Skip to main content
SpringerLink
Log in

On Abstraction of Probabilistic Systems

  • Chapter
Stochastic Model Checking. Rigorous Dependability Analysis Using Model Checking Techniques for Stochastic Systems (ROCKS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8453))

Abstract

Probabilistic model checking extends traditional model checking by incorporating quantitative information about the probability of system transitions. However, probabilistic models that describe interesting behavior are often too complex for straightforward analysis. Abstraction is one way to deal with this complexity: instead of analyzing the (“concrete”) model, a simpler (“abstract”) model that preserves the relevant properties is built and analyzed. This paper surveys various abstraction techniques proposed in the past decade. For each abstraction technique we identify in what sense properties are preserved or provide alternatively suitable boundaries.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. de Alfaro, L., Roy, P.: Magnifying-lens abstraction for Markov decision processes. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 325–338. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  2. Aljazzar, H., Leue, S.: Directed explicit state-space search in the generation of counterexamples for stochastic model checking. IEEE Trans. Software Eng. 36(1), 37–60 (2010)

    Article  Google Scholar 

  3. Aziz, A., Singhal, V., Balarin, F., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: It usually works: The temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  4. Baier, C., Katoen, J.-P.: Principles of model checking. MIT Press, Cambridge (2008)

    Google Scholar 

  5. Baier, C., Katoen, J.-P., Hermanns, H., Wolf, V.: Comparative branching-time semantics for Markov chains. Information and Computation 200(2), 149–214 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  7. Bozzano, M., Cimatti, A., Katoen, J.-P., Nguyen, V.Y., Noll, T., Roveri, M.: Safety, dependability and performance analysis of extended aadl models. Comput. J. 54(5), 754–775 (2011)

    Article  Google Scholar 

  8. Chadha, R., Viswanathan, M.: A counterexample-guided abstraction-refinement framework for markov decision processes. ACM Trans. Comput. Log. 12(1), 1 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, T., Forejt, V., Kwiatkowska, M., Parker, D., Simaitis, A.: Prism-games: A model checker for stochastic multi-player games. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 185–191. Springer, Heidelberg (2013)

    Google Scholar 

  10. Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  11. Condon, A.: The complexity of stochastic games. Information and Computation 96(2), 203–224 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)

    Google Scholar 

  13. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and refinement strategies for probabilistic analysis. In: Hermanns, H., Segala, R. (eds.) PAPM-PROBMIV 2002. LNCS, vol. 2399, pp. 57–76. Springer, Heidelberg (2002)

    Google Scholar 

  14. Dehnert, C., Katoen, J.-P., Parker, D.: SMT-based bisimulation minimisation of markov models. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 28–47. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  15. Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wąsowski, A.: Abstract probabilistic automata. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 324–339. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  16. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. Information and Computation 184(1), 160–200 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Donaldson, A.F., Miller, A.: Symmetry reduction for probabilistic model checking using generic representatives. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 9–23. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Eisentraut, C., Hermanns, H., Schuster, J., Turrini, A., Zhang, L.: The quest for minimal quotients for probabilistic automata. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 16–31. Springer, Heidelberg (2013)

    Google Scholar 

  19. Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Science of Computer Programming 2(3), 241–266 (1982)

    Article  MATH  Google Scholar 

  20. Fecher, H., Leucker, M., Wolf, V.: Don’t know in probabilistic systems. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 71–88. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  21. Feng, L., Kwiatkowska, M.Z., Parker, D.: Compositional verification of probabilistic systems using learning. In: QEST, pp. 133–142. IEEE Computer Society (2010)

    Google Scholar 

  22. Ferrer Fioriti, L.M., Hahn, E.M., Hermanns, H., Wachter, B.: Variable probabilistic abstraction refinement. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 300–316. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  23. Hahn, E.M., Hermanns, H., Wachter, B., Zhang, L.: Pass: Abstraction refinement for infinite probabilistic models. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 353–357. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  24. Han, T., Katoen, J.-P.: Counterexamples in probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 72–86. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  26. Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  27. Hermanns, H., Wachter, B., Zhang, L.: Probabilistic CEGAR. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 162–175. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  28. Huth, M., Jagadeesan, R., Schmidt, D.: Modal transition systems: A foundation for three-valued program analysis. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, pp. 155–169. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  29. Jansen, D.N., Song, L., Zhang, L.: Revisiting weak simulation for substochastic Markov chains. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 209–224. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  30. Jansen, N., Ábrahám, E., Katelaan, J., Wimmer, R., Katoen, J.-P., Becker, B.: Hierarchical counterexamples for discrete-time markov chains. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 443–452. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  31. Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: Proc. LICS 1991, pp. 266–277. IEEE Comp. Soc. Pr. (1991)

