Abstract
In order to facilitate automated reasoning about large Boolean combinations of non-linear arithmetic constraints involving ordinary differential equations (ODEs), we provide a seamless integration of safe numeric overapproximation of initial-value problems into a SAT-modulo-theory (SMT) approach to interval-based arithmetic constraint solving. Interval-based safe numeric approximation of ODEs is used as an interval contractor being able to narrow candidate sets in phase space in both temporal directions: post-images of ODEs (i.e., sets of states reachable from a set of initial values) are narrowed based on partial information about the initial values and, vice versa, pre-images are narrowed based on partial knowledge about post-sets.
In contrast to the related CLP(F) approach of Hickey and Wittenberg [12], we do (a) support coordinate transformations mitigating the wrapping effect encountered upon iterating interval-based overapproximations of reachable state sets and (b) embed the approach into an SMT framework, thus accelerating the solving process through the algorithmic enhancements of recent SAT solving technology.
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References
Alur, R., Pappas, G.J. (eds.): HSCC 2004. LNCS, vol. 2993. Springer, Heidelberg (2004)
Audemard, G., Bozzano, M., Cimatti, A., Sebastiani, R.: Verifying industrial hybrid systems with MathSAT. ENTCS 89(4) (2004)
Benhamou, F., Granvilliers, L.: Continuous and interval constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, Foundations of Artificial Intelligence, ch. 16, pp. 571–603. Elsevier, Amsterdam (2006)
Biere, A., Cimatti, A., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579. Springer, Heidelberg (1999)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5, 394–397 (1962)
Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)
Fehnker, A., Ivančić, F.: Benchmarks for hybrid systems verification. In: Alur, Pappas (eds.) in [1] pp. 326–341
Fränzle, M., Herde, C.: HySAT: An efficient proof engine for bounded model checking of hybrid systems. Formal Methods in Syst. Design 30(3), 179–198 (2007)
Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT Special Issue on Constraint Programming and SAT 1, 209–236 (2007)
Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(t): Fast decision procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114. Springer, Heidelberg (2004)
Henzinger, T.A., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond HYTECH: Hybrid systems analysis using interval numerical methods. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 130–144. Springer, Heidelberg (2000)
Hickey, T., Wittenberg, D.: Rigorous modeling of hybrid systems using interval arithmetic constraints. In: Alur, Pappas (eds.) in [1], pp. 402–416
Lohner, R.J.: Enclosing the solutions of ordinary initial and boundary value problems. In: Computerarithmetic: Scientific Computation and Programming Languages, pp. 255–286. Teubner, Stuttgart (1987)
Moore, R.E.: Automatic local coordinate transformation to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. In: Ball, L.B. (ed.) Error in Digital Computation, vol. II, pp. 103–140. Wiley, New York (1965)
Ramdani, N., Meslem, N., Candau, Y.: Rechability of unvertain nonlinear systems using a nonlinear hybridization. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 415–428. Springer, Heidelberg (2008)
Stauning, O.: Automatic Validation of Numerical Solutions. PhD thesis, Danmarks Tekniske Universitet, Kgs.Lyngby, Denmark (1997)
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Eggers, A., Fränzle, M., Herde, C. (2008). SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems. In: Cha, S.(., Choi, JY., Kim, M., Lee, I., Viswanathan, M. (eds) Automated Technology for Verification and Analysis. ATVA 2008. Lecture Notes in Computer Science, vol 5311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88387-6_14
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DOI: https://doi.org/10.1007/978-3-540-88387-6_14
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