Skip to main content
Springer Nature Link
Log in

Widening Operators for Weakly-Relational Numeric Abstractions

  • Conference paper
Static Analysis (SAS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 3672))

Included in the following conference series:

  • 630 Accesses

Abstract

We discuss the construction of proper widening operators on several weakly-relational numeric abstractions. Our proposal differs from previous ones in that we actually consider the semantic abstract domains, whose elements are geometric shapes, instead of the (more concrete) syntactic abstract domains of constraint networks and matrices. Since the closure by entailment operator preserves geometric shapes, but not their syntactic expressions, our widenings are immune from the divergence issues that could be faced by the previous approaches when interleaving the applications of widening and closure. The new widenings, which are variations of the standard widening for convex polyhedra defined by Cousot and Halbwachs, can be made as precise as the previous proposals working on the syntactic domains. The implementation of each new widening relies on the availability of an effective reduction procedure for the considered constraint description: we provide such an algorithm for the domain of octagonal shapes.

This work has been partly supported by MURST projects “Constraint Based Verification of Reactive Systems” and “AIDA — Abstract Interpretation: Design and Applications,” and by a Royal Society (UK) International Joint Project (ESEP) award.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Aho, A.V., Garey, M.R., Ullman, J.D.: The transitive reduction of a directed graph. SIAM Journal on Computing 1(2), 131–137 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allen, J.F., Kautz, H.A.: A model of naive temporal reasoning. In: Formal Theories of the Commonsense World, Ablex, Norwood, NJ, pp. 251–268 (1985)

    Google Scholar 

  3. Bagnara, R.: Data-Flow Analysis for Constraint Logic-Based Languages. PhD thesis, Dipartimento di Informatica, Università di Pisa, Italy (1997)

    Google Scholar 

  4. Bagnara, R., Giacobazzi, R., Levi, G.: Static analysis of CLP programs over numeric domains. In: Proc. WSA 1992, Bordeaux. Bigre, vol. 81–82, pp. 43–50 (1992)

    Google Scholar 

  5. Bagnara, R., Giacobazzi, R., Levi, G.: An application of constraint propagation to data-flow analysis. In: Proc. CAIA 1993, Orlando, FL, pp. 270–276 (1993)

    Google Scholar 

  6. Bagnara, R., Hill, P.M., Mazzi, E., Zaffanella, E.: Widening operators for weakly-relational numeric abstractions. Quaderno 399, Dipartimento di Matematica, Univ. di Parma, Italy (2005), Available at, http://www.cs.unipr.it/Publications/

  7. Bagnara, R., Hill, P.M., Ricci, E., Zaffanella, E.: Precise widening operators for convex polyhedra. In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694, pp. 337–354. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  8. Bagnara, R., Hill, P.M., Ricci, E., Zaffanella, E.: Precise widening operators for convex polyhedra. Science of Computer Programming (2005) (to appear)

    Google Scholar 

  9. Bagnara, R., Hill, P.M., Zaffanella, E.: Widening operators for powerset domains. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 135–148. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  10. Bagnara, R., Hill, P.M., Zaffanella, E.: The Parma Polyhedra Library User’s Manual. Department of Mathematics, University of Parma, release 0.7 (2004)

    Google Scholar 

  11. Balasundaram, V., Kennedy, K.: A technique for summarizing data access and its use in parallelism enhancing transformations. In: Proc. PLDI 1989, OR. ACM SIGPLAN Notices, Portland, vol. 24(7), pp. 41–53 (1989)

    Google Scholar 

  12. Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  13. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  14. Blanchet, B., Cousot, P., Cousot, R., Feret, J., et al.: A static analyzer for large safety-critical software. In: Proc. PLDI 2003, San Diego, CA, pp. 196–207 (2003)

    Google Scholar 

  15. Clarisó, R., Cortadella, J.: The octahedron abstract domain. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 312–327. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  16. Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: Proc. ISOP 1976, Paris, France, pp. 106–130 (1976)

    Google Scholar 

  17. Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proc. POPL 1977, New York, pp. 238–252 (1977)

    Google Scholar 

  18. Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proc. POPL 1979, New York, pp. 269–282 (1979)

    Google Scholar 

  19. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Proc. POPL 1978, Tucson, AR, pp. 84–96 (1978)

    Google Scholar 

  20. Davis, E.: Constraint propagation with interval labels. Artificial Intelligence 32(3), 281–331 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990)

    Google Scholar 

  22. Halbwachs, N.: Détermination Automatique de Relations Linéaires Vérifiées par les Variables d’un Programme. PhD thesis, Université de Grenoble, France (1979)

    Google Scholar 

  23. Halbwachs, N., Proy, Y.-E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Form. Method Syst. Des. 11(2), 157–185 (1997)

    Article  Google Scholar 

  24. Larsen, K., Larsson, F., Pettersson, P., Yi, W.: Efficient verification of real-time systems: Compact data structure and state-space reduction. In: Proc. RTSS 1997, San Francisco, CA, pp. 14–24 (1997)

    Google Scholar 

  25. Miné, A.: A new numerical abstract domain based on difference-bound matrices. In: Danvy, O., Filinski, A. (eds.) PADO 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  26. Miné, A.: The octagon abstract domain. In: Proc. WCRE 2001, Stuttgart, pp. 310–319 (2001)

    Google Scholar 

  27. Miné, A.: A few graph-based relational numerical abstract domains. In: Hermenegildo, M.V., Puebla, G. (eds.) SAS 2002. LNCS, vol. 2477, pp. 117–132. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  28. Miné, A.: The Octagon Abstract Domain Library. École Normale Supérieure, Paris, France, release 0.9.6 (2002), Available at, http://www.di.ens.fr/~mine/oct/

  29. 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 

  30. Miné, A.: Weakly Relational Numerical Abstract Domains. PhD thesis, École Polytechnique, Paris, France (2005)

    Google Scholar 

  31. Sankaranarayanan, S., Sipma, H., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  32. Shaham, R., Kolodner, E.K., Sagiv, S.: Automatic removal of array memory leaks in java. In: Watt, D.A. (ed.) CC 2000. LNCS, vol. 1781, pp. 50–66. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  33. Simon, A., King, A., Howe, J.M.: 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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Parma, Italy

    Roberto Bagnara, Elena Mazzi & Enea Zaffanella

  2. School of Computing, University of Leeds, UK

    Patricia M. Hill

Authors
  1. Roberto Bagnara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patricia M. Hill
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elena Mazzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Enea Zaffanella
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computing, Imperial College London,  

    Chris Hankin

  2. Department of Computing, Imperial College London, 180 Queen’s Gate, SW7 2BZ, London, UK

    Igor Siveroni

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bagnara, R., Hill, P.M., Mazzi, E., Zaffanella, E. (2005). Widening Operators for Weakly-Relational Numeric Abstractions. In: Hankin, C., Siveroni, I. (eds) Static Analysis. SAS 2005. Lecture Notes in Computer Science, vol 3672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11547662_3

Download citation

  • DOI: https://doi.org/10.1007/11547662_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28584-7

  • Online ISBN: 978-3-540-31971-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation