Skip to main content
SpringerLink
Log in
Book cover

Formal System Verification pp 73–121Cite as

Probabilistic Model Checking: Advances and Applications

  • Chapter
  • First Online:

Abstract

Probabilistic model checking is a powerful technique for formally verifying quantitative properties of systems that exhibit stochastic behaviour. Such systems are found in many application domains: for example, probabilistic behaviour may arise due to the presence of failures in unreliable hardware, message loss in wireless communication channels, or the use of randomisation in distributed protocols. This chapter starts with an introduction to the technique of probabilistic model checking. We then survey some recent advances in the area, including controller synthesis, compositional verification, probabilistic real-time systems and parametric model checking. We illustrate the application of the various techniques with a combination of toy examples and descriptions of larger case studies. The chapter concludes with a discussion of some of the key challenges in the field.

This is a preview of subscription content, 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   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Notes

  1. 1.

    The bounded until operator \(\psi _1{\ {\mathtt U}^{\leqslant k}\ }\psi _2\) is not usually included in the syntax of LTL, but it can be derived from other operators so its inclusion is not problematic.

  2. 2.

    Strictly speaking, the definition of reward operators differs in [28].

  3. 3.

    In fact, state and action-labelled variants of temporal logics are equally expressive [90].

References

  1. E. Ábrahám, B. Becker, C. Dehnert, N. Jansen, J.-P. Katoen, R. Wimmer, Counterexample generation for discrete-time Markov models: an introductory survey, in Formal Methods for the Design of Computer, Communication, and Software Systems (SFM’14), ed. By M. Bernardo, F. Damiani, R. Haehnle, E. Johnsen, I. Schaefer. LNCS, vol. 8483 (Springer, 2014), pp. 65–121

    Google Scholar 

  2. R. Alur, C. Courcoubetis, D. Dill, Model-checking for probabilistic real-time systems, in Proceedings of the 19th International Colloq Automata, Languages and Programming (ICALP’91). LNCS, vol. 510, (Springer, 1991), pp. 115–136

    Google Scholar 

  3. R. Alur, C. Courcoubetis, D. Dill, Model checking in dense real time. Inf. Comput. 104(1), 2–34 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Alur, T. Dang, F. Ivancic, Predicate abstraction for reachability analysis of hybrid systems. ACM Trans. Embed. Comput. Syst. 5(1), 152–199 (2006)

    Article  MATH  Google Scholar 

  5. R. Alur, D. Dill, A theory of timed automata. Theor. Comput. Sci. 126, 183–235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Alur, T. Henzinger, O. Kupferman, Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Alur, T. Henzinger, S. Rajamani, Symbolic exploration of transition hierarchies, in Proceedings of the 4th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’98). LNCS, vol. 1384, (Springer, 1998), pp. 330–344

    Google Scholar 

  8. R. Alur, S. La Torre, G. Pappas, Optimal paths in weighted timed automata. Theor. Comput. Sci. 318(3), 297–322 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Alur, A. Trivedi, Relating average and discounted costs for quantitative analysis of timed systems, in Proceedings of the 11th International Conference Embedded Software (EMSOFT’11) (ACM, 2011), pp. 165–174

    Google Scholar 

  10. E. André, T. Chatain, E. Encrenaz, L. Fribourg, An inverse method for parametric timed automata. Int. J. Found. Comput. Sci. 20(5), 819–836 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. André, L. Fribourg, J. Sproston, An extension of the inverse method to probabilistic timed automata. Form. Methods Syst. Des. 42(2), 119–145 (2013)

    Article  MATH  Google Scholar 

  12. C. Baier, E. Clarke, V. Hartonas-Garmhausen, M. Kwiatkowska, M. Ryan, Symbolic model checking for probabilistic processes, in Proceedings of the 24th International Colloquium Automata, Languages and Programming (ICALP’97), ed. By P. Degano, R. Gorrieri, A. Marchetti-Spaccamela. LNCS, vol. 1256 (Springer, 1997), pp. 430–440

