Advertisement

Synthesis in pMDPs: A Tale of 1001 Parameters

  • Murat Cubuktepe
  • Nils Jansen
  • Sebastian JungesEmail author
  • Joost-Pieter Katoen
  • Ufuk Topcu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11138)

Abstract

This paper considers parametric Markov decision processes (pMDPs) whose transitions are equipped with affine functions over a finite set of parameters. The synthesis problem is to find a parameter valuation such that the instantiated pMDP satisfies a (temporal logic) specification under all strategies. We show that this problem can be formulated as a quadratically-constrained quadratic program (QCQP) and is non-convex in general. To deal with the NP-hardness of such problems, we exploit a convex-concave procedure (CCP) to iteratively obtain local optima. An appropriate interplay between CCP solvers and probabilistic model checkers creates a procedure—realized in the tool PROPheSY—that solves the synthesis problem for models with thousands of parameters.

References

  1. 1.
    Aflaki, S., Volk, M., Bonakdarpour, B., Katoen, J.P., Storjohann, A.: Automated fine tuning of probabilistic self-stabilizing algorithms. In: SRDS, pp. 94–103. IEEE CS (2017)Google Scholar
  2. 2.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amato, C., Bernstein, D.S., Zilberstein, S.: Solving POMDPs using quadratically constrained linear programs. In: AAMAS, pp. 341–343. ACM (2006)Google Scholar
  4. 4.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)Google Scholar
  5. 5.
    Bartocci, E., Grosu, R., Katsaros, P., Ramakrishnan, C.R., Smolka, S.A.: Model repair for probabilistic systems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 326–340. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19835-9_30CrossRefzbMATHGoogle Scholar
  6. 6.
    Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng. 8(1) (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefGoogle Scholar
  8. 8.
    Burer, S., Saxena, A.: The MILP road to MIQCP. Mixed Integer Nonlinear Programming, pp. 373–405 (2012)Google Scholar
  9. 9.
    Calinescu, R., Ghezzi, C., Kwiatkowska, M., Mirandola, R.: Self-adaptive software needs quantitative verification at runtime. Commun. ACM 55(9), 69–77 (2012)CrossRefGoogle Scholar
  10. 10.
    Chen, T., Hahn, E.M., Han, T., Kwiatkowska, M., Qu, H., Zhang, L.: Model repair for Markov decision processes. In: TASE, pp. 85–92. IEEE CS (2013)Google Scholar
  11. 11.
    Cubuktepe, M., Jansen, N., Junges, S., Katoen, J.P., Topcu, U.: Synthesis in pMDPs: a tale of 1001 parameters. CoRR abs/1803.02884 (2018)Google Scholar
  12. 12.
    Cubuktepe, M., et al.: Sequential convex programming for the efficient verification of parametric MDPs. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 133–150. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54580-5_8CrossRefGoogle Scholar
  13. 13.
    Daws, C.: Symbolic and parametric model checking of discrete-time Markov chains. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 280–294. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31862-0_21CrossRefzbMATHGoogle Scholar
  14. 14.
    Dehnert, C., et al.: PROPhESY: a probabilistic parameter synthesis tool. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 214–231. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21690-4_13CrossRefGoogle Scholar
  15. 15.
    Dehnert, C., Junges, S., Katoen, J.-P., Volk, M.: A storm is coming: a modern probabilistic model checker. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 592–600. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63390-9_31CrossRefGoogle Scholar
  16. 16.
    Diamond, S., Boyd, S.: CVXPY: a python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17(83), 1–5 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duflot, M., et al.: Probabilistic model checking of the CSMA/CD protocol using PRISM and APMC. Electr. Notes TCS 128(6), 195–214 (2005)zbMATHGoogle Scholar
  18. 18.
    Filieri, A., Tamburrelli, G., Ghezzi, C.: Supporting self-adaptation via quantitative verification and sensitivity analysis at run time. IEEE Trans. Softw. Eng. 42(1), 75–99 (2016)CrossRefGoogle Scholar
  19. 19.
    Gainer, P., Hahn, E.M., Schewe, S.: Incremental verification of parametric and reconfigurable Markov chains. CoRR abs/1804.01872 (2018)Google Scholar
  20. 20.
    Gurobi Optimization Inc.: Gurobi optimizer reference manual. http://www.gurobi.com (2013)
  21. 21.
    Hahn, E.M., Hermanns, H., Wachter, B., Zhang, L.: PARAM: a model checker for parametric markov models. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 660–664. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14295-6_56CrossRefGoogle Scholar
  22. 22.
    Hahn, E.M., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. STTT 13(1), 3–19 (2010)CrossRefGoogle Scholar
  23. 23.
    Hahn, E.M., Li, Y., Schewe, S., Turrini, A., Zhang, L.: iscasMc: a web-based probabilistic model checker. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 312–317. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-06410-9_22CrossRefGoogle Scholar
  24. 24.
    Hutschenreiter, L., Baier, C., Klein, J.: Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination. GandALF. EPTCS 256, 16–30 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jovanović, D., de 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).  https://doi.org/10.1007/978-3-642-31365-3_27CrossRefGoogle Scholar
  26. 26.
    Junges, S., et al.: Finite-state controllers of POMDPs using parameter synthesis. In: UAI. AUAI Press, Canada (2018), to appearGoogle Scholar
  27. 27.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22110-1_47CrossRefGoogle Scholar
  28. 28.
    Lanotte, R., Maggiolo-Schettini, A., Troina, A.: Parametric probabilistic transition systems for system design and analysis. Form. Asp. Comput. 19(1), 93–109 (2007)CrossRefGoogle Scholar
  29. 29.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lipp, T., Boyd, S.: Variations and extension of the convex-concave procedure. Optim. Eng. 17(2), 263–287 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    O’Donoghue, B., Chu, E., Parikh, N., Boyd, S.: Conic optimization via operator splitting and homogeneous self-dual embedding. J. Optim. Theory Appl. 169(3), 1042–1068 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Park, J., Boyd, S.: General heuristics for nonconvex quadratically constrained quadratic programming. arXiv preprint arXiv:1703.07870 (2017)
  33. 33.
    Shen, X., Diamond, S., Gu, Y., Boyd, S.: Disciplined convex-concave programming. In: CDC, pp. 1009–1014. IEEE (2016)Google Scholar
  34. 34.
    Su, G., Rosenblum, D.S., Tamburrelli, G.: Reliability of run-time quality-of-service evaluation using parametric model checking. In: ICSE, pp. 073–84. ACM (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Murat Cubuktepe
    • 1
  • Nils Jansen
    • 2
  • Sebastian Junges
    • 3
    Email author
  • Joost-Pieter Katoen
    • 3
  • Ufuk Topcu
    • 1
  1. 1.The University of Texas at AustinAustinUSA
  2. 2.Radboud UniversityNijmegenThe Netherlands
  3. 3.RWTH Aachen UniversityAachenGermany

Personalised recommendations