Invariant Checking of NRA Transition Systems via Incremental Reduction to LRA with EUF

Alessandro Cimatti; Alberto Griggio; Ahmed Irfan; Marco Roveri; Roberto Sebastiani

doi:10.1007/978-3-662-54577-5_4

International Conference on Tools and Algorithms for the Construction and Analysis of Systems

TACAS 2017: Tools and Algorithms for the Construction and Analysis of Systems pp 58-75

Invariant Checking of NRA Transition Systems via Incremental Reduction to LRA with EUF

Authors
Authors and affiliations

Alessandro Cimatti
Alberto Griggio
Ahmed IrfanEmail author
Marco Roveri
Roberto Sebastiani

Conference paper

First Online:: 31 March 2017

DOI: 10.1007/978-3-662-54577-5_4

160 Downloads

Part of the Lecture Notes in Computer Science book series (LNCS, volume 10205)

Cite this paper as:: Cimatti A., Griggio A., Irfan A., Roveri M., Sebastiani R. (2017) Invariant Checking of NRA Transition Systems via Incremental Reduction to LRA with EUF. In: Legay A., Margaria T. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2017. Lecture Notes in Computer Science, vol 10205. Springer, Berlin, Heidelberg

Abstract

Model checking invariant properties of designs, represented as transition systems, with non-linear real arithmetic (NRA), is an important though very hard problem. On the one hand NRA is a hard-to-solve theory; on the other hand most of the powerful model checking techniques lack support for NRA. In this paper, we present a counterexample-guided abstraction refinement (CEGAR) approach that leverages linearization techniques from differential calculus to enable the use of mature and efficient model checking algorithms for transition systems on linear real arithmetic (LRA) with uninterpreted functions (EUF). The results of an empirical evaluation confirm the validity and potential of this approach.

References

1.
Ábrahám, E., Corzilius, F., Loup, U., Sturm, T.: A lazy SMT-solver for a non-linear subset of real algebra. In: Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum f A1/4r Informatik (2010)Google Scholar
2.
Bak, S., Bogomolov, S., Johnson, T.T.: HYST: a source transformation and translation tool for hybrid automaton models. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 128–133. ACM (2015)Google Scholar
3.
Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999). doi:10.1007/3-540-49059-0_14 CrossRefGoogle Scholar
4.
Birgmeier, J., Bradley, A.R., Weissenbacher, G.: Counterexample to induction-guided abstraction-refinement (CTIGAR). In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 831–848. Springer, Cham (2014). doi:10.1007/978-3-319-08867-9_55 Google Scholar
5.
Brain, M., D’Silva, V., Griggio, A., Haller, L., Kroening, D.: Interpolation-based verification of floating-point programs with abstract CDCL. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 412–432. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38856-9_22 CrossRefGoogle Scholar
6.
Brat, G., Bushnell, D., Davies, M., Giannakopoulou, D., Howar, F., Kahsai, T.: Verifying the safety of a flight-critical system. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 308–324. Springer, Heidelberg (2015). doi:10.1007/978-3-319-19249-9_20 CrossRefGoogle Scholar
7.
Cavada, R., et al.: The nuXmv symbolic model checker. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 334–342. Springer, Cham (2014). doi:10.1007/978-3-319-08867-9_22 Google Scholar
8.
Champion, A., Gurfinkel, A., Kahsai, T., Tinelli, C.: CoCoSpec: a mode-aware contract language for reactive systems. In: De Nicola, R., Kühn, E. (eds.) SEFM 2016. LNCS, vol. 9763, pp. 347–366. Springer, Cham (2016). doi:10.1007/978-3-319-41591-8_24 Google Scholar
9.
Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: IC3 modulo theories via implicit predicate abstraction. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 46–61. Springer, Heidelberg (2014). doi:10.1007/978-3-642-54862-8_4 CrossRefGoogle Scholar
10.
Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: HyComp: an SMT-based model checker for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 52–67. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46681-0_4 Google Scholar
11.
Cimatti, A., Mover, S., Tonetta, S.: A quantifier-free SMT encoding of non-linear hybrid automata. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 187–195. IEEE (2012)Google Scholar
12.
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition-preliminary report. SIGSAM Bull. 8(3), 80–90 (1974). http://doi.acm.org/10.1145/1086837.1086852 CrossRefGoogle Scholar
13.
Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). doi:10.1007/978-3-540-78800-3_24 CrossRefGoogle Scholar
14.
Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Heidelberg (2014). doi:10.1007/978-3-319-08867-9_49 Google Scholar
15.
Eén, N., Sörensson, N.: Temporal induction by incremental SAT solving. Electron. Notes Theor. Comput. Sci. 89(4), 543–560 (2003)CrossRefMATHGoogle Scholar
16.
Gao, S., Kong, S., Clarke, E.M.: dReal: an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38574-2_14 CrossRefGoogle Scholar
17.
Gario, M., Micheli, A.: PySMT: a solver-agnostic library for fast prototyping of SMT-based algorithms. In: Proceedings of the 13th International Workshop on Satisfiability Modulo Theories (SMT), pp. 373–384 (2015)Google Scholar
18.
Hoder, K., Bjørner, N.: Generalized property directed reachability. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 157–171. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31612-8_13 CrossRefGoogle Scholar
19.
Hueschen, R.M.: Development of the Transport Class Model (TCM) aircraft simulation from a sub-scale Generic Transport Model (GTM) simulation. Technical report, NASA Langley Research Center (2011)Google Scholar
20.
Jovanović, D., Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 339–354. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31365-3_27 CrossRefGoogle Scholar
21.
Komuravelli, A., Gurfinkel, A., Chaki, S.: SMT-based model checking for recursive programs. Form. Methods Syst. Des. 48(3), 175–205 (2016)CrossRefMATHGoogle Scholar
22.
Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \({\delta }\)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46681-0_15 Google Scholar
23.
Kupferschmid, S., Becker, B.: Craig interpolation in the presence of non-linear constraints. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 240–255. Springer, Heidelberg (2011). doi:10.1007/978-3-642-24310-3_17 CrossRefGoogle Scholar
24.
Mahdi, A., Scheibler, K., Neubauer, F., Fränzle, M., Becker, B.: Advancing software model checking beyond linear arithmetic theories. In: Bloem, R., Arbel, E. (eds.) HVC 2016. LNCS, vol. 10028, pp. 186–201. Springer, Heidelberg (2016). doi:10.1007/978-3-319-49052-6_12 CrossRefGoogle Scholar
25.
Maréchal, A., Fouilhé, A., King, T., Monniaux, D., Périn, M.: Polyhedral approximation of multivariate polynomials using Handelman’s theorem. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 166–184. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49122-5_8 CrossRefGoogle Scholar
26.
McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45069-6_1 CrossRefGoogle Scholar
27.
Nuzzo, P., Puggelli, A., Seshia, S.A., Sangiovanni-Vincentelli, A.: CalCS: SMT solving for non-linear convex constraints. In: Proceedings of the 2010 Conference on Formal Methods in Computer-Aided Design, pp. 71–80. FMCAD Inc. (2010)Google Scholar
28.
Scheibler, K., Kupferschmid, S., Becker, B.: Recent improvements in the SMT solver iSAT. MBMV 13, 231–241 (2013)Google Scholar
29.
Sheeran, M., Singh, S., Stålmarck, G.: Checking safety properties using induction and a SAT-solver. In: Hunt, W.A., Johnson, S.D. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 127–144. Springer, Heidelberg (2000). doi:10.1007/3-540-40922-X_8 CrossRefGoogle Scholar
30.
Tiwari, A.: Time-aware abstractions in HybridSal. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 504–510. Springer, Heidelberg (2015). doi:10.1007/978-3-319-21690-4_34 CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Alessandro Cimatti
- 1
Alberto Griggio
- 1
Ahmed Irfan
- 1
- 2
Email author
Marco Roveri
- 1
Roberto Sebastiani
- 2

1.Fondazione Bruno KesslerTrentoItaly
2.University of TrentoTrentoItaly

Personalised recommendations

Actions

Buy eBook

USD 84.99

Instant download
Readable on all devices
Own it forever
Local sales tax included if applicable

Invariant Checking of NRA Transition Systems via Incremental Reduction to LRA with EUF

Abstract

Copyright information

Authors and Affiliations

Personalised recommendations

Cookies