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Expectation Invariants for Probabilistic Program Loops as Fixed Points

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Static Analysis (SAS 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8723))

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Abstract

We present static analyses for probabilistic loops using expectation invariants. Probabilistic loops are imperative while-loops augmented with calls to random variable generators. Whereas, traditional program analysis uses Floyd-Hoare style invariants to over-approximate the set of reachable states, our approach synthesizes invariant inequalities involving the expected values of program expressions at the loop head. We first define the notion of expectation invariants, and demonstrate their usefulness in analyzing probabilistic program loops. Next, we present the set of expectation invariants for a loop as a fixed point of the pre-expectation operator over sets of program expressions. Finally, we use existing concepts from abstract interpretation theory to present an iterative analysis that synthesizes expectation invariants for probabilistic program loops. We show how the standard polyhedral abstract domain can be used to synthesize expectation invariants for probabilistic programs, and demonstrate the usefulness of our approach on some examples of probabilistic program loops.

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References

  1. Antonopoulos, T., Gorogiannis, N., Haase, C., Kanovich, M., Ouaknine, J.: Foundations for decision problems in separation logic with general inductive predicates. In: Muscholl, A. (ed.) FOSSACS 2014. LNCS, vol. 8412, pp. 411–425. Springer, Heidelberg (2014)

    Google Scholar 

  2. Bouissou, O., Goubault, E., Goubault-Larrecq, J., Putot, S.: A generalization of p-boxes to affine arithmetic. Computing 94(2-4), 189–201 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chakarov, A., Sankaranarayanan, S.: Probabilistic program analysis with martingales. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 511–526. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  4. Chakarov, A., Sankaranarayanan, S.: Expectation invaraiants for probabilistic program loops as fixed points (2014) (extended version) (Draft, Available upon request)

    Google Scholar 

  5. Chung, K.L.: A course in probability theory, vol. 3. Academic Press, New York (1974)

    MATH  Google Scholar 

  6. Colón, M.A., Sankaranarayanan, S., Sipma, H.B.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420–432. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Cousot, P., Cousot, R.: Abstract Interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: ACM Principles of Programming Languages, pp. 238–252 (1977)

    Google Scholar 

  8. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among the variables of a program. In: POPL 1978, pp. 84–97 (January 1978)

    Google Scholar 

  9. Cousot, P., Monerau, M.: Probabilistic abstract interpretation. In: Seidl, H. (ed.) ESOP 2012. LNCS, vol. 7211, pp. 169–193. Springer, Heidelberg (2012)

    Google Scholar 

  10. Dubhashi, D., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press (2009)

    Google Scholar 

  11. Gretz, F., Katoen, J.-P., McIver, A.: Prinsys - on a quest for probabilistic loop invariants. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 193–208. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  12. Halbwachs, N.: Détermination automatique de relations linéaires vérifiées par les variables d’un programme. PhD thesis, Institut National Polytechnique de Grenoble-INPG (1979)

    Google Scholar 

  13. Katoen, J.-P., McIver, A.K., Meinicke, L.A., Morgan, C.C.: Linear-invariant generation for probabilistic programs. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 390–406. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  14. Kozen, D.: Semantics of probabilistic programs. J. Comput. Syst. Sci. 22(3), 328–350 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mardziel, P., Magill, S., Hicks, M., Srivatsa, M.: Dynamic enforcement of knowledge-based security policies. In: 2011 IEEE 24th Computer Security Foundations Symposium (CSF), pp. 114–128. IEEE (2011)

    Google Scholar 

  16. McAdams, H., Arkin, A.: It’s a noisy business! genetic regulation at the nanomolar scale. Trends Genetics 15(2), 65–69 (1999)

    Article  Google Scholar 

  17. McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer (2004)

    Google Scholar 

  18. Monniaux, D.: Abstract interpretation of probabilistic semantics. In: SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  19. Monniaux, D.: Backwards abstract interpretation of probabilistic programs. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, pp. 367–382. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  20. Monniaux, D.: Abstract interpretation of programs as markov decision processes. Science of Computer Programming 58(1), 179–205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (1995)

    Google Scholar 

  22. Sankaranarayanan, S., Chakarov, A., Gulwani, S.: Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In: PLDI, pp. 447–458. ACM (2013)

    Google Scholar 

  23. Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constraint-based linear-relations analysis. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 53–68. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  24. Williams, D.: Probability with Martingales. Cambridge University Press (1991)

    Google Scholar 

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Chakarov, A., Sankaranarayanan, S. (2014). Expectation Invariants for Probabilistic Program Loops as Fixed Points. In: Müller-Olm, M., Seidl, H. (eds) Static Analysis. SAS 2014. Lecture Notes in Computer Science, vol 8723. Springer, Cham. https://doi.org/10.1007/978-3-319-10936-7_6

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  • DOI: https://doi.org/10.1007/978-3-319-10936-7_6

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10935-0

  • Online ISBN: 978-3-319-10936-7

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