    Google Scholar 

  13. C. Baier, M. Größer, M. Leucker, B. Bollig, F. Ciesinski, Controller synthesis for probabilistic systems, in Proceedings of the 3rd IFIP International Conference Theoretical Computer Science (TCS’06), ed. By J.-J. Lévy, E. Mayr, J. Mitchell (Kluwer, 2004), pp. 493–5062

    Google Scholar 

  14. C. Baier, B. Haverkort, H. Hermanns, J.-P. Katoen, Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Softw. Eng. 29(6), 524–541 (2003)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. E. Bartocci, R. Grosu, P. Katsaros, C. Ramakrishnan, S. Smolka, Model repair for probabilistic systems, in Proceedings of the 17th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’11), ed. By P. Abdulla, K. Leino. LNCS, vol. 6605 (Springer, 2011), pp. 326–340

    Google Scholar 

  17. D. Beauquier, Probabilistic timed automata. Theor. Comput. Sci. 292(1), 65–84 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Behrmann, A. Cougnard, A. David, E. Fleury, K. Larsen, D. Lime, UPPAAL-Tiga: time for playing games!, in Proceedings of the 19th International Conference Computer Aided Verification (CAV’07). LNCS, vol. 4590 (Springer, 2007), pp. 121–125

    Google Scholar 

  19. G. Behrmann, A. Fehnker, T. Hune, K. Larsen, P. Pettersson, J. Romijn, Efficient guiding towards cost-optimality in UPPAAL, in Proceedings of the 7th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’01), ed. By T. Margaria, W. Yi. LNCS, vol. 2031 (Springer, 2001), pp. 174–188

    Google Scholar 

  20. R. Bellman, Dynamic Programming (Princeton University Press, New Jersey, 1957)

    MATH  Google Scholar 

  21. P. Billingsley, Probability and Measure (Wiley, New Jersey, 1995)

    MATH  Google Scholar 

  22. P. Bouyer, U. Fahrenberg, K. Larsen, N. Markey, Quantitative analysis of real-time systems using priced timed automata. Commu. ACM 54(9), 78–87 (2011)

    Article  Google Scholar 

  23. T. Brázdil, V. Brožek, V. Forejt, A. Kučera, Stochastic games with branching-time winning objectives, in Proceedings of the 21th IEEE Symposium Logic in Computer Science (LICS’06) (IEEE Computer Society, 2006), pp. 349–358

    Google Scholar 

  24. L. Brim, M. Češka, D.V.S. Dražan, Exploring parameter space of stochastic biochemical systems using quantitative model checking, in Proceedings of the 25th International Conference Computer Aided Verification (CAV’13). LNCS, vol. 8044 (Springer, 2013), pp. 107–123

    Google Scholar 

  25. P. Buchholz, E.M. Hahn, H. Hermanns, L. Zhang, Model checking algorithms for CTMDPs, in Proceedings of the 23rd International Conference Computer Aided Verification (CAV’11), ed. By G. Gopalakrishnan, S. Qadeer. LNCS, vol. 6806 (Springer, 2011), pp. 225–242

    Google Scholar 

  26. M. Češka, F. Dannenberg, M. Kwiatkowska, N. Paoletti, Precise parameter synthesis for stochastic biochemical systems, in Proceedings of the 12th International Conference Computational Methods in Systems Biology (CMSB’14), ed. By P. Mendes, J. Dada, K. Smallbone. LNCS/LNBI, vol. 8859 (Springer, 2014), pp. 86–98

    Google Scholar 

  27. K. Chatterjee, M. Chmelík, R. Gupta, A. Kanodia, Qualitative analysis of POMDPs with temporal logic specifications for robotics applications, in Proceedings of the IEEE International Conference Robotics and Automation, (ICRA’15) (IEEE Computer Society, 2015), pp. 325–330

