Parallel SMT-Based Parameter Synthesis with Application to Piecewise Multi-affine Systems

  • Nikola Beneš
  • Luboš Brim
  • Martin Demko
  • Samuel Pastva
  • David ŠafránekEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9938)


We propose a novel scalable parallel algorithm for synthesis of interdependent parameters from CTL specifications for non-linear dynamical systems. The method employs a symbolic representation of sets of parameter valuations in terms of the first-order theory of the reals. To demonstrate its practicability, we apply the method to a class of piecewise multi-affine dynamical systems representing dynamics of biological systems with complex non-linear behaviour.


Model Check Atomic Proposition Parameter Synthesis Satisfiability Modulo Theory Border State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Arney, D., Pajic, M., Goldman, J.M., Lee, I., Mangharam, R., Sokolsky, O.: Toward patient safety in closed-loop medical device systems. In: ICCPS 2010, pp. 139–148. ACM (2010)Google Scholar
  2. 2.
    Baker, M., Carpenter, B., Shafi, A.: MPJ express: towards thread safe Java HPC. In: IEEE Cluster Computing 2006. IEEE Computer Society (2006)Google Scholar
  3. 3.
    Barnat, J., Brim, L., Krejčí, A., Streck, A., Šafránek, D., Vejnár, M., Vejpustek, T.: On parameter synthesis by parallel model checking. IEEE/ACM Trans. Comput. Biol. Bioinf. 9(3), 693–705 (2012)CrossRefGoogle Scholar
  4. 4.
    Barnat, J., Brim, L., Šafránek, D.: High-performance analysis of biological systems dynamics with the divine model checker. Brief. Bioinf. 11(3), 301–312 (2010)CrossRefGoogle Scholar
  5. 5.
    Barrett, C., Fontaine, P., Tinelli, C.: The SMT-LIB standard: version 2.5. Technical report, Department of Computer Science, The University of Iowa (2015)Google Scholar
  6. 6.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics). Springer-Verlag New York Inc, Secaucus (2006)zbMATHGoogle Scholar
  7. 7.
    Batt, G., Page, M., Cantone, I., Gössler, G., Monteiro, P., de Jong, H.: Efficient parameter search for qualitative models of regulatory networks using symbolic model checking. Bioinformatics 26(18), 603–610 (2010)CrossRefGoogle Scholar
  8. 8.
    Batt, G., Yordanov, B., Weiss, R., Belta, C.: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 23(18), 2415–2422 (2007)CrossRefGoogle Scholar
  9. 9.
    Bogomolov, S., Schilling, C., Bartocci, E., Batt, G., Kong, H., Grosu, R.: Abstraction-based parameter synthesis for multiaffine systems. In: Piterman, N., et al. (eds.) HVC 2015. LNCS, vol. 9434, pp. 19–35. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-26287-1_2 CrossRefGoogle Scholar
  10. 10.
    Brim, L., Dluhoš, P., Šafránek, D., Vejpustek, T.: STL*: extending signal temporal logic with signal-value freezing operator. Inf. Comput. 236, 52–67 (2014). Special Issue on Hybrid Systems and BiologyMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brim, L., Češka, M., Demko, M., Pastva, S., Šafránek, D.: Parameter synthesis by parallel coloured CTL model checking. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 251–263. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  12. 12.
    Brim, L., Demko, M., Pastva, S., Šafránek, D.: High-performance discrete bifurcation analysis for piecewise-affine dynamical systems. In: Abate, A., et al. (eds.) HSB 2015. LNCS, vol. 9271, pp. 58–74. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-26916-0_4 CrossRefGoogle Scholar
  13. 13.
    Brim, L., Yorav, K., Žídková, J.: Assumption-based distribution of CTL model checking. STTT 7(1), 61–73 (2005)CrossRefGoogle Scholar
  14. 14.
    Chiang, H.D., Wang, T.: On the number and types of unstable equilibria in nonlinear dynamical systems with uniformly-bounded stability regions. IEEE Trans. Autom. Control 61(2), 485–490 (2016)MathSciNetGoogle Scholar
  15. 15.
    Collins, P., Habets, L.C., van Schuppen, J.H., Černá, I., Fabriková, J., Šafránek, D.: Abstraction of biochemical reaction systems on polytopes. In: IFAC World Congress. pp. 14869–14875. IFAC (2011)Google Scholar
  16. 16.
    Dang, T., Donze, A., Maler, O., Shalev, N.: Sensitive state-space exploration. In: IEEE Conference on Decision and Control, pp. 4049–4054 (2008)Google Scholar
  17. 17.
    Donzé, A., Fanchon, E., Gattepaille, L.M., Maler, O., Tracqui, P.: Robustness analysis and behavior discrimination in enzymatic reaction networks. PLoS ONE 6(9), e24246 (2011)CrossRefGoogle Scholar
