Abstract
In the present paper we compute numerical invariants of programs by abstract interpretation. For that we consider the abstract domain of quadratic zones recently introduced by Adjé et al. [2]. We use a relaxed abstract semantics which is at least as precise as the relaxed abstract semantics of Adjé et al. [2]. For computing our relaxed abstract semantics, we present a practical strategy improvement algorithm for precisely computing least solutions of fixpoint equation systems, whose right-hand sides use order-concave operators and the maximum operator. These fixpoint equation systems strictly generalize the fixpoint equation systems considered by Gawlitza and Seidl [11].
This work was partially funded by the ANR project ASOPT.
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Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.): ICALP 2008, Part I. LNCS, vol. 5125, pp. 973–978. Springer, Heidelberg (2008)
Adjé, A., Gaubert, S., Goubault, E.: Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 23–42. Springer, Heidelberg (2010) ISBN 978-3-642-11956-9
Björklund, H., Sandberg, S., Vorobyov, S.: Optimization on completely unimodal hypercubes. Technichal report 2002-18, Department of Information Technology, Uppsala University (2002)
Björklund, H., Sandberg, S., Vorobyov, S.: Complexity of Model Checking by Iterative Improvement: the Pseudo-Boolean Framework. In: Broy, M., Zamulin, A.V. (eds.) PSI 2003. LNCS, vol. 2890, pp. 381–394. Springer, Heidelberg (2004)
Borchers, B.: Csdp, a c library for semidefinite programming. In: Optimization Methods and Software, vol. 11, p. 613. Taylor and Francis, Abington (1999)
Costan, A., Gaubert, S., Goubault, E., Martel, M., Putot, S.: A Policy Iteration Algorithm for Computing Fixed Points in Static Analysis of Programs. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 462–475. Springer, Heidelberg (2005)
Esparza, J., Gawlitza, T., Kiefer, S., Seidl, H.: Approximative methods for monotone systems of min-max-polynomial equations. In: Aceto et al. [1], pp. 698–710, ISBN 978-3-540-70574-1
Etessami, K., Yannakakis, M.: Recursive concurrent stochastic games. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 324–335. Springer, Heidelberg (2006) ISBN 3-540-35907-9
Etessami, K., Wojtczak, D., Yannakakis, M.: Recursive stochastic games with positive rewards. In: Aceto et al. [1], pp.711–723, ISBN 978-3-540-70574-1
Gaubert, S., Goubault, E., Taly, A., Zennou, S.: Static analysis by policy iteration on relational domains. In: Nicola [16], pp. 237–252, ISBN 978-3-540-71314-2
Gawlitza, T., Seidl, H.: Precise relational invariants through strategy iteration. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 23–40. Springer, Heidelberg (2007) ISBN 978-3-540-74914-1
Gawlitza, T., Seidl, H.: Precise fixpoint computation through strategy iteration. In: Nicola [16], pp. 300–315 (2007) ISBN 978-3-540-71314-2
Gawlitza, T.M., Seidl, H.: Solving systems of rational equations through strategy iteration. Technical report, TUM (2009)
Miné, A.: A new numerical abstract domain based on difference-bound matrices. In: Danvy, O., Filinski, A. (eds.) PADO 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001) ISBN 3-540-42068-1
Miné, A.: The octagon abstract domain. In: WCRE, p. 310(2001)
De Nicola, R. (ed.): ESOP 2007. LNCS, vol. 4421. Springer, Heidelberg (2007) ISBN 978-3-540-71314-2
Ortega, J., Rheinboldt, W.: Iterative solution of nonlinear equations in several variables. Academic Press, London (1970)
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) ISBN 3-540-24297-X
Todd, M.J.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)
Wojtczak, D., Etessami, K.: Premo: An analyzer for probabilistic recursive models. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 66–71. Springer, Heidelberg (2007) ISBN 978-3-540-71208-4
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Gawlitza, T.M., Seidl, H. (2010). Computing Relaxed Abstract Semantics w.r.t. Quadratic Zones Precisely. In: Cousot, R., Martel, M. (eds) Static Analysis. SAS 2010. Lecture Notes in Computer Science, vol 6337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15769-1_17
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