Abstract
In this work we develop a new technique for over-approximating (in the sense of timed trace inclusion) continuous dynamical systems by timed automata. This technique refines commonly-used discrete abstractions which are often too coarse to be useful. The essence of our technique is the partition of the state space into cubes and the allocation of a clock for each dimension. This allows us to get much better approximations of the behavior. We specialize this technique to multi-affine systems, a class of nonlinear systems of primary importance for the analysis of biochemical systems and demonstrate its applicability on an example taken from synthetic biology.
This work was partially supported by the French-Israeli project Computational Modeling of Incomplete Biological Regulatory Networks.
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References
Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The Algorithmic Analysis of Hybrid Systems. Theoretical Computer Science 138, 3–34 (1995)
Alur, R., Dill, D.L.: A Theory of Timed Automata. Theoretical Computer Science 126, 183–235 (1994)
Alur, R., Feder, T., Henzinger, T.A.: The Benefits of Relaxing Punctuality. Journal of the ACM 43, 116–146 (1996)
Asarin, A., Bournez, O., Dang, T., Maler, O.: Approximate Reachability Analysis of Piecewise-linear Dynamical Systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 20–31. Springer, Heidelberg (2000)
Asarin, E., Dang, T., Frehse, G., Girard, A., Le Guernic, C., Maler, O.: Recent Progress in Continuous and Hybrid Reachability Analysis. In: CACSD (2006)
Asarin, E., Dang, T., Girard, A.: Reachability Analysis of Nonlinear Systems using Conservative Approximation. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 20–35. Springer, Heidelberg (2003)
Asarin, E., Maler, O., Pnueli, A.: Reachability Analysis of Dynamical Systems having Piecewise-Constant Derivatives. Theoretical Computer Science 138, 35–65 (1995)
Batt, G., Ropers, D., de Jong, H., Geiselmann, J., Mateescu, R., Page, M., Schneider, D.: Validation of qualitative models of genetic regulatory networks by model checking: Analysis of the nutritional stress response in Escherichia coli. Bioinformatics 21, 19–28 (2005)
Batt, G., Yordanov, B., Weiss, R., Belta, C.: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 23, 2415–2422 (2007)
Belta, C., Habets, L.C.G.J.M.: Controlling a Class of Nonlinear Systems on Rectangles. IEEE Trans. on Automatic Control 51, 1749–1759 (2006)
Botchkarev, O., Tripakis, S.: Verification of Hybrid Systems with Linear Differential Inclusions using Ellipsoidal Approximations. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 73–88. Springer, Heidelberg (2000)
Bournez, O., Maler, O., Pnueli, A.: Orthogonal Polyhedra: Representation and Computation. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 46–60. Springer, Heidelberg (1999)
Bozga, M., Graf, S., Mounier, L.: IF-2.0: A Validation Environment for Component-Based Real-Time Systems. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 343–348. Springer, Heidelberg (2002)
Chutinan, A., Krogh, B.H.: Computing Polyhedral Approximations to Dynamic Flow Pipes. In: CDC 1998. IEEE, Los Alamitos (1998)
Chutinan, A., Krogh, B.H.: Computational Techniques for Hybrid System Verification. IEEE Trans. on Automatic Control 8, 64–75 (2003)
Dang, T., Maler, O.: Reachability Analysis via Face Lifting. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 96–109. Springer, Heidelberg (1998)
de Jong, H., Page, M., Hernandez, C., Geiselmann, J.: Qualitative Simulation of Genetic Regulatory Networks: Method and Application. In: IJCAI-2001, pp. 67–73 (2001)
Frehse, G.: PHAVer: Algorithmic Verification of Hybrid Systems Past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005)
Halasz, A., Kumar, V., Imielinski, M., Belta, C., Sokolsky, O., Pathak, S.: Analysis of Lactose Metabolism in E.coli using Reachability Analysis of Hybrid Systems. IEE Proceedings - Systems Biology 21, 130–148 (2007)
Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Algorithmic Analysis of Nonlinear Hybrid Systems. IEEE Trans. on Automatic Control 43, 540–554 (1998)
Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s Decidable about Hybrid Automata? Journal of Computer and System Sciences 57, 94–124 (1998)
Hooshangi, S., Thiberge, S., Weiss, R.: Ultrasensitivity and noise propagation in a synthetic transcriptional cascade. PNAS 102, 3581–3586 (2005)
Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Hytech: A Model Checker for Hybrid Systems. Software Tools for Technology Transfer 1, 110–122 (1997)
Koymans, R.: Specifying Real-time Properties with Metric Temporal Logic. Real-time Systems 2, 255–299 (1990)
Larsen, K.G., Pettersson, P., Yi, W.: Uppaal in a Nutshell. Software Tools for Technology Transfer 1, 134–152 (1997)
Olivero, A., Sifakis, J., Yovine, S.: Using abstractions for the verification of linear hybrid systems. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 81–94. Springer, Heidelberg (1994)
Stursberg, O., Kowalewski, S., Engell, S.: On the Generation of Timed Discrete Approximations for Continuous Systems. Mathematical and Computer Models of Dynamical Systems 6, 51–70 (2000)
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Maler, O., Batt, G. (2008). Approximating Continuous Systems by Timed Automata. In: Fisher, J. (eds) Formal Methods in Systems Biology. FMSB 2008. Lecture Notes in Computer Science(), vol 5054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68413-8_6
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