  18. 18.
    Donzé, A., Krogh, B., Rajhans, A.: Parameter synthesis for hybrid systems with an application to simulink models. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 165–179. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Donzé, A., Maler, O., Bartocci, E., Nickovic, D., Grosu, R., Smolka, S.: On temporal logic and signal processing. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 92–106. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. 20.
    Dreossi, T., Dang, T.: Parameter synthesis for polynomial biological models. In: Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control, HSCC 2014, pp. 233–242 (2014)Google Scholar
  21. 21.
    Dvorak, P., Chrast, L., Nikel, P.I., Fedr, R., Soucek, K., Sedlackova, M., Chaloupkova, R., Lorenzo, V., Prokop, Z., Damborsky, J.: Exacerbation of substrate toxicity by IPTG in Escherichia coli BL21(DE3) carrying a synthetic metabolic pathway. Microb. Cell Fact. 14(1), 1–15 (2015)CrossRefGoogle Scholar
  22. 22.
    Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)CrossRefGoogle Scholar
  23. 23.
    Gao, S., Kong, S., Clarke, E.M.: \({\sf dReal{:}}\) an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 208–214. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339–342 (1999)Google Scholar
  25. 25.
    Giacobbe, M., Guet, C.C., Gupta, A., Henzinger, T.A., Paixão, T., Petrov, T.: Model checking gene regulatory networks. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 469–483. Springer, Heidelberg (2015)Google Scholar
  26. 26.
    Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From cardiac cells to genetic regulatory networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  27. 27.
    Jha, S., Shyamasundar, R.K.: Adapting biochemical kripke structures for distributed model checking. In: Priami, C., Ingólfsdóttir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 107–122. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Li, W., Zhong, L., He, Y., Meng, J., Yao, F., Guo, Y., Xu, C.: Multiple steady-states analysis and unstable operating point stabilization in homogeneous azeotropic distillation with intermediate entrainer. Ind. Eng. Chem. Res. 54(31), 7668–7686 (2015)CrossRefGoogle Scholar
  29. 29.
    Li, Y., Albarghouthi, A., Kincaid, Z., Gurfinkel, A., Chechik, M.: Symbolic optimization with SMT solvers. In: POPL 2014, pp. 607–618. ACM (2014)Google Scholar
  30. 30.
    Madsen, C., Shmarov, F., Zuliani, P.: BioPSy: an SMT-based tool for guaranteed parameter set synthesis of biological models. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 182–194. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  31. 31.
    Maler, O., Nickovic, D.: Monitoring temporal properties of continuous signals. In: Lakhnech, Y., Yovine, S. (eds.) FORMATS 2004 and FTRTFT 2004. LNCS, vol. 3253, pp. 152–166. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  32. 32.
    Milias-Argeitis, A., Engblom, S., Bauer, P., Khammash, M.: Stochastic focusing coupled with negative feedback enables robust regulation in biochemical reaction networks. J. R. Soc. Interface 12(113), 20150831 (2015)CrossRefGoogle Scholar
  33. 33.
    de Moura, L., Bjørner, N.S.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  34. 34.
    Raman, V., Donzé, A., Sadigh, D., Murray, R.M., Seshia, S.A.: Reactive synthesis from signal temporal logic specifications. In: HSCC 2015, pp. 239–248. ACM (2015)Google Scholar
  35. 35.
    Rizk, A., Batt, G., Fages, F., Soliman, S.: A general computational method for robustness analysis with applications to synthetic gene networks. Bioinformatics 25(12), i169–i178 (2009)CrossRefGoogle Scholar
  36. 36.
    Rosenfeld, N., Alon, U.: Response delays and the structure of transcription networks. J. Mol. Biol. 329(4), 645–654 (2003)CrossRefGoogle Scholar
  37. 37.
    Swat, M., Kel, A., Herzel, H.: Bifurcation analysis of the regulatory modules of the mammalian G1/S transition. Bioinformatics 20(10), 1506–1511 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Nikola Beneš
    • 1
  • Luboš Brim
    • 1
  • Martin Demko
    • 1
  • Samuel Pastva
    • 1
  • David Šafránek
    • 1
    Email author
  1. 1.Systems Biology Laboratory, Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations