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Counterexample-guided inductive synthesis for probabilistic systems

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Formal Aspects of Computing

Abstract

This paper presents counterexample-guided inductive synthesis (CEGIS) to automatically synthesise probabilistic models. The starting point is a family of finite-stateMarkov chains with related but distinct topologies. Such families can succinctly be described by a sketch of a probabilistic program. Program sketches are programs containing holes. Every hole has a finite repertoire of possible program snippets by which it can be filled.We study several synthesis problems—feasibility, optimal synthesis, and complete partitioning—for a given quantitative specification \(\varphi\). Feasibility amounts to determine a family member satisfying \(\varphi\), optimal synthesis amounts to find a family member that maximises the probability to satisfy \(\varphi\), and complete partitioning splits the family in satisfying and refuting members. Each of these problems can be considered under the additional constraint of minimising the total cost of instantiations, e.g., what are all possible instantiations for \(\varphi\) that are within a certain budget? The synthesis problems are tackled using a CEGIS approach. The crux is to aggressively prune the search space by using counterexamples provided by a probabilistic model checker. Counterexamples can be viewed as sub-Markov chains that rule out all family members that share this sub-chain. Our CEGIS approach leverages efficient probabilisticmodel checking,modern SMT solving, and programsnippets as counterexamples. Experiments on case studies froma diverse nature—controller synthesis, program sketching, and security—show that synthesis among up to a million candidate designs can be done using a few thousand verification queries.

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References

  1. Ábrahám E, Becker B, Dehnert C, Jansen N, Katoen J-P, Wimmer R (2014) Counterexample generation for discrete-time Markov models: An introductory survey, Springer, vol 8483 of LNCS, pp 65–121

  2. Alur R, Bodík R, Dallal E, Fisman D, Garg P, Juniwal G, Kress-Gazit H, Madhusudan P, Martin MMK, Raghothaman M, Saha S, Seshia SA, Singh R, Solar-Lezama A, Torlak E, Udupa A (2015) Syntax-guided synthesis. In: Dependable software systems engineering, IOS Press, vol 40 of NATO Science for Peace and Security Series, pp 1–25

  3. Abate A, David C, Kesseli P, Kroening D, Polgreen E (2018) Counterexample guided inductive synthesis modulo theories. In: CAV (1), Springer, vol 10981 of LNCS, pp 270–288

  4. Antonik, A., Huth, M., Larsen, K.G., Nyman, U., Wasowski, A.: 20 years of modal and mixed specifications. Bulletin of the EATCS 95, 94–129 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Alur, R., Singh, R., Fisman, D., Solar-Lezama, A.: Search-based program synthesis. Commun ACM 61(12), 84–93 (2018)

    Article  Google Scholar 

  6. Benini, L., Bogliolo, A., Paleologo, G.A., De Micheli, G.: Policy optimization for dynamic power management. IEEE Trans CAD Integr Circuits Syst 18(6), 813–833 (1999)

    Article  Google Scholar 

  7. Baier C, de Alfaro L, Forejt V, Kwiatkowska M (2018) Model checking probabilistic systems. In: Handbook of model checking, Springer, pp 963–999

  8. Budde CE, Dehnert C, Hahn EM, Hartmanns A, Junges S, Turrini A (2017) JANI: quantitative model and tool interaction. In: TACAS, vol 10206 of LNCS, pp 151–168

  9. Bartocci E, Grosu R, Katsaros P, Ramakrishnan CR, Smolka SA (2011) Model repair for probabilistic systems. In: TACAS, Springer, vol 6605 of LNCS, pp 326–340

  10. Biere A, Heule M, van Maaren H, Walsh T (eds) (2009) Handbook of Satisfiability, IOS Press, vol 185 of Frontiers in artificial intelligence and applications

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

  12. Benes N, Křetínský J, Larsen KG, Møller MH, Srba J (2012) Dual-priced modal transition systems with time durations. In: LPAR, Springer, vol 7180 of LNCS, pp 122–137

  13. Benes, N., Kretínský, J., Larsen, K.G., Møller, M.H., Sickert, S., Srba, J.: Refinement checking on parametric modal transition systems. Acta Inf 52(2–3), 269–297 (2015)

    Article  MathSciNet  Google Scholar 

  14. Bornholt J, Torlak E, Grossman D, Ceze L (2016) Optimizing synthesis with metasketches. In: POPL, ACM, pp 775–788

  15. Cardelli L, Češka M, Fränzle M, Kwiatkowska M, Laurenti L, Paoletti N, Whitby M (2017) Syntax-guided optimal synthesis for chemical reaction networks. In: CAV, Springer, vol 10427 of LNCS, pp 375–395

  16. Černý P, Chatterjee K, Henzinger TA, Radhakrishna A, Singh R (2011) Quantitative synthesis for concurrent programs. In: CAV, Springer, vol 6806 of LNCS, pp 243–259

  17. Chaudhuri S, Clochard M, Solar-Lezama A (2014) Bridging boolean and quantitative synthesis using smoothed proof search. In: POPL, ACM, pp 207–220

  18. Chrszon, P., Dubslaff, C., Klüppelholz, S., Baier, C.: ProFeat: feature-oriented engineering for family-based probabilistic model checking. Formal Asp Comput 30(1), 45–75 (2018)

    Article  MathSciNet  Google Scholar 

  19. Češka, M., Dannenberg, F., Paoletti, N., Kwiatkowska, M., Brim, L.: Precise parameter synthesis for stochastic biochemical systems. Acta Inf 54(6), 589–623 (2017)

    Article  MathSciNet  Google Scholar 

  20. Calinescu, R., Ghezzi, C., Johnson, K., Pezzé, M., Rafiq, Y., Tamburrelli, G.: Formal verification with confidence intervals to establish quality of service properties of software systems. IEEE Trans Rel 65(1), 107–125 (2016)

    Article  Google Scholar 

  21. Calinescu, R., Ghezzi, C., Kwiatkowska, M.Z., Mirandola, R.: Self-adaptive software needs quantitative verification at runtime. Commun ACM 55(9), 69–77 (2012)

    Article  Google Scholar 

  22. Chen T, Hahn EM, Han T, Kwiatkowska MZ, Qu H, Zhang L (2013) Model repair for Markov decision processes. In: TASE, IEEE, pp 85–92

  23. Češka M, Hensel C, Junges S, Katoen J-P (2019) Counterexample-driven synthesis for probabilistic program sketches. In: Formal methods – the next 30 years, Springer International Publishing, vol 11800 of LNCS, pp 101–120

  24. Chonev V (2017) Reachability in augmented interval Markov chains. CoRR abs/1701.02996

  25. Češka M, Jansen N, Junges S, Katoen J-P (2019) Shepherding hordes of Markov chains. In: TACAS, Springer, vol 11428 of LNCS

  26. Calinescu R, Češka M, Gerasimou S, Kwiatkowska M, Paoletti N (2017) Designing robust software systems through parametric Markov chain synthesis. In: ICSA, IEEE, pp 131–140

  27. Calinescu R, Češka M, Gerasimou S, Kwiatkowska M, Paoletti N (2017) RODES: A robust-design synthesis tool for probabilistic systems. In: QEST, Springer, pp 304–308

  28. Calinescu, R., Češka, M., Gerasimou, S., Kwiatkowska, M., Paoletti, N.: Efficient synthesis of robust models for stochastic systems. J Syst Softw 143, 140–158 (2018)

    Article  Google Scholar 

  29. Dehnert C, Junges S, Katoen J-P, Volk M (2017) A storm is coming: A modern probabilistic model checker. In: CAV, Springer, vol 10427 of LNCS, pp 592–600

  30. Dehnert C, Jansen N, Wimmer R, Ábrahám E, Katoen J-P (2014) Fast debugging of PRISM models. In ATVA, Springer, vol 8837 of LNCS, pp 146–162

  31. Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wasowski, A.: Abstract probabilistic automata. Inf Comput 232, 66–116 (2013)

    Article  MathSciNet  Google Scholar 

  32. de Moura LM, Bjørner N (2008) Z3: an efficient SMT solver. In: TACAS, Springer, vol 4963 of LNCS, pp 337–340

  33. Dureja R, Rozier KY (2018) More scalable LTL model checking via discovering design-space dependencies. In: TACAS (1), Springer, vol 10805 of LNCS, pp 309–327

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

    Article  Google Scholar 

  35. Gulwani, S., Polozov, O., Singh, R.: Program synthesis. Found Trends Program Lang 4(1–2), 1–119 (2017)

    Google Scholar 

  36. Ghezzi, C., Sharifloo, A.M.: Model-based verification of quantitative non-functional properties for software product lines. Inf Softw Technol 55(3), 508–524 (2013)

    Article  Google Scholar 

  37. Gerasimou S, Tamburrelli G, Calinescu R (2015) Search-based synthesis of probabilistic models for quality-of-service software engineering. In: ASE, IEEE Computer Society, pp 319–330

  38. Henzinger, T.A.: Quantitative reactive modeling and verification. Comput Sci - R&D 28(4), 331–344 (2013)

    Google Scholar 

  39. Hensel C (2018) The probabilistic model checker storm: Symbolic methods for probabilistic model checking. PhD thesis, RWTH Aachen University, Germany

  40. Hartmanns A, Hermanns H (2014) The modest toolset: An integrated environment for quantitative modelling and verification. In: TACAS, Springer, pp 593–598

  41. Hahn, E.M., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. Softw Tools Technol Transf 13(1), 3–19 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  43. Hartmanns A, Klauck M, Parker D, Quatmann T, Ruijters E (2019) The quantitative verification benchmark set. In: TACAS (1), Springer, vol 11427 of Lecture Notes in Computer Science, pp 344–350

  44. Jansen N, Humphrey L, Tumova J, Topcu U (2019) Structured synthesis for probabilistic systems. In: NFM, Springer, vol 11460 of LNCS, pp 237–254

  45. Junges S, Jansen N, Dehnert C, Topcu U, Katoen J-P (2016) Safety-constrained reinforcement learning for MDPs. In: TACAS, Springer, vol 9636 of LNCS, pp 130–146

  46. Junges S, Jansen N, Wimmer R, Quatmann T, Winterer L, Katoen J-P, Becker B (2018) Finite-state controllers of POMDPs using parameter synthesis. In: UAI, AUAI Press, pp 519–529

  47. Jha, S., Seshia, S.A.: A theory of formal synthesis via inductive learning. Acta Inf 54(7), 693–726 (2017)

    Article  MathSciNet  Google Scholar 

  48. Junges S (2020) Parameter synthesis in Markov models. PhD thesis, RWTH Aachen University, Germany, to appear

  49. Katoen J-P (2016) The probabilistic model checking landscape. In: LICS, ACM, pp 31–45

  50. Kaelbling, L.P., Littman, M.L., Cassandra, A.R.: Planning and acting in partially observable stochastic domains. Artif Intell 101(1–2), 99–134 (1998)

    Article  MathSciNet  Google Scholar 

  51. Kwiatkowska M, Norman G, Parker D (2011) Prism 4.0: Verification of probabilistic real-time systems. In: CAV, vol 6806 of LNCS, Springer, pp 585–591

  52. Kwiatkowska, M.Z., Norman, G., Parker, D., Vigliotti, M.G.: Probabilistic mobile ambients. Theor Comput Sci 410(12–13), 1272–1303 (2009)

    Article  MathSciNet  Google Scholar 

  53. Kretínský J (2017) 30 years of modal transition systems: Survey of extensions and analysis. In: Models, algorithms, logics and tools, Springer, vol 10460 of LNCS, pp 36–74

  54. Lanna, A., Castro, T., Alves, V., Rodrigues, G., Schobbens, P.-Y., Apel, S.: Feature-family-based reliability analysis of software product lines. Inform Softw Technol 94, 59–81 (2018)

    Article  Google Scholar 

  55. Larsen KG, Thomsen B (1988) A modal process logic. In: LICS, IEEE Computer Society, pp 203–210

  56. Meuleau N, Kim K-E, Kaelbling LP, Cassandra AR (1999) Solving POMDPs by searching the space of finite policies. In: UAI, Morgan Kaufmann Publishers Inc., pp 417–426

  57. Morgan, C., McIver, A., Seidel, K.: Probabilistic predicate transformers. ACM Trans Program Lang Syst 18(3), 325–353 (1996)

    Article  Google Scholar 

  58. Nori AV, Ozair S, Rajamani SK, Vijaykeerthy D (2015) Efficient synthesis of probabilistic programs. In: PLDI, ACM, pp 208–217

