Abstract
The polyhedra abstract domain is one of the most powerful and commonly used numerical abstract domains in the field of static program analysis based on abstract interpretation. In this paper, we present an implementation of the polyhedra domain using floating-point arithmetic without sacrificing soundness. Floating-point arithmetic allows a compact memory representation and an efficient implementation on current hardware, at the cost of some loss of precision due to rounding. Our domain is based on a constraint-only representation and employs sound floating-point variants of Fourier-Motzkin elimination and linear programming. The preliminary experimental results of our prototype are encouraging. To our knowledge, this is the first time that the polyhedra domain is adapted to floating-point arithmetic in a sound way.
This work is supported by the INRIA project-team Abstraction common to the CNRS and the École Normale Supérieure. This work is partially supported by the Fund of the China Scholarship Council and National Natural Science Foundation of China under Grant No.60725206.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
APRON numerical abstract domain library, http://apron.cri.ensmp.fr/library/
Alexander, S.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1998)
Bagnara, R., Hill, P.M., Zaffanella, E.: The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Quaderno 457, Dipartimento di Matematica, Università di Parma, Italy (2006)
Bemporad, A., Fukuda, K., Torrisi, F.D.: Convexity recognition of the union of polyhedra. Computational Geometry 18(3), 141–154 (2001)
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, San Diego, California, USA, June 2003, pp. 196–207. ACM Press, New York (2003)
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, Los Angeles, California, pp. 238–252. ACM Press, New York (1977)
Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: ACM POPL 1978, pp. 84–96. ACM, New York (1978)
Goubault, E.: Static analyses of floating-point operations. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, pp. 234–259. Springer, Heidelberg (2001)
Halbwachs, N.: Détermination automatique de relations linéaires vérifiées par les variables d’un programme. Ph.D thesis, Thèse de 3ème cycle d’informatique, Université scientifique et médicale de Grenoble, Grenoble, France (March 1979)
Huynh, T., Lassez, C., Lassez, J.-L.: Practical issues on the projection of polyhedral sets. Annals of Mathematics and Artificial Intelligence 6(4), 295–315 (1992)
Imbert, J.-L.: Fourier’s elimination: Which to choose? In: PCPP 1993, pp. 117–129 (1993)
Lalire, G., Argoud, M., Jeannet, B.: Interproc., http://pop-art.inrialpes.fr/people/bjeannet/bjeannet-forge/interproc/
LeVerge, H.: A note on Chernikova’s algorithm. Technical Report 635, IRISA, France (1992)
Makhorin, A.: The GNU Linear Programming Kit (2000), http://www.gnu.org/software/glpk/
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)
Miné, A.: Field-sensitive value analysis of embedded C programs with union types and pointer arithmetics. In: LCTES 2006, Ottawa, Ontario, Canada, pp. 54–63. ACM Press, New York (2006)
Miné, A.: The octagon abstract domain. Higher-Order and Symbolic Computation 19(1), 31–100 (2006)
Moore, R.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Program. 99(2), 283–296 (2004)
Que, D.N.: Robust and generic abstract domain for static program analysis: the polyhedral case. Technical report, École des Mines de Paris (July 2006)
Rohn, J.: Solvability of systems of interval linear equations and inequalities. In: Linear Optimization Problems with Inexact Data, pp. 35–77. Springer, Heidelberg (2006)
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)
Simon, A., King, A.: Exploiting sparsity in polyhedral analysis. In: Hankin, C. (ed.) SAS 2005. LNCS, vol. 3672, pp. 336–351. Springer, Heidelberg (2005)
Simon, A., King, A., Howe, J.M.: Two variables per linear inequality as an abstract domain. In: Leuschel, M.A. (ed.) LOPSTR 2002. LNCS, vol. 2664, pp. 71–89. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, L., Miné, A., Cousot, P. (2008). A Sound Floating-Point Polyhedra Abstract Domain. In: Ramalingam, G. (eds) Programming Languages and Systems. APLAS 2008. Lecture Notes in Computer Science, vol 5356. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89330-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-89330-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89329-5
Online ISBN: 978-3-540-89330-1
eBook Packages: Computer ScienceComputer Science (R0)