    Google Scholar 

  28. T. Chen, V. Forejt, M. Kwiatkowska, D. Parker, A. Simaitis, Automatic verification of competitive stochastic systems. Form. Methods Syst. Des. 43(1), 61–92 (2013)

    Article  MATH  Google Scholar 

  29. S. Cheshire, B. Adoba, E. Gutterman, Dynamic configuration of IPv4 link local addresses. http://www.ietf.org/rfc/rfc3927.txtwww.ietf.org/rfc/rfc3927.txt

  30. E. Clarke, O. Grumberg, S. Jha, Y. Lu, H. Veith, Counterexample-guided abstraction refinement, in Proceedings of the 12th International Conference Computer Aided Verification (CAV’00), ed. By A. Emerson, A. Sistla. LNCS, vol. 1855 (Springer, 2000), pp. 154–169

    Google Scholar 

  31. A. Condon, The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Condon, On algorithms for simple stochastic games, Advances in computational complexity theory. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 13, 51–73 (1993)

    Google Scholar 

  33. M. Daniele, F. Giunchiglia, M. Vardi, Improved automata generation for linear temporal logic, in Proceedings of the 11th International Conference Computer Aided Verification (CAV’99), ed. By N. Halbwachs, D. Peled. LNCS, vol. 1633 (Springer, 1999), pp. 249–260

    Google Scholar 

  34. P. D’Argenio, B. Jeannet, H. Jensen, K. Larsen, in Reachability analysis of probabilistic systems by successive refinements, Proceedings of the 1st Joint International Workshop Process Algebra and Probabilistic Methods, Performance Modelling and Verification (PAPM/PROBMIV’01), ed. By L. de Alfaro, S. Gilmore. LNCS, vol. 2165 (Springer, 2001), pp. 39–56

    Google Scholar 

  35. C. Daws, Symbolic and parametric model checking of discrete-time Markov chains, in Proceedings of the 1st International Colloquium Theoretical Aspects of Computing (ICTAC’04), ed. By Z. Liu, K. Araki. LNCS, vol. 3407 (Springer, 2004), pp. 280–294

    Google Scholar 

  36. L. de Alfaro, Formal Verification of Probabilistic Systems. Ph.D. thesis, Stanford University, 1997

    Google Scholar 

  37. C. Dehnert, S. Junges, N. Jansen, F. Corzilius, M. Volk, H. Bruintjes, J.-P. Katoen, E. Ábrahám, PROPhESY: a PRObabilistic ParamEter SYnthesis tool, in Proceedings of the 27th International Conference Computer Aided Verification (CAV’15). LNCS, vol. 9206 (Springer, 2015), pp. 214–231

    Google Scholar 

  38. J. Desharnais, J. Assouramou, Analysis of non-linear probabilistic hybrid systems, in Proceedings of the 9th Workshop Quantitative Aspects of Programming Languages (QAPL’11). EPTCS, vol. 57 (2011), pp. 104–119

    Google Scholar 

  39. A. Donaldson, A. Miller, Symmetry reduction for probabilistic model checking using generic representatives, in Proceedings of the 4th International Symposium Automated Technology for Verification and Analysis (ATVA’06), ed. By S. Graf, W. Zhang. LNCS, vol. 4218 (Springer, 2006), pp. 9–23

    Google Scholar 

  40. S. Donatelli, S. Haddad, J. Sproston, Model checking timed and stochastic properties with CSL\({}^{\text{ta}}\). IEEE Trans. Softw. Eng. 35(2), 224–240 (2008)

    Google Scholar 

  41. C. Eisentraut, H. Hermanns, L. Zhang, On probabilistic automata in continuous time, in Proceedings of the 25th Annual IEEE Symposium Logic in Computer Science (LICS’10) (IEEE Computer Society, 2010), pp. 342–351

