (2015) PROPhESY: a PRObabilistic ParamEter SYnthesis tool. In: CAV ’15, volume 9206 of LNCS. Springer, pp 214–231Google Scholar
  17. 17.
    Dubhashi DP, Panconesi A (2009) Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  18. 18.
    Feret J (2001) Abstract interpretation-based static analysis of mobile ambients. In: SAS ’01, volume 2126 of LNCS. Springer, pp 412–430Google Scholar
  19. 19.
    Ferson S, Kreinovick V, Ginzburg L, Sentz F (2003) Constructing probability boxes and Dempster-Shafer structures. Technical report, SAND2002-4015Google Scholar
  20. 20.
    Filieri A, Păsăreanu CS, Visser W (2013) Reliability analysis in symbolic pathfinder. In: ICSE ’13. IEEE Press, pp 622–631Google Scholar
  21. 21.
    Filieri A, Păsăreanu CS, Visser W, Geldenhuys J (2014) Statistical symbolic execution with informed sampling. In: FSE ’14. ACM, pp 437–448Google Scholar
  22. 22.
    Geldenhuys J, Dwyer MB, Visser W (2012) Probabilistic symbolic execution. In: ISSTA ’12. ACM, pp 166–176Google Scholar
  23. 23.
    Grimmett G, Stirzaker D (2001) Probability and random processes. Oxford University Press, OxfordzbMATHGoogle Scholar
  24. 24.
    Grïßlinger A (2003) Extending the polyhedron model to inequality systems with non-linear parameters using quantifier elimination. Master thesis, University of PassauGoogle Scholar
  25. 25.
    Hahn E, Hermanns H, Zhang L (2011) Probabilistic reachability for parametric Markov models. Int J Softw Tools Technol Transf 13(1):3–19CrossRefGoogle Scholar
  26. 26.
    Jeannet B, Miné A (2009) Apron: a library of numerical abstract domains for static analysis. In: CAV ’09, volume 5643 of LNCS. Springer, pp 661–667Google Scholar
  27. 27.
    Kemeney JG, Snell JL (1976) Finite Markov chains. Springer, BerlinGoogle Scholar
  28. 28.
    Kozen D (1985) A probabilistic PDL. J Comput Syst Sci 30(2):162–178MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kwiatkowska M, Norman G, Parker D (2011) PRISM 4.0: verification of probabilistic real-time systems. In: CAV ’11, volume 6806 of LNCS. Springer, pp 585–591Google Scholar
  30. 30.
    Le Gall T, Jeannet B (2007) Lattice automata: a representation for languages on infinite alphabets, and some applications to verification. In: SAS ’07, volume 4634 of LNCS. Springer, pp 52–68Google Scholar
  31. 31.
    Lesens D, Halbwachs N, Raymond P (2001) Automatic verification of parameterized networks of processes. Theor Comput Sci 256(1–2):113–144MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Luckow K, Păsăreanu CS, Dwyer MB, Filieri A, Visser W (2014) Exact and approximate probabilistic symbolic execution for nondeterministic programs. In: ASE ’14. ACM, pp 575–586Google Scholar
  33. 33.
    McIver A, Morgan C (2004) Abstraction, refinement and proof for probabilistic systems. Monographs in Computer Science. Springer, BerlinGoogle Scholar
  34. 34.
    Miné A (2006) The octagon abstract domain. Higher Ord Symb Comput (HOSC) 19(1):31–100MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Monniaux D (2000) Abstract interpretation of probabilistic semantics. In: SAS ’00, volume 1824 of LNCS. Springer, pp 322–339Google Scholar
  36. 36.
    Monniaux D (2001) Backwards abstract interpretation of probabilistic programs. In: ESOP ’01, volume 2028 of LNCS. Springer, pp 367–382Google Scholar
  37. 37.
    Monniaux D (2005) Abstract interpretation of programs as Markov decision processes. Sci Comput Program 58:179–205MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Necula G, McPeak S, Rahul S, Weimer W (2002) CIL: intermediate language and tools for analysis and transformation of C programs. In: CC ’02, pp 213–228Google Scholar
  39. 39.
    Ouadjaout A, Miné A (2017) Quantitative static analysis of communication protocols using abstract Markov chains. In: SAS ’17, volume 10422 of LNCS. Springer, pp 277–229Google Scholar
  40. 40.
    Parikh R (1966) On context-free languages. J ACM 13(4):570–581CrossRefzbMATHGoogle Scholar
  41. 41.
    Puterman ML (1994) Markov decision processes: discrete stochastic dynamic programming. Wiley Series in Probability and Statistics. Wiley, HobokenCrossRefGoogle Scholar
  42. 42.
    Ross S (1996) Stochastic processes. Wiley, HobokenzbMATHGoogle Scholar
  43. 43.
    Sankaranarayanan S, Chakarov A, Gulwani S (2013) Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In: PLDI ’13. ACM, pp 447–458Google Scholar
  44. 44.
    Shafer G (1976) A mathematical theory of evidence. Princeton University Press, PrincetonzbMATHGoogle Scholar
  45. 45.
    Soudjani SEZ, Abate A (2014) Precise approximations of the probability distribution of a Markov process in time: an application to probabilistic invariance. In: TACAS ’14, volume 8413 of LNCS. Springer, pp 547–561 (2014)Google Scholar
  46. 46.
    Suriana P (2016) Fourier–Motzkin with non-linear symbolic constant coefficients. Master thesis, Massachusetts Institute of TechnologyGoogle Scholar
  47. 47.
    Van Hentenryck P, Cortesi A, Le Charlier B (1995) Type analysis of Prolog using type graphs. J Log Program 22(3):179–209CrossRefzbMATHGoogle Scholar
  48. 48.
    Venet A (1999) Automatic analysis of pointer aliasing for untyped programs. Sci Comput Program 35(2):223–248MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Villemot S (2002) Automates finis et intérpretation abstraite: application à l’analyse statique de protocoles de communication. Rapport de DEA, École normale supérieureGoogle Scholar
  50. 50.
    Wang D, Hoffmann J, Reps T (2018) PMAF: an algebraic framework for static analysis of probabilistic programs. In: PLDI ’18. ACM, pp 513–528Google Scholar
  51. 51.
    Wolfram Research, Inc. (2017) Mathematica, Version 11.2, Champaign, ILGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire d’Informatique de Paris 6, LIP6, CNRSSorbonne UniversitéParisFrance
  2. 2.Institut Universitaire de FranceParisFrance

Personalised recommendations