Abstract
Knowing that two numerical variables always hold different values, at some point of a program, can be very useful, especially for analyzing aliases: if i ≠ j, then A[i] and A[j] are not aliased, and this knowledge is of great help for many other program analyses. Surprisingly, disequalities are seldom considered in abstract interpretation, most of the proposed numerical domains being restricted to convex sets. In this paper, we propose to combine simple ordering properties with disequalities. “Difference-bound matrices” (or DBMs) is a domain proposed by David Dill, for expressing relations of the form “x − y ≤ c” or “c 1 ≤ x ≤ c 2”. We define dDBMs (“disequalities DBMs”) as conjunctions of DBMs with simple disequalities of the form “x ≠ y” or “x ≠ 0”. We give algorithms on dDBMs, for deciding the emptiness, computing a normal form, and performing the usual operations of an abstract domain. These algorithms have the same complexity (O(n 3), where n is the number of variables) than those for classical DBMs, if the variables are considered to be valued in a dense set (ℝ or ℚ). In the arithmetic case, the emptiness decision is NP-complete, and other operations run in O(n 5).
This work has been partially supported by the APRON project of the “ACI Sécurité et Informatique” of the French Ministry of Research.
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
Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking in dense real-time. Information and Computation 104(1), 2–34 (1993)
Bourdoncle, F.: Abstract debugging of higher-order imperative languages. In: PLDI’93, pp. 46–55. ACM, New York (1993)
Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: 2nd Int. Symp. on Programming, pp. 106–130. Dunod, Paris (1976)
Clarisó, R.C., Cortadella, J.: Verification of parametric timed circuits using octahedra. In: Designing correct circuits, DCC’04, Barcelona (2004)
Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: POPL’78, pp. 84–96 (January 1978)
Cormen, T.H., et al.: Introduction to Algorithms. The MIT Press, Cambridge (1990)
Deutsch, A.: Interprocedural may-alias analysis for pointers: beyond k-limiting. In: PLDI’94, pp. 230–241 (1994)
Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) Automatic Verification Methods for Finite State Systems. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990)
Goralciková, A., Koubek, V.: A reduct-and-closure algorithm for graphs. In: Becvar, J. (ed.) Mathematical Foundations of Computer Science 1979. LNCS, vol. 74, pp. 301–307. Springer, Heidelberg (1979)
Granger, P.: Static analysis of linear congruence equalities among variables of a program. In: Abramsky, S. (ed.) CAAP 1991 and TAPSOFT 1991. LNCS, vol. 493, pp. 169–192. Springer, Heidelberg (1991)
Harvey, W., Stuckey, P.J.: A unit two variable per inequality integer constraint solver for constraint logic programming. In: Twentieth Australasian Computer Science Conference, ACSC’97, pp. 102–111 (February 1997)
Harvey, W., Stuckey, P.J.: Improving linear constraint propagation by changing constraint representation. Constraints 8(2), 173–207 (2003)
Imbert, J.-L.: Variable elimination for generalized linear constraints. In: ICLP’93, pp. 499–516 (1993)
Jeannet, B.: The NBAC verification/slicing tool, http://www.inrialpes.fr/pop-art/people/bjeannet/nbac/
Jeannet, B., Halbwachs, N., Raymond, P.: Dynamic partitioning in analyses of numerical properties. In: Cortesi, A., Filé, G. (eds.) SAS 1999. LNCS, vol. 1694, Springer, Heidelberg (1999)
Karr, M.: Affine relationships among variables of a program. Acta Informatica 6, 133–151 (1976)
Lassez, J.-L., McAloon, K.: A canonical form for generalized linear constraints. J. Symb. Comput. 13(1), 1–24 (1992)
Larsen, K.G., et al.: Clock difference diagrams. Nordic J. of Computing 6(3), 271–298 (1999)
Masdupuy, F.: Semantic analysis of interval congruences. In: International Conference on Formal Methods in Programming and Their Applications, pp. 142–155 (1993)
Miné, A.: The octagon abstract domain. In: AST 2001 in WCRE 2001, pp. 310–319. IEEE Computer Society Press, Los Alamitos (2001)
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)
Møller, J.B., et al.: Difference decision diagrams. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 111–125. Springer, Heidelberg (1999)
Mauborgne, L., Rival, X.: Trace partitioning in abstract interpretation based static analyzers. In: Sagiv, M. (ed.) ESOP 2005. LNCS, vol. 3444, Springer, Heidelberg (2005)
Puget, J.-F.: A fast algorithm for the bound consistency of alldiff constraints. In: Mittal, V.O., et al. (eds.) Assistive Technology and Artificial Intelligence. LNCS (LNAI), vol. 1458, pp. 359–366. Springer, Heidelberg (1998)
Pugh, W., Wonnacott, D.: Constraint-based array dependence analysis. TOPLAS 20(3), 635–678 (1998)
Rosenkrantz, D.J., Hunt III, H.B.: Processing conjunctive predicates and ueries. In: VLDB, pp. 64–72 (1980)
Simon, K., Crippa, D., Collenberg, F.: On the distribution of the transitive closure in a random acyclic digraph. In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 345–356. Springer, Heidelberg (1993)
Simon, K.: An improved algorithm for transitive closure on acyclic digraphs. TCS 58(1-3), 325–346 (1988)
Sankaranarayanan, S., et al.: Static analysis in disjunctive numerical domains. In: Yi, K. (ed.) SAS 2006. LNCS, vol. 4134, Springer, Heidelberg (2006)
Seater, R., Wonnacott, D.: Efficient Manipulation of Disequalities During Dependence Analysis. In: Pugh, B., Tseng, C.-W. (eds.) LCPC 2002. LNCS, vol. 2481, pp. 295–308. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Péron, M., Halbwachs, N. (2007). An Abstract Domain Extending Difference-Bound Matrices with Disequality Constraints. In: Cook, B., Podelski, A. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2007. Lecture Notes in Computer Science, vol 4349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69738-1_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-69738-1_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69735-0
Online ISBN: 978-3-540-69738-1
eBook Packages: Computer ScienceComputer Science (R0)