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Statistical verification of PCTL using antithetic and stratified samples

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Abstract

In this work, we study the problem of statistically verifying Probabilistic Computation Tree Logic (PCTL) formulas on discrete-time Markov chains (DTMCs) with stratified and antithetic samples. We show that by properly choosing the representation of the DTMCs, semantically negatively correlated samples can be generated for a fraction of PCTL formulas via the stratified or antithetic sampling techniques. Using stratified or antithetic samples, we propose statistical verification algorithms with asymptotic correctness guarantees based on sequential probability ratio tests, and show that these algorithms are more sample-efficient than the algorithms using independent Monte Carlo sampling. Finally, the efficiency of the statistical verification algorithm with stratified and antithetic samples is demonstrated by numerical experiments on several benchmarks.

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Notes

  1. We did not use the ‘Exact’ engine, since it does not support bounded \(\mathcal {U}\) formulas

References

  1. Agresti A, Coull BA (1998) Approximate is better than “exact” for interval estimation of binomial proportions. Am Stat 52(2):119–126

    MathSciNet  Google Scholar 

  2. Clarke EM, Zuliani P (2011) Statistical model checking for cyber-physical systems. Automated technology for verification and analysis. Springer, Berlin, pp 1–12

    MATH  Google Scholar 

  3. D’Argenio P, Jeannet B, Jensen H, Larsen K (2001) Reachability analysis of probabilistic systems by successive refinements. In: de Alfaro L, Gilmore S (eds) Proceedings of 1st joint international workshop on process algebra and probabilistic methods, performance modelling and verification (PAPM/PROBMIV’01). Springer, LNCS, vol 2165, pp 39–56

  4. Even S, Goldreich O, Lempel A (1985) A randomized protocol for signing contracts. Commun ACM 28(6):637–647

    Article  MathSciNet  Google Scholar 

  5. Helmink L, Sellink M, Vaandrager F (1994) Proof-checking a data link protocol. In: Barendregt H, Nipkow T (eds) Proceedings of international workshop on types for proofs and programs (TYPES’93). Springer, LNCS, vol 806, pp 127–165

  6. Henriques D, Martins JG, Zuliani P, Platzer A, Clarke EM (2012) Statistical model checking for markov decision processes. In: 2012 Ninth international conference on quantitative evaluation of systems, pp 84–93

  7. Hermanns H, Nielson F, Jansen DN, Zhang L (2012) Efficient csl model checking using stratification. Log Methods Comput Sci 8:1–18

    MathSciNet  MATH  Google Scholar 

  8. Kwiatkowska M, Norman G, Parker D (2011) Prism 4.0: Verification of probabilistic real-time systems. In: International conference on computer aided verification. Springer, pp 585–591

  9. Larsen KG, Legay A (2016) Statistical model checking: past, present, and future. Leveraging applications of formal methods, verification and validation: foundational techniques. Springer, Cham, pp 3–15

    Chapter  Google Scholar 

  10. Liu J (2008) Monte Carlo strategies in scientific computing. Springer, Cham

    MATH  Google Scholar 

  11. Maginnis PA, West M, Dullerud GE (2016) Variance-reduced simulation of lattice discrete-time markov chains with applications in reaction networks. J Comput Phys 322:400–414

    Article  MathSciNet  Google Scholar 

  12. Norman G, Shmatikov V (2006) Analysis of probabilistic contract signing. J Comput Secur 14(6):561–589

    Article  Google Scholar 

  13. Reiter M, Rubin A (1998) Crowds: anonymity for web transactions. ACM Trans Inf Syst Secur (TISSEC) 1(1):66–92

    Article  Google Scholar 

  14. Roohi N, Wang Y, West M, Dullerud GE, Viswanathan M (2017) Statistical verification of the Toyota powertrain control verification benchmark. In: Proceedings of the 20th international conference on hybrid systems: computation and control. ACM, pp 65–70

  15. Sen K, Viswanathan M, Agha G (2004) Statistical model checking of black-box probabilistic systems. In: Alur R, Peled DA (eds) computer aided verification. Springer, Berlin, Heidelberg, no. 3114 in Lecture Notes in Computer Science, pp 202–215

    Chapter  Google Scholar 

  16. Sen K, Viswanathan M, Agha G (2005) On statistical model checking of stochastic systems. In: Etessami K, Rajamani SK (eds) Computer aided verification. Springer, Berlin, Heidelberg, no. 3576 in Lecture Notes in Computer Science, pp 266–280

    Chapter  Google Scholar 

  17. Sen K, Viswanathan M, Agha G (2005) Vesta: A statistical model-checker and analyzer for probabilistic systems. In: Second international conference on the quantitative evaluation of systems, 2005, pp 251–252

  18. Shmatikov V (2002) Probabilistic analysis of anonymity. In: Proceedings of the 15th IEEE computer security foundations workshop (CSFW’02). IEEE Computer Society Press, pp 119–128

  19. Shmatikov V (2004) Probabilistic model checking of an anonymity system. J Comput Secur 12(3/4):355–377

    Article  Google Scholar 

  20. Tony Cai T (2005) One-sided confidence intervals in discrete distributions. J Stat Plan Inference 131(1):63–88

    Article  MathSciNet  Google Scholar 

  21. Wang Y, Roohi N, West M, Viswanathan M, Dullerud GE (2015) A mori-zwanzig and mitl based approach to statistical verification of continuous-time dynamical systems. IFAC-PapersOnLine 48(27):267–273

    Article  Google Scholar 

  22. Wang Y, Roohi N, West M, Viswanathan M, Dullerud GE (2015) Statistical verification of dynamical systems using set oriented methods. In: Proceedings of the 18th international conference on hybrid systems: computation and control. ACM, New York, HSCC ’15, pp 169–178

  23. Wang Y, Roohi N, West M, Viswanathan M, Dullerud GE (2016) Verifying continuous-time stochastic hybrid systems via mori-zwanzig model reduction. In: 2016 IEEE 55th conference on decision and control (CDC), pp 3012–3017

  24. Wang Y, Roohi N, West M, Viswanathan M, Dullerud GE (2018) Statistical verification of pctl using stratified samples. IFAC-PapersOnLine 51(16):85–90

    Article  Google Scholar 

  25. Younes HLS (2005) Ymer: a statistical model checker. In: Etessami K, Rajamani SK (eds) Computer aided verification. Springer, Berlin, no. 3576 in Lecture Notes in Computer Science, pp 429–433

    Chapter  Google Scholar 

  26. Younes HLS, Simmons RG (2006) Statistical probabilistic model checking with a focus on time-bounded properties. Inf Comput 204(9):1368–1409

    Article  MathSciNet  Google Scholar 

  27. Zuliani P, Baier C, Clarke EM (2012) Rare-event verification for stochastic hybrid systems. In: Proceedings of the 15th ACM international conference on hybrid systems: computation and control. ACM, New York, HSCC ’12, pp 217–226

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Acknowledgements

This work was supported by NSF CPS Grant 1329991 and AFOSR Grant FA9550-15-1-0059.

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Authors and Affiliations

  1. Electrical and Computer Engineering, Duke University, Durham, USA

    Yu Wang

  2. Computer Science and Engineering, University of California San Diego, San Diego, USA

    Nima Roohi

  3. Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, USA

    Matthew West

  4. Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA

    Mahesh Viswanathan

  5. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, USA

    Geir E. Dullerud

Authors
  1. Yu Wang
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  2. Nima Roohi
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  3. Matthew West
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  4. Mahesh Viswanathan
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  5. Geir E. Dullerud
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Correspondence to Yu Wang.

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Wang, Y., Roohi, N., West, M. et al. Statistical verification of PCTL using antithetic and stratified samples. Form Methods Syst Des 54, 145–163 (2019). https://doi.org/10.1007/s10703-019-00339-8

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  • DOI: https://doi.org/10.1007/s10703-019-00339-8

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