    Google Scholar 

  42. K. Etessami, M. Kwiatkowska, M. Vardi, M. Yannakakis, Multi-objective model checking of Markov decision processes. Logical Methods Comput. Sci. 4(4), 1–21 (2008)

    MathSciNet  MATH  Google Scholar 

  43. A. Filieri, G. Tamburrelli, C. Ghezzi, Supporting self-adaptation via quantitative verification and sensitivity analysis at run time. IEEE Trans. Softw. Eng. 42(1), 75–99 (2016)

    Article  Google Scholar 

  44. V. Forejt, M. Kwiatkowska, G. Norman, D. Parker, Automated verification techniques for probabilistic systems, in Formal Methods for Eternal Networked Software Systems (SFM’11), ed. By M. Bernardo, V. Issarny. LNCS, vol. 6659 (Springer, 2011), pp. 53–113

    Google Scholar 

  45. V. Forejt, M. Kwiatkowska, G. Norman, D. Parker, H. Qu, Quantitative multi-objective verification for probabilistic systems, in Proceedings of the 17th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’11), ed. By P. Abdulla, K. Leino. LNCS, vol. 6605 (Springer, 2011), pp. 112–127

    Google Scholar 

  46. V. Forejt, M. Kwiatkowska, D. Parker, Pareto curves for probabilistic model checking, in Proceedings of the 10th International Symposium Automated Technology for Verification and Analysis (ATVA’12), ed. By S. Chakraborty, M. Mukund. LNCS, vol. 7561 (Springer, 2012), pp. 317–332

    Google Scholar 

  47. M. Fränzle, T. Teige, A. Eggers, Engineering constraint solvers for automatic analysis of probabilistic hybrid automata. J. Logic Algebr. Progr. 79(7), 436–466 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Graf, H. Saidi, Construction of abstract state graphs with PVS, in Proceedings of the 9th International Conference Computer Aided Verification (CAV’97), ed. By O. Grumberg. LNCS, vol. 1254 (Springer, 1997), pp. 72–83

    Google Scholar 

  49. D. Gross, D. Miller, The randomization technique as a modeling tool and solution procedure for transient Markov processes. Oper. Res. 32(2), 343–361 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Größer, C. Baier, Partial order reduction for Markov decision processes: a survey, in Proceedings of the 4th International Symposium Formal Methods for Component and Objects (FMCO’05), ed. By F. de Boer, M. Bonsangue, S. Graf, W.-P. de Roever. LNCS, vol. 4111 (Springer, 2006), pp. 408–427

    Google Scholar 

  51. E.M. Hahn, T. Han, L. Zhang, Synthesis for PCTL in parametric Markov decision processes, in Proceedings of the 3rd NASA Formal Methods Symposium (NFM’11). LNCS, vol. 6617 (Springer, 2011)

    Google Scholar 

  52. E.M. Hahn, H. Hermanns, B. Wachter, L. Zhang, PARAM: a model checker for parametric Markov models, in Proceedings of the 22nd International Conference Computer Aided Verification (CAV’10). LNCS, vol. 6174 (Springer, 2010), pp. 660–664

    Google Scholar 

  53. E.M. Hahn, H. Hermanns, B. Wachter, L. Zhang, PASS: abstraction refinement for infinite probabilistic models, in Proceedings of the 16th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’10), ed. By J. Esparza, R. Majumdar. LNCS, vol. 6105 (Springer, 2010), pp. 353–357

    Google Scholar 

  54. E.M. Hahn, H. Hermanns, L. Zhang, Probabilistic reachability for parametric Markov models. Int. J. Softw. Tools Technol. Trans. (STTT) 13(1), 3–19 (2011)

    Article  Google Scholar 

  55. E.M. Hahn, Y. Li, S. Schewe, A. Turrini, L. Zhang, iscasMc: a web-based probabilistic model checker, in Proceedings of the 19th International Symposium on Formal Methods (FM’14) (2014), pp. 312–317