    Google Scholar 

  32. Katoen, J.-P., Kemna, T., Zapreev, I., Jansen, D.N.: Bisimulation minimisation mostly speeds up probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 87–101. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  33. Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for probabilistic systems. JLAP 81(4), 356–389 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Kattenbelt, M., Kwiatkowska, M., Norman, G., Parker, D.: A game-based abstraction-refinement framework for markov decision processes. Formal Methods in System Design 36(3), 246–280 (2010)

    Article  MATH  Google Scholar 

  35. Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: A game-based abstraction-refinement framework for markov decision processes. Formal Methods in System Design 36(3), 246–280 (2010)

    Article  MATH  Google Scholar 

  36. Komuravelli, A., Păsăreanu, C.S., Clarke, E.M.: Assume-guarantee abstraction refinement for probabilistic systems. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 310–326. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  37. Kwiatkowska, M., Norman, G., Parker, D.: Symmetry reduction for probabilistic model checking. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 234–248. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  38. 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)

    Chapter  Google Scholar 

  39. Kwiatkowska, M., Norman, G., Parker, D., Qu, H.: Assume-guarantee verification for probabilistic systems. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 23–37. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  40. Larsen, K.G., Thomsen, B.: A modal process logic. In: Proc. LICS 1988, pp. 203–210. Los Alamitos, Calif (1988)

    Google Scholar 

  41. Larsen, K.G., Thomsen, B.: A modal process logic. In: LICS, pp. 203–210 (1988)

    Google Scholar 

  42. McMillan, K.L.: Lazy abstraction with interpolants. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 123–136. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  43. Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. Nordic Journal of Computing 2, 250–273 (1995)

    MathSciNet  MATH  Google Scholar 

  44. Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS, pp. 327–338. IEEE Comp. Soc. Pr., Washington, DC (1985)

    Google Scholar 

  45. Wachter, B.: Refined probabilistic abstraction. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2011)

    Google Scholar 

  46. Wachter, B., Zhang, L.: Best probabilistic transformers. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 362–379. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  47. Wimmer, R., Herbstritt, M., Hermanns, H., Strampp, K., Becker, B.: Sigref- a symbolic bisimulation tool box. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 477–492. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  48. Wimmer, R., Jansen, N., Ábrahám, E., Becker, B., Katoen, J.-P.: Minimal critical subsystems for discrete-time markov models. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 299–314. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  49. Wimmer, R., Jansen, N., Vorpahl, A., Ábrahám, E., Katoen, J.-P., Becker, B.: High-level counterexamples for probabilistic automata. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 39–54. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  50. Zhang, L.: Decision algorithms for probabilistic simulations. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Software Modeling and Verification Group, RWTH Aachen University, Ahornstraße 55, 52056, Aachen, Germany

    Christian Dehnert

  2. Department of Computer Science, VU University Amsterdam, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    Daniel Gebler

  3. Institute for Computing and Information Sciences, Radboud University, Heyendaalseweg 135, 6500 GL, Nijmegen, The Netherlands

    Michele Volpato & David N. Jansen

Authors
  1. Christian Dehnert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Gebler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michele Volpato
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David N. Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Design and Analysis of Communication Systems, University of Twente, The Netherlands

    Anne Remke

  2. Department of Computer Science, University of Twente, The Netherlands

    Mariëlle Stoelinga

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dehnert, C., Gebler, D., Volpato, M., Jansen, D.N. (2014). On Abstraction of Probabilistic Systems. In: Remke, A., Stoelinga, M. (eds) Stochastic Model Checking. Rigorous Dependability Analysis Using Model Checking Techniques for Stochastic Systems. ROCKS 2012. Lecture Notes in Computer Science, vol 8453. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45489-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-45489-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45488-6

  • Online ISBN: 978-3-662-45489-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation