Advertisement

Probabilistic Analysis of Programs: A Weak Limit Approach

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8552)

Abstract

We present an approach to probabilistic analysis which is based on program semantics and exploits the mathematical properties of the semantical operators to ensure a form of optimality for the analysis. As in the algorithmic setting, where the analysis results are used the help the design of efficient algorithms, the purposes of our framework are to offer static analysis techniques usable for resource optimisation.

Keywords

Hilbert Space Abstract Interpretation Discrete Time Markov Chain Abstract Domain Galois Connection 
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.

References

  1. 1.
    Di Pierro, A., Wiklicky, H.: Concurrent constraint programming: towards probabilistic abstract interpretation. In: PPDP’00, 127–138. ACM (2000)Google Scholar
  2. 2.
    Di Pierro, A., Hankin, C., Wiklicky, H.: A systematic approach to probabilistic pointer analysis. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 335–350. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Di Pierro, A., Sotin, P., Wiklicky, H.: Relational analysis and precision via probabilistic abstract interpretation. In: QAPL’08. Volume 220(3) of ENTCS., pp. 23–42. Elsevier (2008)Google Scholar
  4. 4.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proceedings of POPL’79, pp. 269–282 (1979)Google Scholar
  5. 5.
    Kubrusly, C.S.: The Elements of Operator Theory, 2nd edn. Birkhäuser, Boston (2011)CrossRefMATHGoogle Scholar
  6. 6.
    Groetsch, C.W.: Stable Approximate Evaluation of Unbounded Operators. Lecture Notes in Mathematics. Springer, Berlin (2007)MATHGoogle Scholar
  7. 7.
    Kozen, D.: Semantics of probabilistic programs. J. Comput. Syst. Sci. 22(3), 328–350 (1981)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Nielson, F., Nielson, H.R., Hankin, C.: Principles of Program Analysis. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Roman, S.: Advanced Linear Algebra, 2nd edn. Springer, New York (2005)MATHGoogle Scholar
  10. 10.
    Kadison, R., Ringrose, J.: Fundamentals of the Theory of Operator Algebras: Elementary Theory. AMS (1997) reprint from Academic Press edition (1983)Google Scholar
  11. 11.
    Di Pierro, A., Wiklicky, H.: Semantics of probabilistic programs: a weak limit approach. In: Shan, C. (ed.) APLAS 2013. LNCS, vol. 8301, pp. 241–256. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Di Pierro, A., Wiklicky, H.: Measuring the precision of abstract interpretations. In: Lau, K.-K. (ed.) LOPSTR 2000. LNCS, vol. 2042, pp. 147–164. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Measuring the confinement of probabilistic systems. Theor. Comput. Sci. 340(1), 3–56 (2005)CrossRefMATHGoogle Scholar
  14. 14.
    Ben-Israel, A., Greville, T.N.E.: Gereralized Inverses - Theory and Applications. CMS Books in Mathematics, 2nd edn. Springer, New York (2003)Google Scholar
  15. 15.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)CrossRefMATHGoogle Scholar
  16. 16.
    Deutsch, F.: Best Approximation in Inner-Product Spaces. Springer, New York (2001)CrossRefMATHGoogle Scholar
  17. 17.
    Pinkus, A.M.: On \(L^1\)-Approximation. Cambridge University Press, London (1989)Google Scholar
  18. 18.
    Di Pierro, A., Wiklicky, H.: Probabilistic data flow analysis: a linear equational approach. In: Proceedings of GandALF’13. Volume 119 of EPTCS, pp. 150–165 (2013)Google Scholar
  19. 19.
    Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory - The Basis for Linear and Nonlinear Analysis. Springer, New York (2011)MATHGoogle Scholar
  20. 20.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis - A Functional Analysis Framework, 3rd edn. Springer, New York (2009)MATHGoogle Scholar
  21. 21.
    Nailin, D.: Finite-dimensional approximation settings for infinite-dimensional Moore-Penrose inverses. SIAM J. of Numer. Anal. 46(3), 1454–1482 (2008)CrossRefMATHGoogle Scholar
  22. 22.
    Kulkarni, S., Ramesh, G.: Projection methods for computing Moore-Penrose inverses of unbounded operators. Indian J. Pure Appl. Math. 41(5), 647–662 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Groetsch, C.: Spectral methods for linear inverse problems with unbounded operators. J. Approx. Theory 70, 16–28 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Groetsch, C.: Dykstra’s algorithm and a representation of the Moore-Penrose inverse. J. Approx. Theory 117, 179–184 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Groetsch, C.: An iterative stabilization method for the evaluation of unbounded operators. Proc. AMS 134, 1173–1181 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Plateau, B., Atif, K.: Stochastic automata network of modeling parallel systems. IEEE Trans. Softw. Eng. 17(10), 1093–1108 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Fourneau, J.M., Plateau, B., Stewart, W.: Product form for stochastic automata networks. In: Proceedings of ValueTools ’07, ICST, pp. 32:1–32:10 (2007)Google Scholar
  28. 28.
    Gugercin, S., Antoulas, A.: Model reduction of large-scale systems by least squares. Linear Algebra Appl. 415, 290–321 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Buchholz, P., Kriege, J.: Aggregation of markovian models - an alternating least squares approach. In: QEST, pp. 43–52 (2012)Google Scholar
  30. 30.
    Cousot, P., Monerau, M.: Probabilistic abstract interpretation. In: Seidl, H. (ed.) ESOP 2012. LNCS, vol. 7211, pp. 169–193. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly
  2. 2.Department of ComputingImperial CollegeLondonUK

Personalised recommendations