Skip to main content
SpringerLink
Log in

New Approaches for Almost-Sure Termination of Probabilistic Programs

  • Conference paper
  • First Online:
Programming Languages and Systems (APLAS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11275))

Included in the following conference series:

Abstract

We study the almost-sure termination problem for probabilistic programs. First, we show that supermartingales with lower bounds on conditional absolute difference provide a sound approach for the almost-sure termination problem. Moreover, using this approach we can obtain explicit optimal bounds on tail probabilities of non-termination within a given number of steps. Second, we present a new approach based on Central Limit Theorem for the almost-sure termination problem, and show that this approach can establish almost-sure termination of programs which none of the existing approaches can handle. Finally, we discuss algorithmic approaches for the two above methods that lead to automated analysis techniques for almost-sure termination of probabilistic programs.

A full version is available in http://arxiv.org/abs/1806.06683.

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. Agrawal, S., Chatterjee, K., Novotný, P.: Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs. PACMPL 2(POPL), 34:1–34:32 (2018). https://doi.org/10.1145/3158122

    Article  Google Scholar 

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

    MATH  Google Scholar 

  3. Bournez, O., Garnier, F.: Proving positive almost-sure termination. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 323–337. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-32033-3_24

    Chapter  Google Scholar 

  4. Bradley, A.R., Manna, Z., Sipma, H.B.: Linear ranking with reachability. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 491–504. Springer, Heidelberg (2005). https://doi.org/10.1007/11513988_48

    Chapter  Google Scholar 

  5. Brázdil, T., Kiefer, S., Kucera, A., Vareková, I.H.: Runtime analysis of probabilistic programs with unbounded recursion. J. Comput. Syst. Sci. 81(1), 288–310 (2015)

    Article  MathSciNet  Google Scholar 

  6. Chakarov, A., Sankaranarayanan, S.: Probabilistic program analysis with martingales. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 511–526. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_34

    Chapter  Google Scholar 

  7. Chatterjee, K., Fu, H.: Termination of nondeterministic recursive probabilistic programs. CoRR abs/1701.02944, January 2017

    Google Scholar 

  8. Chatterjee, K., Fu, H., Goharshady, A.K.: Termination analysis of probabilistic programs through positivstellensatz’s. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 3–22. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41528-4_1

    Chapter  Google Scholar 

  9. Chatterjee, K., Fu, H., Novotný, P., Hasheminezhad, R.: Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs. In: POPL, pp. 327–342 (2016)

    Article  Google Scholar 

  10. Chatterjee, K., Novotný, P., Žikelić, Đ.: Stochastic invariants for probabilistic termination. In: POPL, pp. 145–160 (2017)

    Google Scholar 

  11. Colóon, M.A., Sipma, H.B.: Synthesis of linear ranking functions. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 67–81. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45319-9_6

    Chapter  Google Scholar 

  12. Esparza, J., Gaiser, A., Kiefer, S.: Proving termination of probabilistic programs using patterns. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 123–138. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31424-7_14

    Chapter  Google Scholar 

  13. Fioriti, L.M.F., Hermanns, H.: Probabilistic termination: soundness, completeness, and compositionality. In: POPL, pp. 489–501 (2015)

    Google Scholar 

  14. Floyd, R.W.: Assigning meanings to programs. Math. Aspects Comput. Sci. 19, 19–33 (1967)

    Article  MathSciNet  Google Scholar 

  15. Foster, F.G.: On the stochastic matrices associated with certain queuing processes. Ann. Math. Stat. 24(3), 355–360 (1953)

    Article  MathSciNet  Google Scholar 

  16. Gordon, A.D., Henzinger, T.A., Nori, A.V., Rajamani, S.K.: Probabilistic programming. In: Herbsleb, J.D., Dwyer, M.B. (eds.) FOSE, pp. 167–181. ACM (2014)

    Google Scholar 

  17. Kaelbling, L.P., Littman, M.L., Cassandra, A.R.: Planning and acting in partially observable stochastic domains. Artif. Intell. 101(1), 99–134 (1998)

    Article  MathSciNet  Google Scholar 

  18. Kaelbling, L.P., Littman, M.L., Moore, A.W.: Reinforcement learning: a survey. JAIR 4, 237–285 (1996)

    Article  Google Scholar 

  19. Kaminski, B.L., Katoen, J.-P.: On the hardness of almost–sure termination. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9234, pp. 307–318. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48057-1_24

    Chapter  Google Scholar 

  20. Kaminski, B.L., Katoen, J.-P., Matheja, C., Olmedo, F.: Weakest precondition reasoning for expected run–times of probabilistic programs. In: Thiemann, P. (ed.) ESOP 2016. LNCS, vol. 9632, pp. 364–389. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49498-1_15

    Chapter  Google Scholar 

  21. Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains. D. Van Nostrand Company, Princeton (1966)

    Google Scholar 

  22. 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 

  23. McIver, A., Morgan, C.: Developing and reasoning about probabilistic programs in pGCL. In: Cavalcanti, A., Sampaio, A., Woodcock, J. (eds.) PSSE 2004. LNCS, vol. 3167, pp. 123–155. Springer, Heidelberg (2006). https://doi.org/10.1007/11889229_4

    Chapter  Google Scholar 

  24. McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer, New York (2005). https://doi.org/10.1007/b138392

    Book  MATH  Google Scholar 

  25. McIver, A., Morgan, C.: A new rule for almost-certain termination of probabilistic and demonic programs. CoRR abs/1612.01091, December 2016

    Google Scholar 

  26. McIver, A., Morgan, C., Kaminski, B.L., Katoen, J.: A new proof rule for almost-sure termination. PACMPL 2(POPL), 33:1–33:28 (2018). https://doi.org/10.1145/3158121

    Article  Google Scholar 

  27. van de Meent, J., Yang, H., Mansinghka, V., Wood, F.: Particle gibbs with ancestor sampling for probabilistic programs. In: AISTATS (2015)

    Google Scholar 

  28. Ngo, V.C., Carbonneaux, Q., Hoffmann, J.: Bounded expectations: resource analysis for probabilistic programs. In: Foster, J.S., Grossman, D. (eds.) Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI 2018, Philadelphia, PA, USA, 18–22 June 2018, pp. 496–512. ACM (2018). https://doi.org/10.1145/3192366.3192394

  29. Olmedo, F., Kaminski, B.L., Katoen, J.P., Matheja, C.: Reasoning about recursive probabilistic programs. In: LICS, pp. 672–681 (2016)

    Google Scholar 

  30. Paz, A.: Introduction to Probabilistic Automata (Computer Science and Applied Mathematics). Academic Press, Cambridge (1971)

    Google Scholar 

  31. Podelski, A., Rybalchenko, A.: A complete method for the synthesis of linear ranking functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239–251. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24622-0_20

    Chapter  Google Scholar 

  32. Rabin, M.: Probabilistic automata. Inf. Control 6, 230–245 (1963)

    Article  Google Scholar 

  33. Sankaranarayanan, S., Chakarov, A., Gulwani, S.: Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In: PLDI, pp. 447–458 (2013)

    Google Scholar 

  34. Sohn, K., Gelder, A.V.: Termination detection in logic programs using argument sizes. In: PODS, pp. 216–226 (1991)

    Google Scholar 

  35. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

Download references

Acknowledgements

This work was financially supported by NSFC (Grant No. 61772336, 61472239), Notional Key Research and Development Program of China (Grant No. 2017YFB0701900), Austrian Science Fund (FWF) grant S11407-N23 (RiSE/SHiNE) and Vienna Science and Technology Fund (WWTF) project ICT15-003.

Author information

Authors and Affiliations

  1. BASICS Lab, Shanghai Jiao Tong University, Shanghai, China

    Mingzhang Huang

  2. Shanghai Jiao Tong University, Shanghai, China

    Hongfei Fu

  3. IST Austria, Klosterneuburg, Austria

    Krishnendu Chatterjee

Authors
  1. Mingzhang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongfei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Krishnendu Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongfei Fu .

Editor information

Editors and Affiliations

  1. Korea Advanced Institute of Science and Technology, Daejeon, Korea (Republic of)

    Sukyoung Ryu

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

Huang, M., Fu, H., Chatterjee, K. (2018). New Approaches for Almost-Sure Termination of Probabilistic Programs. In: Ryu, S. (eds) Programming Languages and Systems. APLAS 2018. Lecture Notes in Computer Science(), vol 11275. Springer, Cham. https://doi.org/10.1007/978-3-030-02768-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-02768-1_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02767-4

  • Online ISBN: 978-3-030-02768-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation