Static Analysis of Programs with Imprecise Probabilistic Inputs

  • Assale Adje
  • Olivier Bouissou
  • Jean Goubault-Larrecq
  • Eric Goubault
  • Sylvie Putot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8164)


Having a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.


Arithmetic Operation Operational Semantic Probabilistic Choice Abstract Interpretation Program Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arce, G.: Nonlinear Signal Processing: A Statistical Approach. Wiley (2005)Google Scholar
  2. 2.
    Auer, E., Luther, W., Rebner, G., Limbourg, P.: A verified matlab toolbox for the dempster-shafer theory. In: Workshop on the Theory of Belief Functions (2010)Google Scholar
  3. 3.
    Berleant, D., Goodman-Strauss, C.: Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. Reliable Computing 4(2), 147–165 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Berleant, D., Xie, L., Zhang, J.: Statool: A tool for distribution envelope determination (denv), an interval-based algorithm for arithmetic on random variables. Reliable Computing 9, 91–108 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bouissou, O., Goubault, E., Goubault-Larrecq, J., Putot, S.: A generalization of p-boxes to affine arithmetic. Computing, 1–13 (2011), 10.1007/s00607-011-0182-8Google Scholar
  6. 6.
    Bouissou, O., Goubault, E., Putot, S., Tekkal, K., Vedrine, F.: Hybridfluctuat: A static analyzer of numerical programs within a continuous environment. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 620–626. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Busaba, J., Suwan, S., Kosheleva, O.: A faster algorithm for computing the sum of p-boxes. Journal of Uncertain Systems 4(4) (2010)Google Scholar
  8. 8.
    Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: SEBGRAPI 1993 (1993)Google Scholar
  10. 10.
    Cousot, P., Monerau, M.: Probabilistic abstract interpretation. In: Seidl, H. (ed.) ESOP 2012. LNCS, vol. 7211, pp. 169–193. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Destercke, S., Dubois, D., Chojnacki, E.: Unifying practical uncertainty representations - I: Generalized p-boxes. J. of Approximate Reasoning 49(3) (2008)Google Scholar
  12. 12.
    Destercke, S., Dubois, D., Chojnacki, E.: Unifying practical uncertainty representations. II: Clouds. Intl. J. of Approximate Reasoning 49(3) (2008)Google Scholar
  13. 13.
    Enszer, J.A., Lin, Y., Ferson, S., Corliss, G.F., Stadtherr, M.A.: Probability bounds analysis for nonlinear dynamic process models. AIChE Journal 57(2) (2011)Google Scholar
  14. 14.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley (1968)Google Scholar
  15. 15.
    Feret, J.: Static analysis of digital filters. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 33–48. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Ferson, S.: RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain Numbers. Lewis Publishers (2002)Google Scholar
  17. 17.
    Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D., Sentz, K.: Constructing probability boxes and Dempster-Shafer structures. Tech. Rep. SAND2002-4015, Sandia National Laboratories (2003)Google Scholar
  18. 18.
    Ferson, S.: What Monte-Carlo methods cannot do. Human and Ecological Risk Assessment 2, 990–1007 (1996)CrossRefGoogle Scholar
  19. 19.
    Fuchs, M., Neumaier, A.: Potential based clouds in robust design optimization. J. Stat. Theory Practice 3, 225–238 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ghorbal, K., Goubault, E., Putot, S.: A logical product approach to zonotope intersection. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 212–226. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Goubault, E., Putot, S.: A zonotopic framework for functional abstractions. CoRR abs/0910.1763 (2009)Google Scholar
  22. 22.
    Goubault, E., Putot, S.: Static analysis of finite precision computations. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 232–247. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  23. 23.
    Goubault-Larrecq, J.: Continuous capacities on continuous state spaces. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 764–776. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Goubault-Larrecq, J.: Continuous previsions. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 542–557. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Goubault-Larrecq, J.: Prevision domains and convex powercones. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 318–333. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  26. 26.
    Goubault-Larrecq, J., Keimel, K.: Choquet-Kendall-Matheron theorems for non-Hausdorff spaces. MSCS 21(3), 511–561 (2011)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Kwiatkowska, M., Norman, G., Parker, D.: Prism 4.0: Verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  28. 28.
    Lin, Y., Stadtherr, M.A.: Validated solution of initial value problems for odes with interval parameters. In: NSF Workshop on Reliable Engineering Computing (2006)Google Scholar
  29. 29.
    Mancini, R., Carter, B.: Op Amps for Everyone. Electronics & Electrical (2009)Google Scholar
  30. 30.
    McIver, A., Morgan, C.: Demonic, angelic and unbounded probabilistic choices in sequential programs. Acta Informatica 37(4/5), 329–354 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Mislove, M.W.: Nondeterminism and probabilistic choice: Obeying the laws. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 350–364. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  32. 32.
    Monniaux, D.: Abstract interpretation of probabilistic semantics. In: Palsberg, J. (ed.) SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  33. 33.
    Neumaier, A.: Clouds, fuzzy sets and probability intervals. Reliable Computing (2004)Google Scholar
  34. 34.
    Rump, S.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers (1999)Google Scholar
  35. 35.
    Sankaranarayanan, S., Chakarov, A., Gulwani, S.: Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In: Boehm, H.J., Flanagan, C. (eds.) PLDI, pp. 447–458. ACM (2013)Google Scholar
  36. 36.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)Google Scholar
  37. 37.
    Sun, J., Huang, Y., Li, J., Wang, J.M.: Chebyshev affine arithmetic based parametric yield prediction under limited descriptions of uncertainty. In: ASP-DAC 2008, pp. 531–536. IEEE Computer Society Press (2008)Google Scholar
  38. 38.
    Terejanu, G., Singla, P., Singh, T., Scott, P.D.: Approximate interval method for epistemic uncertainty propagation using polynomial chaos and evidence theory. In: 2010 American Control Conference, Baltimore, Maryland (2010)Google Scholar
  39. 39.
    Tix, R.: Continuous D-Cones: Convexity and Powerdomain Constructions. Ph.D. thesis, Technische Universität Darmstadt (1999)Google Scholar
  40. 40.
    Tix, R., Keimel, K., Plotkin, G.: Semantic domains for combining probability and non-determinism. ENTCS 129, 1–104 (2005)MathSciNetGoogle Scholar
  41. 41.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman Hall (1991)Google Scholar
  42. 42.
    Williamson, R.C., Downs, T.: Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds. J. Approximate Reasoning (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Assale Adje
    • 2
  • Olivier Bouissou
    • 1
  • Jean Goubault-Larrecq
    • 2
  • Eric Goubault
    • 1
  • Sylvie Putot
    • 1
  1. 1.CEA LIST, CEA SaclayGif-sur-Yvette CEDEXFrance
  2. 2.LSV, ENS CachanCachanFrance

Personalised recommendations