Skip to main content
SpringerLink
Log in

Widening Operators for Powerset Domains

  • Conference paper
Verification, Model Checking, and Abstract Interpretation (VMCAI 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2937))

Abstract

The finite powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. In this paper we define two generic widening operators for the finite powerset abstract domain. Both widenings are obtained by lifting any widening operator defined on the base-level abstract domain and are parametric with respect to the specification of a few additional operators. We illustrate the proposed techniques by instantiating our widenings on powersets of convex polyhedra, a domain for which no non-trivial widening operator was previously known.

This work has been partly supported by MURST projects “Aggregate- and Number-Reasoning for Computing: from Decision Algorithms to Constraint Programming with Multisets, Sets, and Maps” and “Constraint Based Verification of Reactive Systems.”

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, ch. 1, pp. 1–168. Clarendon Press, Oxford (1994)

    Google Scholar 

  2. Bagnara, R.: A hierarchy of constraint systems for data-flow analysis of constraint logic-based languages. Science of Computer Programming 30(1–2), 119–155 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagnara, R., Hill, P.M., Ricci, E., Zaffanella, E.: The Parma Polyhedra Library User’s Manual. Department of Mathematics, University of Parma, Parma, Italy, release 0.5 edition (April 2003), Available at http://www.cs.unipr.it/ppl/

  4. 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 

  5. Bagnara, R., Hill, P.M., Zaffanella, E.: Widening operators for powerset domains. Quaderno, Dipartimento di Matematica, Università di Parma, Italy (2003) (to appear)

    Google Scholar 

  6. Bagnara, R., Ricci, E., Zaffanella, E., Hill, P.M.: Possibly not closed convex polyhedra and the Parma Polyhedra Library. In: Hermenegildo, M.V., Puebla, G. (eds.) SAS 2002. LNCS, vol. 2477, pp. 213–229. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Bultan, T., Gerber, R., Pugh, W.: Model-checking concurrent systems with unbounded integer variables: Symbolic representations, approximations, and experimental results. ACM Transactions on Programming Languages and Systems 21(4), 747–789 (1999)

    Article  Google Scholar 

  8. Cortesi, A., Le Charlier, B., Van Hentenryck, P.: Combinations of abstract domains for logic programming: Open product and generic pattern construction. Science of Computer Programming 38(1–3), 27–71 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: Robinet, B. (ed.) Proceedings of the Second International Symposium on Programming, Dunod, Paris, France, pp. 106–130 (1976)

    Google Scholar 

  10. Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the Fourth Annual ACM Symposium on Principles of Programming Languages, pp. 238–252. ACM Press, New York (1977)

    Google Scholar 

  11. Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Programming Languages, pp. 269–282. ACM Press, New York (1979)

    Chapter  Google Scholar 

  12. Cousot, P., Cousot, R.: Abstract interpretation and applications to logic programs. Journal of Logic Programming 13(2&3), 103–179 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cousot, P., Cousot, R.: Abstract interpretation frameworks. Journal of Logic and Computation 2(4), 511–547 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cousot, P., Cousot, R.: Comparing the Galois connection and widening/ narrowing approaches to abstract interpretation. In: Bruynooghe, M., Wirsing, M. (eds.) PLILP 1992. LNCS, vol. 631, pp. 269–295. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  15. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Conference Record of the Fifth Annual ACM Symposium on Principles of Programming Languages, pp. 84–96. ACM Press, Tucson (1978)

    Google Scholar 

  16. Delzanno, G., Podelski, A.: Model checking in CLP. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 223–239. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  17. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Communications of the ACM 22(8), 465–476 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Filé, G., Ranzato, F.: The powerset operator on abstract interpretations. Theoretical Computer Science 222, 77–111 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Halbwachs, N.: Détermination Automatique de Relations Linéaires Vérifiées par les Variables d’un Programme. Thèse de 3ème cycle d’informatique, Université scientifique et médicale de Grenoble, Grenoble, France (March 1979)

    Google Scholar 

  20. Halbwachs, N.: Delay analysis in synchronous programs. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 333–346. Springer, Heidelberg (1993)

    Google Scholar 

  21. Halbwachs, N., Proy, Y.-E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Formal Methods in System Design 11(2), 157–185 (1997)

    Article  Google Scholar 

  22. Holzbaur, C.: OFAI clp(q,r) manual, edition 1.3.3. Technical Report TR-95-09, Austrian Research Institute for Artificial Intelligence, Vienna (1995)

    Google Scholar 

  23. Kelly, W., Maslov, V., Pugh, W., Rosser, E., Shpeisman, T., Wonnacott, D.: The Omega library interface guide. Technical Report CS-TR-3445, Department of Computer Science, University of Maryland, College Park, MD, USA (1995)

    Google Scholar 

  24. Le Verge, H.: A note on Chernikova’s algorithm. Publication interne 635, IRISA, Campus de Beaulieu, Rennes, France (1992)

    Google Scholar 

  25. Loechner, V.: PolyLib: A library for manipulating parameterized polyhedra (March 1999), Available at http://icps.u-strasbg.fr/~loechner/polylib/ ; Declares itself to be a continuation of [28]

  26. Pugh, W.: A practical algorithm for exact array dependence analysis. Communications of the ACM 35(8), 102–114 (1992)

    Article  Google Scholar 

  27. Srivastava, D.: Subsumption and indexing in constraint query languages with linear arithmetic constraints. Annals of Mathematics and Artificial Intelligence 8(3-4), 315–343 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  28. Wilde, D.K.: A library for doing polyhedral operations. Master’s thesis, Oregon State University, Corvallis, Oregon (December 1993); Also published as IRISA Publication interne 785, Rennes, France (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Parma, Italy

    Roberto Bagnara & 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. Enea Zaffanella
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Chair Programming Systems, Universität Dortmund, Otto-Hahn-Str. 14, 44227, Dortmund, Germany

    Bernhard Steffen

  2. Dipartimento di Informatica, Università di Pisa, Via Buonarroti, 2, 56100, Pisa, Italy

    Giorgio Levi

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bagnara, R., Hill, P.M., Zaffanella, E. (2004). Widening Operators for Powerset Domains. In: Steffen, B., Levi, G. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2004. Lecture Notes in Computer Science, vol 2937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24622-0_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24622-0_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20803-7

  • Online ISBN: 978-3-540-24622-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation