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Interval Slopes as a Numerical Abstract Domain for Floating-Point Variables

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Static Analysis (SAS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 6337))

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Abstract

The design of embedded control systems is mainly done with model-based tools such as Matlab/Simulink. Numerical simulation is the central technique of development and verification of such tools. Floating-point arithmetic, which is well-known to only provide approximated results, is omnipresent in this activity. In order to validate the behaviors of numerical simulations using abstract interpretation-based static analysis, we present, theoretically and with experiments, a new partially relational abstract domain dedicated to floating-point variables. It comes from interval expansion of non-linear functions using slopes and it is able to mimic all the behaviors of the floating-point arithmetic. Hence it is adapted to prove the absence of run-time errors or to analyze the numerical precision of embedded control systems.

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References

  1. Bischof, C.H., Hovland, P.D., Norris, B.: Implementation of automatic differentiation tools. In: Partial Evaluation and Semantics-Based Program Manipulation, pp. 98–107. ACM Press, New York (2002)

    Google Scholar 

  2. Boldo, S., Nguyen, T.: Hardware-independant proofs of numerical programs. In: NASA Formal Methods Symposium (2010)

    Google Scholar 

  3. Chapoutot, A., Martel, M.: Abstract simulation: a static analysis of Simulink models. In: International Conference on Embedded Systems and Software, pp. 83–92. IEEE Press, Los Alamitos (2009)

    Chapter  Google Scholar 

  4. Chapoutot, A., Martel, M.: Automatic differentiation and Taylor forms in static analysis of numerical programs. Technique et Science Informatiques 28(4), 503–531 (2009) (in French)

    Article  Google Scholar 

  5. Chen, L., Miné, A., Patrick, C.: A sound floating-point polyhedra abstract domain. In: Ramalingam, G. (ed.) APLAS 2008. LNCS, vol. 5356, pp. 3–18. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  6. Chen, L., Miné, A., Wang, J., Cousot, P.: Interval polyhedra: an abstract domain to infer interval linear relationships. In: Palsberg, J., Su, Z. (eds.) Static Analysis. LNCS, vol. 5673, pp. 309–325. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  7. Chen, L., Miné, A., Wang, J., Cousot, P.: An abstract domain to discover interval linear equalities. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 112–128. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  8. Clarisó, R., Cortadella, J.: The Octahedron abstract domain. Science Computer Programming 64(1), 115–139 (2007)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  10. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Principles of Programming Languages, pp. 84–97. ACM Press, New York (1978)

    Google Scholar 

  11. Férêt, J.: Static analysis of digital filter. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 33–48. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Ghorbal, K., Goubault, E., Putot, S.: The zonotope abstract domain Taylor1 +. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 627–633. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. Goubault, E.: Static analyses of floating-point operations. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, pp. 234–259. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  14. Goubault, E., Putot, S.: Static analysis of numerical algorithms. In: Yi, K. (ed.) SAS 2006. LNCS, vol. 4134, pp. 18–34. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. Granger, P.: Improving the results of static analyses programs by local decreasing iteration. In: Shyamasundar, R.K. (ed.) FSTTCS 1992. LNCS, vol. 652, pp. 68–79. Springer, Heidelberg (1992)

    Google Scholar 

  16. Granger, P.: Static analysis of linear congruence equalities among variables of a program. In: Abramsky, S. (ed.) CAAP 1991 and TAPSOFT 1991. LNCS, vol. 493, pp. 169–192. Springer, Heidelberg (1991)

    Google Scholar 

  17. Higham, N.: Accuracy and stability of numerical algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002)

    MATH  Google Scholar 

  18. IEEE Task P754: IEEE 754-2008, Standard for Floating-Point Arithmetic. Institute of Electrical, and Electronic Engineers (2008)

    Google Scholar 

  19. Karr, M.: Affine relationships among variables of a program. Acta Informatica 6, 133–151 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Krawczyk, R., Neumaier, A.: Interval slopes for rational functions and associated centered forms. SIAM Journal on Numerical Analysis 22(3), 604–616 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Laviron, V., Logozzo, F.: Subpolyhedra: a (more) scalable approach to infer linear inequalities. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 229–244. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  22. Logozzo, F., Fähndrich, M.: Pentagons: a weakly relational abstract domain for the efficient validation of array accesses. In: Symposium on Applied Computing, pp. 184–188. ACM, New York (2008)

    Google Scholar 

  23. Martel, M.: Semantics of roundoff error propagation in finite precision computations. Higher Order and Symbolic Computation 19(1), 7–30 (2004)

    Article  Google Scholar 

  24. Miné, A.: Relational abstract domains for the detection of floating-point run-time errors. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 3–17. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  25. Miné, A.: The Octagon abstract domain. Journal of Higher-Order and Symbolic Computation 19(1), 31–100 (2006)

    Article  MATH  Google Scholar 

  26. Monniaux, D.: Compositional analysis of floating-point linear numerical filters. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 199–212. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  27. Moore, R.: Interval analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  28. Muller, J.M., Brisebarre, N., De Dinechin, F., Jeannerod, C.P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of floating-point arithmetic. Birkhauser, Boston (2009)

    Google Scholar 

  29. Péron, M., Halbwachs, N.: An abstract domain extending difference-bound matrices with disequality constraints. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 268–282. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  30. Rump, S.: Expansion and estimation of the range of nonlinear functions. Mathematics of Computation 65(216), 1503–1512 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sankaranarayanan, S., Colon, M., Sipma, H., Manna, Z.: Efficient strongly relational polyhedral analysis. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 111–125. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  32. Simon, A., King, A., Howe, J.: Two variables per linear inequality as an abstract domain. In: Leuschel, M. (ed.) LOPSTR 2002. LNCS, vol. 2664, pp. 71–89. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. LIP6, Université Pierre et Marie Curie, 4, place Jussieur, F-75252, Paris Cedex 05, France

    Alexandre Chapoutot

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  1. Alexandre Chapoutot
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Editors and Affiliations

  1. Départment d’Informatique, École normale supérieure, rue d’Ulm 45, 75230, Paris cedex, France

    Radhia Cousot

  2. ELIAUS-DALI Laboratory, Université de Perpignan Via Domitia, 52, avenue Paul Alduy, 66860, Perpignan cedex, France

    Matthieu Martel

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Chapoutot, A. (2010). Interval Slopes as a Numerical Abstract Domain for Floating-Point Variables. In: Cousot, R., Martel, M. (eds) Static Analysis. SAS 2010. Lecture Notes in Computer Science, vol 6337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15769-1_12

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  • DOI: https://doi.org/10.1007/978-3-642-15769-1_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15768-4

  • Online ISBN: 978-3-642-15769-1

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