Abstract
The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiability-modulo-theory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) quantification over discrete variables within the mixed Boolean-arithmetic constraint system. This provides the technological basis for a fully symbolic analysis of probabilistic hybrid automata. Generalizing SMT-based bounded model-checking of hybrid automata [2,11], stochastic SMT permits the direct and fully symbolic analysis of probabilistic bounded reachability problems of probabilistic hybrid automata without resorting to approximation by intermediate finite-state abstractions.
This work has been partially supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS, www.avacs.org).
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References
Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley - Interscience (1974)
Audemard, G., Bozzano, M., Cimatti, A., Sebastiani, R.: Verifying industrial hybrid systems with MathSAT. BMC, ENTCS 119, 17–32 (2004)
Biere, A., Cimatti, A., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) ETAPS 1999 and TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)
Blom, H.A.P., Krystul, J., Bakker, G.J.: A particle system for safety verification of free flight in air traffic. In: Decision and Control, pp. 1574–1579. IEEE, Los Alamitos (2006)
Bordeaux, L., Samulowitz, H.: On the stochastic constraint satisfaction framework. In: SAC, pp. 316–320. ACM Press, New York (2007)
Bujorianu, L., Lygeros, J.: Toward a general theory of stochastic hybrid systems. In: Stochastic Hybrid Systems: Theory and Safety Critical Applications, LNCIS, vol. 337, pp. 3–30 (2006)
Bujorianu, M.L., Lygeros, J.: Reachability questions in piecewise deterministic Markov processes. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 126–140. Springer, Heidelberg (2003)
Davis, M.: Markov Models and Optimization. Chapman & Hall, London (1993)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Comm. 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)
Fränzle, M., Herde, C.: HySAT: An efficient proof engine for bounded model checking of hybrid systems. Formal Methods in System Design 30, 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. Journal on Satisfiability, Boolean Modeling and Computation 1, 209–236 (2007)
Groote, J.F., Koorn, J.W.C., van Vlijmen, S.F.M.: The Safety Guaranteeing System at Station Hoorn-Kersenboogerd. In: Conference on Computer Assurance, pp. 57–68. National Institute of Standards and Technology (1995)
Hehner, E.C.R.: Predicative programming. Comm. of the ACM 27, 134–151 (1984)
Hespanha, J.P.: Polynomial stochastic hybrid systems. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 322–338. Springer, Heidelberg (2005)
Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 160–173. Springer, Heidelberg (2000)
Koutsoukos, X.D., Riley, D.: Computational methods for reachability analysis of stochastic hybrid systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 377–391. Springer, Heidelberg (2006)
Littman, M.L., Majercik, S.M., Pitassi, T.: Stochastic Boolean satisfiability. Journal of Automated Reasoning 27(3), 251–296 (2001)
Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT Modulo Theories: from an Abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T). Journal of the ACM 53(6), 937–977 (2006)
Sproston, J.: Model Checking of Probabilistic Timed and Hybrid Systems. PhD thesis, University of Birmingham (2000)
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Fränzle, M., Hermanns, H., Teige, T. (2008). Stochastic Satisfiability Modulo Theory: A Novel Technique for the Analysis of Probabilistic Hybrid Systems. In: Egerstedt, M., Mishra, B. (eds) Hybrid Systems: Computation and Control. HSCC 2008. Lecture Notes in Computer Science, vol 4981. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78929-1_13
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DOI: https://doi.org/10.1007/978-3-540-78929-1_13
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