Symbolic Methods to Enhance the Precision of Numerical Abstract Domains

  • Antoine Miné
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3855)


We present lightweight and generic symbolic methods to improve the precision of numerical static analyses based on Abstract Interpretation. The main idea is to simplify numerical expressions before they are fed to abstract transfer functions. An important novelty is that these simplifications are performed on-the-fly, using information gathered dynamically by the analyzer.

A first method, called “linearization,” allows abstracting arbitrary expressions into affine forms with interval coefficients while simplifying them. A second method, called “symbolic constant propagation,” enhances the simplification feature of the linearization by propagating assigned expressions in a symbolic way. Combined together, these methods increase the relationality level of numerical abstract domains and make them more robust against program transformations. We show how they can be integrated within the classical interval, octagon and polyhedron domains. These methods have been incorporated within the Astrée static analyzer that checks for the absence of run-time errors in embedded critical avionics software. We present an experimental proof of their usefulness.


Interval Arithmetic Abstract Interpretation Relational Domain Program Transformation Abstract Domain 
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.
    Blanchet, B., Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: A static analyzer for large safety-critical software. In: ACM PLDI 2003, vol. 548030, pp. 196–207. ACM Press, New York (2003)CrossRefGoogle Scholar
  2. 2.
    Bourdoncle, F.: Efficient chaotic iteration strategies with widenings. In: Pottosin, I.V., Bjorner, D., Broy, M. (eds.) FMP&TA 1993. LNCS, vol. 735, pp. 128–114. Springer, Heidelberg (1993)Google Scholar
  3. 3.
    Clarisó, R., Cortadella, J.: The octahedron abstract domain. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 312–327. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Colby, C.: Semantics-Based Program Analysis via Symbolic Composition of Transfer Relations. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA (1996)Google Scholar
  5. 5.
    Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: ISOP 1976, Dunod, Paris, France, pp. 106–130 (1976)Google Scholar
  6. 6.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: ACM POPL 1977, pp. 238–252. ACM Press, New York (1977)Google Scholar
  7. 7.
    Cousot, P., Cousot, R.: Abstract interpretation and application to logic programs. Journal of Logic Programming 13(2–3), 103–179 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: ACM POPL 1978, pp. 84–97. ACM Press, New York (1978)Google Scholar
  9. 9.
    Feret, J.: Static analysis of digital filters. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 33–48. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Kildall, G.: A unified approach to global program optimization. In: ACM POPL 1973, pp. 194–206. ACM Press, New York (1973)Google Scholar
  12. 12.
    Miné, A.: The octagon abstract domain. In: AST 2001 in WCRE 2001, pp. 310–319. IEEE CS Press, Los Alamitos (2001)Google Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Miné, A.: Weakly Relational Numerical Abstract Domains. PhD thesis, École Polytechnique, Palaiseau, France (december 2004)Google Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Vinícius, M., Andrade, A., Comba, J.L.D., Stolfi, J.: Affine arithmetic. In: INTERVAL 1994 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Antoine Miné
    • 1
  1. 1.École Normale SupérieureParisFrance

Personalised recommendations