Article

Widening operators for weakly-relational numeric abstractions

Authors:

Roberto Bagnara,

Patricia M. Hill,

Enea ZaffanellaAuthors Info & Claims

SAS'05: Proceedings of the 12th international conference on Static Analysis

Pages 3 - 18

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

Published: 07 September 2005 Publication History

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.

References

[1]

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

Digital Library

[2]

J. F. Allen and H. A. Kautz. A model of naive temporal reasoning. In Formal Theories of the Commonsense World, pp. 251-268. Ablex, Norwood, NJ, 1985.

[3]

R. Bagnara. Data-Flow Analysis for Constraint Logic-Based Languages. PhD thesis, Dipartimento di Informatica, Universit`a di Pisa, Italy, 1997.

[4]

R. Bagnara, R. Giacobazzi, and G. Levi. Static analysis of CLP programs over numeric domains. In Proc. WSA 1992, vol. 81-82 of Bigre, pp. 43-50, Bordeaux.

[5]

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

[6]

R. Bagnara, P. M. Hill, E. Mazzi, and E. Zaffanella. 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]

R. Bagnara, P. M. Hill, E. Ricci, and E. Zaffanella. Precise widening operators for convex polyhedra. In Proc. SAS 2003, vol. 2694 of LNCS, pp. 337-354, San Diego.

Digital Library

[8]

R. Bagnara, P. M. Hill, E. Ricci, and E. Zaffanella. Precise widening operators for convex polyhedra. Science of Computer Programming, 2005. To appear.

Digital Library

[9]

R. Bagnara, P. M. Hill, and E. Zaffanella. Widening operators for powerset domains. In Proc. VMCAI 2004, vol. 2937 of LNCS, pp. 135-148, Venice, Italy.

[10]

R. Bagnara, P. M. Hill, and E. Zaffanella. The Parma Polyhedra Library User's Manual. Department of Mathematics, University of Parma, release 0.7, 2004.

[11]

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

Digital Library

[12]

R. Bellman. Dynamic Programming. Princeton University Press, 1957.

[13]

G. Birkhoff. Lattice Theory. American Mathematical Society, 3rd edition, 1967.

[14]

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

Digital Library

[15]

R. Clarisó and J. Cortadella. The octahedron abstract domain. In Proc. SAS 2004, vol. 3148 of LNCS, pp. 312-327, Verona, Italy.

[16]

P. Cousot and R. Cousot. Static determination of dynamic properties of programs. In Proc. ISOP 1976, pp. 106-130, Paris, France.

[17]

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

Digital Library

[18]

P. Cousot and R. Cousot. Systematic design of program analysis frameworks. In Proc. POPL 1979, pp. 269-282, New York.

Digital Library

[19]

P. Cousot and N. Halbwachs. Automatic discovery of linear restraints among variables of a program. In Proc. POPL 1978, pp. 84-96, Tucson, AR.

Digital Library

[20]

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

Digital Library

[21]

D. L. Dill. Timing assumptions and verification of finite-state concurrent systems. In Proc. AVMFSS 1989, vol. 407 of LNCS, pp. 197-212, Grenoble, France.

Digital Library

[22]

N. Halbwachs. Détermination Automatique de Relations Linéaires Vérifiées par les Variables d'un Programme. PhD thesis, Université de Grenoble, France, 1979.

[23]

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

Digital Library

[24]

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

Digital Library

[25]

A. Miné. A new numerical abstract domain based on difference-bound matrices. In Proc. PADO 2001, vol. 2053 of LNCS, pp. 155-172, Aarhus, Denmark.

Digital Library

[26]

A. Miné. The octagon abstract domain. In Proc. WCRE'01, pp. 310-319, Stuttgart.

Digital Library

[27]

A. Miné. A few graph-based relational numerical abstract domains. In Proc. SAS 2002, vol. 2477 of LNCS, pp. 117-132, Madrid, Spain.

Digital Library

[28]

A. Miné. 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]

A. Miné. Relational abstract domains for the detection of floating-point run-time errors. In Proc. ESOP 2004, vol. 2986 of LNCS, pp. 3-17, Barcelona, Spain.

[30]

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

[31]

S. Sankaranarayanan, H. Sipma, and Z. Manna. Scalable analysis of linear systems using mathematical programming. In Proc. VMCAI 2005, pp. 25-41, Paris, France.

Digital Library

[32]

R. Shaham, E. K. Kolodner, and S. Sagiv. Automatic removal of array memory leaks in Java. In Proc. CC 2000, vol. 1781 of LNCS, pp. 50-66, Berlin, Germany.