    Google Scholar 

  56. E.M. Hahn, G. Norman, D. Parker, B. Wachter, L. Zhang, Game-based abstraction and controller synthesis for probabilistic hybrid systems, in Proceedings of the 8th International Conference Quantitative Evaluation of SysTems (QEST’11) (IEEE Computer Society Press, 2011), pp. 69–78

    Google Scholar 

  57. J. Han, P. Jonker, A system architecture solution for unreliable nanoelectronic devices. IEEE Trans. Nanotechnol. 1, 201–208 (2002)

    Article  Google Scholar 

  58. T. Han, J.-P. Katoen, B. Damman, Counterexample generation in probabilistic model checking. IEEE Trans. Softw. Eng. 35(2), 241–257 (2009)

    Google Scholar 

  59. T. Han, J.-P. Katoen, A. Mereacre, Approximate parameter synthesis for probabilistic time-bounded reachability, in Proceedings of the IEEE Real-Time Systems Symposium (RTSS 08) (IEEE Computer Society Press, 2008), pp. 173–182

    Google Scholar 

  60. H. Hansson, B. Jonsson, A logic for reasoning about time and reliability. Form. Asp. Comput. 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  61. A. Hartmanns, H. Hermanns, A modest approach to checking probabilistic timed automata, in Proceedings of the 6th International Conference Quantitative Evaluation of Systems (QEST’09) (2009). To appear

    Google Scholar 

  62. H. Hermanns, Interactive Markov Chains and the Quest for Quantified Quality. LNCS, vol. 2428 (Springer, New York, 2002)

    Google Scholar 

  63. H. Hermanns, B. Wachter, L. Zhang, Probabilistic CEGAR, in Proceedings of the 20th International Conference Computer Aided Verification (CAV’08), ed. By A. Gupta, S. Malik. LNCS, vol. 5123 (Springer, 2008), pp. 162–175

    Google Scholar 

  64. R. Howard, Dynamic Programming and Markov Processes (The MIT Press, Cambridge, 1960)

    Google Scholar 

  65. N. Jansen, F. Corzilius, M. Volk, R. Wimmer, E. Ábrahám, J.-P. Katoen, B. Becker, Accelerating parametric probabilistic verification, in Proceedings of the 11th International Conference Quantitative Evaluation of Systems (QEST’14) (2014), pp. 404–420

    Google Scholar 

  66. B. Jeannet, P. D’Argenio, K. Larsen, Rapture: a tool for verifying Markov decision processes, in Proceedings of the Tools Day, affiliated to 13th International Conference Concurrency Theory (CONCUR’02), ed. By I. Cerna. Technical Report FIMU-RS-2002-05, Faculty of Informatics Masaryk University (2002), pp. 84–98

    Google Scholar 

  67. A. Jensen, Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift 36, 87–91 (1953)

    MathSciNet  MATH  Google Scholar 

  68. H. Jensen, Model checking probabilistic real time systems, in Proceedings of the 7th Nordic Workshop Programming Theory (1996), pp. 247–261

    Google Scholar 

  69. A. Jovanovic, M. Kwiatkowska, Parameter synthesis for probabilistic timed automata using stochastic games, in Proceedings of the 8th International Workshop Reachability Problems (RP’14), ed. By J. Ouaknine, I. Potapov, J. Worrell. LNCS, vol. 8762, (Springer, 2014), pp. 176–189

    Google Scholar 

  70. M. Jurdziński, M. Kwiatkowska, G. Norman, A. Trivedi, Concavely-priced probabilistic timed automata, in Proceedings of the 20th International Conference Concurrency Theory (CONCUR’09), ed. By M. Bravetti, G. Zavattaro. LNCS, vol. 5710 (Springer, 2009), pp. 415–430

    Google Scholar 

  71. J.-P. Katoen, Probabilistic programming: a true challenge in verification, in Proceedings of the 13th International Symposium on Automated Technology for Verification and Analysis (ATVA’15). LNCS (Springer, 2015), pp. 1–3

