Formal Methods in System Design

, Volume 54, Issue 2, pp 145–163 | Cite as

Statistical verification of PCTL using antithetic and stratified samples

  • Yu WangEmail author
  • Nima Roohi
  • Matthew West
  • Mahesh Viswanathan
  • Geir E. Dullerud


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.


Markov chains Temporal logic Variance reduction Sequential probability ratio test 



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


  1. 1.
    Agresti A, Coull BA (1998) Approximate is better than “exact” for interval estimation of binomial proportions. Am Stat 52(2):119–126MathSciNetGoogle Scholar
  2. 2.
    Clarke EM, Zuliani P (2011) Statistical model checking for cyber-physical systems. Automated technology for verification and analysis. Springer, Berlin, pp 1–12zbMATHGoogle Scholar
  3. 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 39–56Google Scholar
  4. 4.
    Even S, Goldreich O, Lempel A (1985) A randomized protocol for signing contracts. Commun ACM 28(6):637–647MathSciNetCrossRefGoogle Scholar
  5. 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 127–165Google Scholar
  6. 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 84–93Google Scholar
  7. 7.
    Hermanns H, Nielson F, Jansen DN, Zhang L (2012) Efficient csl model checking using stratification. Log Methods Comput Sci 8:1–18MathSciNetzbMATHGoogle Scholar
  8. 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 585–591Google Scholar
  9. 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 3–15CrossRefGoogle Scholar
  10. 10.
    Liu J (2008) Monte Carlo strategies in scientific computing. Springer, ChamzbMATHGoogle Scholar
  11. 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 322:400–414MathSciNetCrossRefGoogle Scholar
  12. 12.
    Norman G, Shmatikov V (2006) Analysis of probabilistic contract signing. J Comput Secur 14(6):561–589CrossRefGoogle Scholar
  13. 13.
    Reiter M, Rubin A (1998) Crowds: anonymity for web transactions. ACM Trans Inf Syst Secur (TISSEC) 1(1):66–92CrossRefGoogle Scholar
  14. 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 65–70Google Scholar
  15. 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 202–215CrossRefGoogle Scholar
  16. 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 266–280CrossRefGoogle Scholar
  17. 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 251–252Google Scholar
  18. 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 119–128Google Scholar
  19. 19.
    Shmatikov V (2004) Probabilistic model checking of an anonymity system. J Comput Secur 12(3/4):355–377CrossRefGoogle Scholar
  20. 20.
    Tony Cai T (2005) One-sided confidence intervals in discrete distributions. J Stat Plan Inference 131(1):63–88MathSciNetCrossRefGoogle Scholar
  21. 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):267–273CrossRefGoogle Scholar
  22. 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 169–178Google Scholar
  23. 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 3012–3017Google Scholar
  24. 24.
    Wang Y, Roohi N, West M, Viswanathan M, Dullerud GE (2018) Statistical verification of pctl using stratified samples. IFAC-PapersOnLine 51(16):85–90CrossRefGoogle Scholar
  25. 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 429–433CrossRefGoogle Scholar
  26. 26.
    Younes HLS, Simmons RG (2006) Statistical probabilistic model checking with a focus on time-bounded properties. Inf Comput 204(9):1368–1409MathSciNetCrossRefGoogle Scholar
  27. 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 217–226Google Scholar

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Authors and Affiliations

  1. 1.Electrical and Computer EngineeringDuke UniversityDurhamUSA
  2. 2.Computer Science and EngineeringUniversity of California San DiegoSan DiegoUSA
  3. 3.Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  5. 5.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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