Digital Library

[33]

A. Simon, A. King, and J. M. Howe. Two variables per linear inequality as an abstract domain. In Proc. LOPSTR 2002, vol. 2664 of LNCS, pp. 71-89, Madrid.

Digital Library

Cited By

Gange GMa ZNavas JSchachte PSøndergaard HStuckey P(2021)A Fresh Look at Zones and OctagonsACM Transactions on Programming Languages and Systems10.1145/345788543:3(1-51)Online publication date: 3-Sep-2021
https://dl.acm.org/doi/10.1145/3457885
Bagnara RHill PZaffanella E(2010)Exact join detection for convex polyhedra and other numerical abstractionsComputational Geometry: Theory and Applications10.1016/j.comgeo.2009.09.00243:5(453-473)Online publication date: 1-Jul-2010
https://dl.acm.org/doi/10.1016/j.comgeo.2009.09.002
Monniaux D(2009)Automatic modular abstractions for linear constraintsACM SIGPLAN Notices10.1145/1594834.148089944:1(140-151)Online publication date: 21-Jan-2009
https://dl.acm.org/doi/10.1145/1594834.1480899
Show More Cited By

Recommendations

Precise widening operators for convex polyhedra
Special issue: Static analysis symposium (SAS 2003)

In the context of static analysis via abstract interpretation, convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it ...
Widening operators for powerset domains

The finite powerset construction upgrades an abstract domain by allowing for the representation of finite disjunctions of its elements. While most of the operations on the finite powerset abstract domain are easily obtained by "lifting" the ...
Precise widening operators for convex polyhedra
SAS'03: Proceedings of the 10th international conference on Static analysis

Convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it has to be used in conjunction with appropriate mechanisms for ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

SAS'05: Proceedings of the 12th international conference on Static Analysis

September 2005

368 pages

ISBN:3540285849

Editors:
Chris Hankin
Department of Computing, Imperial College London
,
Igor Siveroni
Department of Computing, Imperial College London, 180 Queen's Gate, London, UK

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 07 September 2005

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 16 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Gange GMa ZNavas JSchachte PSøndergaard HStuckey P(2021)A Fresh Look at Zones and OctagonsACM Transactions on Programming Languages and Systems10.1145/345788543:3(1-51)Online publication date: 3-Sep-2021
https://dl.acm.org/doi/10.1145/3457885
Bagnara RHill PZaffanella E(2010)Exact join detection for convex polyhedra and other numerical abstractionsComputational Geometry: Theory and Applications10.1016/j.comgeo.2009.09.00243:5(453-473)Online publication date: 1-Jul-2010
https://dl.acm.org/doi/10.1016/j.comgeo.2009.09.002
Monniaux D(2009)Automatic modular abstractions for linear constraintsACM SIGPLAN Notices10.1145/1594834.148089944:1(140-151)Online publication date: 21-Jan-2009
https://dl.acm.org/doi/10.1145/1594834.1480899
Jhala RMajumdar R(2009)Software model checkingACM Computing Surveys10.1145/1592434.159243841:4(1-54)Online publication date: 9-Oct-2009
https://dl.acm.org/doi/10.1145/1592434.1592438
Monniaux DShao ZPierce B(2009)Automatic modular abstractions for linear constraintsProceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages10.1145/1480881.1480899(140-151)Online publication date: 21-Jan-2009
https://dl.acm.org/doi/10.1145/1480881.1480899
Monniaux D(2009)A minimalistic look at widening operatorsHigher-Order and Symbolic Computation10.1007/s10990-009-9046-822:2(145-154)Online publication date: 1-Jun-2009
https://dl.acm.org/doi/10.1007/s10990-009-9046-8
Cousot PCousot RFeret JMauborgne LMiné ARival X(2009)Why does Astrée scale up?Formal Methods in System Design10.1007/s10703-009-0089-635:3(229-264)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.1007/s10703-009-0089-6
Bagnara RHill PZaffanella E(2009)Weakly-relational shapes for numeric abstractionsFormal Methods in System Design10.1007/s10703-009-0073-135:3(279-323)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.1007/s10703-009-0073-1
Bagnara RHill PZaffanella E(2008)An improved tight closure algorithm for integer octagonal constraintsProceedings of the 9th international conference on Verification, model checking, and abstract interpretation10.5555/1787526.1787532(8-21)Online publication date: 7-Jan-2008
https://dl.acm.org/doi/10.5555/1787526.1787532
Logozzo FFähndrich MWainwright RHaddad H(2008)PentagonsProceedings of the 2008 ACM symposium on Applied computing10.1145/1363686.1363736(184-188)Online publication date: 16-Mar-2008
https://dl.acm.org/doi/10.1145/1363686.1363736
Show More Cited By

View Options

View options

Figures

Tables

Media

View Table of Conten