    Google Scholar 

  72. J.-P. Katoen, T. Kemna, I. Zapreev, D. Jansen, Bisimulation minimisation mostly speeds up probabilistic model checking, in Proceedings of the 13th International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’07), ed. By O. Grumberg, M. Huth. LNCS, vol. 4424 (Springer, 2007), pp. 87–101

    Google Scholar 

  73. J.-P. Katoen, I. Zapreev, E.M. Hahn, H. Hermanns, D. Jansen, The ins and outs of the probabilistic model checker MRMC, in Proceedings of the 6th International Conference Quantitative Evaluation of Systems (QEST’09) (IEEE Computer Society Press, 2009), pp. 167–176

    Google Scholar 

  74. M. Kattenbelt, M. Kwiatkowska, G. Norman, D. Parker, Abstraction refinement for probabilistic software, in Proceedings of the 10th International Conference Verification, Model Checking, and Abstract Interpretation (VMCAI’09), ed. By N. Jones, M. Muller-Olm. LNCS, vol. 5403 (Springer, 2009), pp. 182–197

    Google Scholar 

  75. M. Kattenbelt, M. Kwiatkowska, G. Norman, D. Parker, A game-based abstraction-refinement framework for Markov decision processes. Form. Methods Syst. Des. 36(3), 246–280 (2010)

    Article  MATH  Google Scholar 

  76. J. Kemeny, J. Snell, A. Knapp, Denumerable Markov Chains, 2nd edn. (Springer, Heidelberg, 1976)

    Google Scholar 

  77. M. Kwiatkowska, G. Norman, D. Parker, Symmetry reduction for probabilistic model checking, in Proceedings of the 18th International Conference Computer Aided Verification (CAV’06), ed. By T. Ball, R. Jones. LNCS, vol. 4114 (Springer, 2006), pp. 234–248

    Google Scholar 

  78. M. Kwiatkowska, G. Norman, D. Parker, Stochastic model checking, in Formal Methods for the Design of Computer, Communication and Software Systems: Performance Evaluation (SFM’07), ed. By M. Bernardo, J. Hillston. LNCS (Tutorial Volume), vol. 4486 (Springer, 2007), pp. 220–270

    Google Scholar 

  79. M. Kwiatkowska, G. Norman, D. Parker, Stochastic games for verification of probabilistic timed automata, in Proceedings of the 7th International Conference Formal Modelling and Analysis of Timed Systems (FORMATS’09), ed. By J. Ouaknine, F. Vaandrager. LNCS, vol. 5813 (Springer, 2009), pp. 212–227

    Google Scholar 

  80. M. Kwiatkowska, G. Norman, D. Parker, PRISM 4.0: verification of probabilistic real-time systems, in Proceedings of the 23rd International Conference Computer Aided Verification (CAV’11), ed. By G. Gopalakrishnan, S. Qadeer. LNCS, vol. 6806 (Springer, 2011), pp. 585–591

    Google Scholar 

  81. M. Kwiatkowska, G. Norman, D. Parker, H. Qu, Compositional probabilistic verification through multi-objective model checking. Inf. Comput. 232, 38–65 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  82. M. Kwiatkowska, G. Norman, D. Parker, J. Sproston, Performance analysis of probabilistic timed automata using digital clocks. Form. Methods Syst. Des. 29, 33–78 (2006)

    Article  MATH  Google Scholar 

  83. M. Kwiatkowska, G. Norman, R. Segala, J. Sproston, Verifying quantitative properties of continuous probabilistic timed automata, in In Proceedings of the 11th International Conference Concurrency Theory (CONCUR’00), ed. By C. Palamidessi. LNCS, vol. 1877 (Springer, 2000), pp. 123–137

    Google Scholar 

  84. M. Kwiatkowska, G. Norman, R. Segala, J. Sproston, Automatic verification of real-time systems with discrete probability distributions. Theor. Comput. Sci. 282, 101–150 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  85. M. Kwiatkowska, G. Norman, J. Sproston, F. Wang, Symbolic model checking for probabilistic timed automata. Inf. Comput. 205(7), 1027–1077 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  86. M. Kwiatkowska, D. Parker, C. Wiltsche, PRISM-games 2.0: a tool for multi-objective strategy synthesis for stochastic games, in Proceedings of the 22nd International Conference Tools and Algorithms for the Construction and Analysis of Systems (TACAS’16). LNCS (Springer, 2016)

    Google Scholar 

  87. O. Madani, S. Hanks, A. Condon, On the undecidability of probabilistic planning and related stochastic optimization problems. Artif. Intell. 147(1–2), 5–34 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  88. O. Maler, K. Larsen, B. Krogh, On zone-based analysis of duration probabilistic automata, in Proceedings of the 12th International Workshop Verification of Infinite-State Systems (INFINITY’10). EPTCS, vol. 39 (2010), pp. 33–46

    Google Scholar 

  89. R. Milner, Calculi for synchrony and asynchrony. Theor. Comput. Sci. 25(3), 267–310 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  90. R. Nicola, F. Vaandrager, Action versus state based logics for transition systems, in Proceedings of the LITP Spring School on Theoretical Computer Science: Semantics of Systems of Concurrent Processes, ed. By I. Guessarian (Springer, 1990), pp. 407–419

    Google Scholar 

  91. G. Norman, D. Parker, M. Kwiatkowska, S. Shukla, Evaluating the reliability of NAND multiplexing with PRISM. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24(10), 1629–1637 (2005)

    Article  Google Scholar 

  92. G. Norman, D. Parker, J. Sproston, Model checking for probabilistic timed automata. Form. Methods Syst. Des. 43(2), 164–190 (2013)

    Article  MATH  Google Scholar 

  93. G. Norman, D. Parker, X. Zou, Verification and control of partially observable probabilistic real-time systems, in Proceedings of the 13th International Conference Formal Modelling and Analysis of Timed Systems (FORMATS’15), ed. By S. Sankaranarayanan, E. Vicario. LNCS, vol. 9268 (Springer, 2015), pp. 240–255

    Google Scholar 

  94. D. Parker, Implementation of Symbolic Model Checking for Probabilistic Systems. Ph.D. thesis, University of Birmingham, 2002

    Google Scholar 

  95. A. Pnueli, The temporal semantics of concurrent programs. Theor. Comput. Sci. 13, 45–60 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  96. A. Puggelli, W. Li, A. Sangiovanni-Vincentelli, S. Seshia, Polynomial-time verification of PCTL properties of MDPs with convex uncertainties, in Proceedings of the 25th International Conference Computer Aided Verification (CAV’13). LNCS, vol. 8044 (Springer, 2013), pp. 527–542

    Google Scholar 

  97. M. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming (Wiley, New Jersey, 1994)

    Google Scholar 

  98. S. Ratschan, Z. She, Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Trans. Embed. Comput. Syst. 6(1) (2007)

    Google Scholar 

  99. A.W. Roscoe, The Theory and Practice of Concurrency (Prentice-Hall, New Jersey, 1997)

    Google Scholar 

  100. R. Segala, Modelling and verification of randomized distributed real time systems. Ph.D. thesis, Massachusetts Institute of Technology, 1995

    Google Scholar 

  101. R. Segala, N. Lynch, Probabilistic simulations for probabilistic processes. Nordic J. Comput. 2(2), 250–273 (1995)

    MathSciNet  MATH  Google Scholar 

  102. J. Sproston, Decidable model checking of probabilistic hybrid automata, in Proceedings of the International Symposium on Formal Techniques in Real-Time and Fault Tolerant Systems (FTRTFT’00), ed. By M. Joseph. LNCS, vol. 1926 (Springer, 2000), pp. 31–45

