Abstract
An abstract interpretation based static analyzer depends on the choice of both an abstract domain and a methodology to compute fixpoints of monotonic functions. Abstract domains are almost always representations of convex sets that must provide efficient algorithms to perform both numerical and order-theoretic computations. In this paper, we present a new abstract domain that uses support functions to represent convex sets. We define the order-theoretic operations and, using a predefined set of directions, we define an efficient method to compute the fixpoint of linear and non-linear programs. Experiments show the efficiency and precision of our methods.
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References
Allamigeon, X., Gaubert, S., Goubault, É.: Inferring Min and Max Invariants Using Max-plus Polyhedra. In: Alpuente, M., Vidal, G. (eds.) SAS 2008. LNCS, vol. 5079, pp. 189–204. Springer, Heidelberg (2008)
Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific Series in Optimization and Neural Computation. Athena Scientific (2003)
Chen, L., Miné, A., Wang, J., Cousot, P.: Interval polyhedra: An abstract domain to infer interval linear relationships. In: Palsberg, J., Su, Z. (eds.) SAS 2009. LNCS, vol. 5673, pp. 309–325. Springer, Heidelberg (2009)
Colón, M.A., Sankaranarayanan, S.: Generalizing the template polyhedral domain. In: Barthe, G. (ed.) ESOP 2011. LNCS, vol. 6602, pp. 176–195. Springer, Heidelberg (2011)
Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Conference Record of the Fourth ACM Symposium on Principles of Programming Languages (POPL 1977), pp. 238–252. ACM Press (1977)
Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Conference Record of the Fifth ACM Symposium on Principles of Programming Languages (POPL 1978), pp. 84–97. ACM Press (1978)
Cox, A., Sankaranarayanan, S., Chang, B.-Y.E.: A bit too precise? Bounded verification of quantized digital filters. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 33–47. Springer, Heidelberg (2012)
Gaubert, S., McEneaney, W.M., Qu, Z.: Curse of dimensionality reduction in max-plus based approximation methods: Theoretical estimates and improved pruning algorithms. In: CDC-ECE (2011)
Goubault, E., Putot, S.: Perturbed affine arithmetic for invariant computation in numerical program analysis. CoRR, abs/0807.2961 (2008)
Le Guernic, C., Girard, A.: Reachability analysis of hybrid systems using support functions. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 540–554. Springer, Heidelberg (2009)
Hiriart-Urrut, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer (2004)
Lakhdar-Chaouch, L., Jeannet, B., Girault, A.: Widening with thresholds for programs with complex control graphs. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 492–502. Springer, Heidelberg (2011)
Miné, A.: Weakly relational numerical abstract domains. PhD thesis, École Polytechnique (2004), http://www.di.ens.fr/~mine/these/these-color.pdf
Miné, A.: The octagon abstract domain. Higher-Order and Symbolic Computation 19 (2006)
Sankaranarayanan, S., Sipma, H.B., 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)
Seladji, Y., Bouissou, O.: Fixpoint computation in the polyhedra abstract domain using convex and numerical analysis tools. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 149–168. Springer, Heidelberg (2013)
Seladji, Y., Bouissou, O.: Numerical abstract domain using support functions (extended version) (2013), http://www.lix.polytechnique.fr/~bouissou/pdf/publications/NFM13_extended.pdf
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Seladji, Y., Bouissou, O. (2013). Numerical Abstract Domain Using Support Functions. In: Brat, G., Rungta, N., Venet, A. (eds) NASA Formal Methods. NFM 2013. Lecture Notes in Computer Science, vol 7871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38088-4_11
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DOI: https://doi.org/10.1007/978-3-642-38088-4_11
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