  59. Quatmann T, Dehnert C, Jansen N, Junges S, Katoen J-P (2016) Parameter synthesis for Markov models: Faster than ever. In: ATVA, vol 9938 of LNCS, pp 50–67

  60. Quatmann T, Jansen N, Dehnert C, Wimmer R, Ábrahám E, Katoen J-P, Becker B (2015) Counterexamples for expected rewards. In: FM, Springer, vol 9109 of LNCS, pp 435–452

  61. Rodrigues GN, Alves V, Nunes V, Lanna A, Cordy M, Schobbens P-Y, Sharifloo AM, Legay A (2015) Modeling and verification for probabilistic properties in software product lines. In: HASE, IEEE pp 173–180

  62. Rosenblum DS (2016) The power of probabilistic thinking. In: ASE, ACM, p 3

  63. Sesic, A., Dautovic, S., Malbasa, V.: Dynamic power management of a system with a two-priority request queue using probabilistic-model checking. IEEE Trans CAD Integr Circuits Syst 27(2), 403–407 (2008)

    Article  Google Scholar 

  64. Solar-Lezama A, Jones CG, Bodik R (2008) Sketching concurrent data structures. In: PLDI, ACM, pp 136–148

  65. Solar-Lezama A, Tancau L, Bodik R, Seshia S, Saraswat V (2006) Combinatorial sketching for finite programs. In: ASPLOS, ACM, pp 404–415

  66. Solar-Lezama, A.: Program sketching. STTT 15(5–6), 475–495 (2013)

    Google Scholar 

  67. Solar-Lezama A, Rabbah RM, Bodík R, Ebcioglu K (2005) Programming by sketching for bit-streaming programs. In: PLDI, ACM, pp 281–294

  68. Varshosaz M, Khosravi R (2013) Discrete time Markov chain families: modeling and verification of probabilistic software product lines. In: SPLC Workshops, ACM, pp 34–41

  69. Vandin A, ter Beek MH, Legay A, Lluch-Lafuente A (2018) Qflan: A tool for the quantitative analysis of highly reconfigurable systems. In: FM, Springer, vol 10951 of LNCS, pp 329–337

  70. Wimmer R, Jansen N, Ábrahám E, Becker B, Katoen J-P (2012) Minimal critical subsystems for discrete-time Markov models. In TACAS, Springer, vol 7214 of LNCS, pp 299–314

  71. Wimmer, R., Jansen, N., Ábrahám, E., Katoen, J.-P., Becker, B.: Minimal counterexamples for linear-time probabilistic verification. Theor Comput Sci 549, 61–100 (2014)

    Article  MathSciNet  Google Scholar 

  72. Wimmer, R., Jansen, N., Vorpahl, A., Ábrahám, E., Katoen, J.-P., Becker, B.: High-level counterexamples for probabilistic automata. Log Methods Comput Sci 11(1), (2015)

  73. Zhou W, Li W (2018) Safety-aware apprenticeship learning. In CAV'18, Springer, vol 10981 of LNCS, pp 662–680

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Acknowledgements

We thank the reviewers for their detailed and constructive comments. We thank Roman Andriushchenko for help with some experiments. M. Češka is funded by the Czech Science Foundation grant GJ20-02328Y. S. Junges was supported in part by NSF grants 1545126 (VeHICaL) and 1646208, by theDARPA Assured Autonomy program and the DARPA SDCPS program (Contract: FA8750-20-C-0156), by Berkeley Deep Drive, and by Toyota under the iCyPhy center. J.P. Katoen is funded by the ERC Advanced Research Grant 787914 FRAPPANT.

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Authors and Affiliations

  1. Brno University of Technology, Brno, Czech Republic

    Milan Češka

  2. RWTH Aachen University, Aachen, Germany

    Christian Hensel & Joost-Pieter Katoen

  3. University of California, Berkeley, CA, USA

    Sebastian Junges

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  1. Milan Češka
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  2. Christian Hensel
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Correspondence to Sebastian Junges.

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Češka, M., Hensel, C., Junges, S. et al. Counterexample-guided inductive synthesis for probabilistic systems. Form Asp Comp 33, 637–667 (2021). https://doi.org/10.1007/s00165-021-00547-2

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