    Google Scholar 

  103. J. Sun, Y. Liu, J.S. Dong, J. Pang, Pat: towards flexible verification under fairness, in Proceedings of the 21st International Conference Computer Aided Verification (CAV’09). LNCS, vol. 5643 (Springer, 2009), pp. 709–714

    Google Scholar 

  104. M. Svorenova, M. Kwiatkowska, Quantitative verification and strategy synthesis for stochastic games. Eur. J. Control 30, 15–30 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  105. M. Svoreňová, M. Chmelík, K. Leahy, H. Eniser, K. Chatterjee, I. Černá, C. Belta, Temporal logic motion planning using POMDPs with parity objectives: case study paper, in Proceedings of the 18th International Conference Hybrid Systems: Computation and Control (HSCC’15) (ACM, 2015), pp. 233–238

    Google Scholar 

  106. S. Tripakis, The analysis of timed systems in practice. Ph.D. thesis, Université Joseph Fourier, Grenoble, 1998

    Google Scholar 

  107. S. Tripakis, S. Yovine, A. Bouajjan, Checking timed Buchi automata emptiness efficiently. Form. Methods Syst. Des. 26(3), 267–292 (2005)

    Article  MATH  Google Scholar 

  108. M. Vardi, P. Wolper, Reasoning about infinite computations. Inf. Comput. 115(1), 1–37 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  109. J. von Neumann, Probabilistic logics and synthesis of reliable organisms from unreliable components, in Automata Studies, ed. By C. Shannon, J. McCarthy (Princeton University Press, 1956), pp. 43–98

    Google Scholar 

  110. B. Wachter, L. Zhang, H. Hermanns, Probabilistic model checking modulo theories, in Proceedings of the 4th International Conference Quantitative Evaluation of Systems (QEST’07) (IEEE Computer Society Press, 2007), pp. 129–140

    Google Scholar 

  111. C. Wiltsche, Assume-Guarantee Strategy Synthesis for Stochastic Games. Ph.D thesis, University of Oxford, 2015

    Google Scholar 

  112. E. Wolff, U. Topcu, R. Murray, Robust control of uncertain Markov decision processes with temporal logic specifications, in Proceedings of the IEEE 51st Annual Conference Decision and Control (CDC’12) (Computer Society Press, 2012), pp. 3372–3379

    Google Scholar 

  113. L. Zhang, Z. She, S. Ratschan, H. Hermanns, E.M. Hahn, Safety verification for probabilistic hybrid systems. Eur. J. Control 18(6), 572–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  114. http://www.prismmodelchecker.org

  115. http://www.prismmodelchecker.org/files/fsv-pmc/

  116. http://www.prismmodelchecker.org/games

  117. http://www.prismmodelchecker.org/other-tools.php

Download references

Acknowledgements

This work was supported by the ERC Advanced Investigators Grant VERIWARE, the EPSRC Mobile Autonomy Programme Grant EP/M019918/1, the EU FP7-funded project HIERATIC and the DARPA-funded BRASS project.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Marta Kwiatkowska

  2. School of Computing Science, University of Glasgow, Glasgow, UK

    Gethin Norman

  3. School of Computer Science, University of Birmingham, Birmingham, UK

    David Parker

Authors
  1. Marta Kwiatkowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gethin Norman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Parker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marta Kwiatkowska .

Editor information

Editors and Affiliations

  1. DFKI, University of Bremen, Bremen, Germany

    Rolf Drechsler

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Kwiatkowska, M., Norman, G., Parker, D. (2018). Probabilistic Model Checking: Advances and Applications. In: Drechsler, R. (eds) Formal System Verification. Springer, Cham. https://doi.org/10.1007/978-3-319-57685-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57685-5_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57683-1

  • Online ISBN: 978-3-319-57685-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation