Skip to main content
SpringerLink
Log in

Synthesis in pMDPs: A Tale of 1001 Parameters

  • Conference paper
  • First Online:
Automated Technology for Verification and Analysis (ATVA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,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.

Supported by the grants ONR N000141613165, NASA NNX17AD04G and AFRL FA8650-15-C-2546

Supported by the CDZ project CAP (GZ 1023), and the DFG RTG 2236 “UnRAVeL”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MathSciNet  Google Scholar 

  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. Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)

    Google Scholar 

  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_30

    Chapter  MATH  Google Scholar 

  6. Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng. 8(1) (2007)

    Article  MathSciNet  Google Scholar 

  7. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  8. Burer, S., Saxena, A.: The MILP road to MIQCP. Mixed Integer Nonlinear Programming, pp. 373–405 (2012)

    Google Scholar 

  9. Calinescu, R., Ghezzi, C., Kwiatkowska, M., Mirandola, R.: Self-adaptive software needs quantitative verification at runtime. Commun. ACM 55(9), 69–77 (2012)

    Article  Google Scholar 

  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. 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. 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_8

    Chapter  Google Scholar 

  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_21

    Chapter  MATH  Google Scholar 

  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_13

    Chapter  Google Scholar 

  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_31

    Chapter  Google Scholar 

  16. Diamond, S., Boyd, S.: CVXPY: a python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17(83), 1–5 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  19. Gainer, P., Hahn, E.M., Schewe, S.: Incremental verification of parametric and reconfigurable Markov chains. CoRR abs/1804.01872 (2018)

    Google Scholar 

  20. Gurobi Optimization Inc.: Gurobi optimizer reference manual. http://www.gurobi.com (2013)

  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_56

    Chapter  Google Scholar 

  22. Hahn, E.M., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. STTT 13(1), 3–19 (2010)

    Article  Google Scholar 

  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_22

    Chapter  Google Scholar 

  24. Hutschenreiter, L., Baier, C., Klein, J.: Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination. GandALF. EPTCS 256, 16–30 (2017)

    Article  MathSciNet  Google Scholar 

  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_27

    Chapter  Google Scholar 

  26. Junges, S., et al.: Finite-state controllers of POMDPs using parameter synthesis. In: UAI. AUAI Press, Canada (2018), to appear

    Google Scholar 

  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_47

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  29. Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)

    Article  MathSciNet  Google Scholar 

  30. Lipp, T., Boyd, S.: Variations and extension of the convex-concave procedure. Optim. Eng. 17(2), 263–287 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  32. Park, J., Boyd, S.: General heuristics for nonconvex quadratically constrained quadratic programming. arXiv preprint arXiv:1703.07870 (2017)

  33. Shen, X., Diamond, S., Gu, Y., Boyd, S.: Disciplined convex-concave programming. In: CDC, pp. 1009–1014. IEEE (2016)

    Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. The University of Texas at Austin, Austin, TX, USA

    Murat Cubuktepe & Ufuk Topcu

  2. Radboud University, Nijmegen, The Netherlands

    Nils Jansen

  3. RWTH Aachen University, Aachen, Germany

    Sebastian Junges & Joost-Pieter Katoen

Authors
  1. Murat Cubuktepe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nils Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebastian Junges
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joost-Pieter Katoen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ufuk Topcu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastian Junges .

Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Shuvendu K. Lahiri

  2. University of Southern California, Los Angeles, CA, USA

    Chao Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cubuktepe, M., Jansen, N., Junges, S., Katoen, JP., Topcu, U. (2018). Synthesis in pMDPs: A Tale of 1001 Parameters. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-01090-4_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-01089-8

  • Online ISBN: 978-3-030-01090-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation