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Parameterized Verification of Systems with Precise (0,1)-Counter Abstraction

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Verification, Model Checking, and Abstract Interpretation (VMCAI 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15529))

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Abstract

We introduce a new framework for verifying systems with a parametric number of concurrently running processes. The systems we consider are well-structured with respect to a specific well-quasi order. This allows us to decide a wide range of verification problems, including control-state reachability, coverability, and target, in a fixed finite abstraction of the infinite state-space, called a 01-counter system. We show that several systems from the parameterized verification literature fall into this class, including reconfigurable broadcast networks (or systems with lossy broadcast), disjunctive systems, synchronizations and systems with a fixed number of shared finite-domain variables. Our framework provides a simple and unified explanation for the properties of these systems, which have so far been investigated separately. Additionally, it extends and improves on a range of the existing results, and gives rise to other systems with similar properties.

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Notes

  1. 1.

    To see this, note first that if a system has multiple controllers, we can encode all of them as a single controller by simply considering their (finite-state) product. To support k different types of user processes with state sets \(Q_1,\ldots ,Q_k\) such that \(Q_i \cap Q_j = \emptyset \) for all \(i \ne j\), we simply construct one big user process with state set \(Q_1 \cup \cdots \cup Q_k\), and similarly let the union of all individual initial states be the initial states of the constructed system.

  2. 2.

    This is sometimes called strong compatibility in the literature.

  3. 3.

    The restriction to a single variable is for simplicity, our results extend to multiple finite-domain variables.

  4. 4.

    Internal steps can be seen as a special case of lossy broadcast, disjunctive guard, or ASM steps.

  5. 5.

    E.g., for RBN without a controller, CRP for \(CC[\ge 1]\) is in PTIME, and for \(CC[\ge 1,=0]\) it is in NP  [19].

References

  1. Abdulla, P.A., Cerans, K., Jonsson, B., Tsay, Y.: General decidability theorems for infinite-state systems. In: Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey, USA, 27–30 July 1996, pp. 313–321. IEEE Computer Society (1996). https://doi.org/10.1109/LICS.1996.561359

  2. Abdulla, P.A., Cerans, K., Jonsson, B., Tsay, Y.: Algorithmic analysis of programs with well quasi-ordered domains. Inf. Comput. 160(1–2), 109–127 (2000). https://doi.org/10.1006/INCO.1999.2843

    Article  MathSciNet  MATH  Google Scholar 

  3. Aminof, B., Kotek, T., Rubin, S., Spegni, F., Veith, H.: Parameterized model checking of rendezvous systems. Distrib. Comput. 31(3), 187–222 (2018). https://doi.org/10.1007/S00446-017-0302-6

    Article  MathSciNet  MATH  Google Scholar 

  4. André, É., Eichler, P., Jacobs, S., Karra, S.L.: Parameterized verification of disjunctive timed networks. In: Dimitrova, R., Lahav, O., Wolff, S. (eds.) Verification, Model Checking, and Abstract Interpretation - 25th International Conference, VMCAI 2024, London, United Kingdom, 15–16 January 2024, Proceedings, Part I. Lecture Notes in Computer Science, vol. 14499, pp. 124–146. Springer, Cham (2024). https://doi.org/10.1007/978-3-031-50524-9_6

  5. Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distrib. Comput. 20(4), 279–304 (2007). https://doi.org/10.1007/S00446-007-0040-2

    Article  MATH  Google Scholar 

  6. Apt, K.R., Kozen, D.: Limits for automatic verification of finite-state concurrent systems. Inf. Process. Lett. 22(6), 307–309 (1986). https://doi.org/10.1016/0020-0190(86)90071-2

    Article  MathSciNet  MATH  Google Scholar 

  7. Außerlechner, S., Jacobs, S., Khalimov, A.: Tight cutoffs for guarded protocols with fairness. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 476–494. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49122-5_23

    Chapter  MATH  Google Scholar 

  8. Balasubramanian, A.R., Bertrand, N., Markey, N.: Parameterized verification of synchronization in constrained reconfigurable broadcast networks. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 38–54. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89963-3_3

    Chapter  MATH  Google Scholar 

  9. Balasubramanian, A.R., Guillou, L., Weil-Kennedy, C.: Parameterized analysis of reconfigurable broadcast networks. In: FoSSaCS 2022. LNCS, vol. 13242, pp. 61–80. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-99253-8_4

    Chapter  MATH  Google Scholar 

  10. Balasubramanian, A.R., Weil-Kennedy, C.: Reconfigurable broadcast networks and asynchronous shared-memory systems are equivalent. In: Ganty, P., Bresolin, D. (eds.) Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2021, Padua, Italy, 20–22 September 2021. EPTCS, vol. 346, pp. 18–34 (2021). https://doi.org/10.4204/EPTCS.346.2

  11. Baumeister, T., Eichler, P., Jacobs, S., Sakr, M., Völp, M.: Parameterized verification of round-based distributed algorithms via extended threshold automata. In: Platzer, A., Rozier, K.Y., Pradella, M., Rossi, M. (eds.) Formal Methods - 26th International Symposium, FM 2024, Milan, Italy, 9–13 September 2024, Proceedings, Part I. Lecture Notes in Computer Science, vol. 14933, pp. 638–657. Springer, Cham (2024). https://doi.org/10.1007/978-3-031-71162-6_33

  12. Bertrand, N., Dewaskar, M., Genest, B., Gimbert, H., Godbole, A.A.: Controlling a population. Log. Methods Comput. Sci. 15(3) (2019). https://doi.org/10.23638/LMCS-15(3:6)2019

  13. Bertrand, N., Fournier, P., Sangnier, A.: Playing with probabilities in reconfigurable broadcast networks. In: Muscholl, A. (ed.) FoSSaCS 2014. LNCS, vol. 8412, pp. 134–148. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54830-7_9

    Chapter  MATH  Google Scholar 

  14. Bloem, R., et al.: Decidability of Parameterized Verification. Synthesis Lectures on Distributed Computing Theory, Morgan & Claypool Publishers (2015). https://doi.org/10.2200/S00658ED1V01Y201508DCT013

  15. Bouyer, P., Markey, N., Randour, M., Sangnier, A., Stan, D.: Reachability in networks of register protocols under stochastic schedulers. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, 11–15 July 2016, Rome, Italy. LIPIcs, vol. 55, pp. 106:1–106:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016). https://doi.org/10.4230/LIPIcs.ICALP.2016.106

  16. Colcombet, T., Fijalkow, N., Ohlmann, P.: Controlling a random population. Log. Methods Comput. Sci. 17(4) (2021). https://doi.org/10.46298/LMCS-17(4:12)2021

  17. Delzanno, G., Sangnier, A., Traverso, R., Zavattaro, G.: On the complexity of parameterized reachability in reconfigurable broadcast networks. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, 15–17 December 2012, Hyderabad, India. LIPIcs, vol. 18, pp. 289–300. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012). https://doi.org/10.4230/LIPICS.FSTTCS.2012.289

  18. Delzanno, G., Sangnier, A., Zavattaro, G.: Parameterized verification of ad hoc networks. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 313–327. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15375-4_22

    Chapter  MATH  Google Scholar 

  19. Delzanno, G., Sangnier, A., Zavattaro, G.: Verification of ad hoc networks with node and communication failures. In: Giese, H., Rosu, G. (eds.) FMOODS/FORTE -2012. LNCS, vol. 7273, pp. 235–250. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30793-5_15

    Chapter  MATH  Google Scholar 

  20. Eichler, P., Jacobs, S., Weil-Kennedy, C.: Parameterized verification of systems with precise (0,1)-counter abstraction (2024). https://arxiv.org/abs/2408.05954

  21. Emerson, E.A., Kahlon, V.: Reducing model checking of the many to the few. In: McAllester, D. (ed.) CADE 2000. LNCS (LNAI), vol. 1831, pp. 236–254. Springer, Heidelberg (2000). https://doi.org/10.1007/10721959_19

    Chapter  MATH  Google Scholar 

  22. Emerson, E.A., Kahlon, V.: Model checking guarded protocols. In: 18th IEEE Symposium on Logic in Computer Science (LICS 2003), 22–25 June 2003, Ottawa, Canada, Proceedings, pp. 361–370. IEEE Computer Society (2003). https://doi.org/10.1109/LICS.2003.1210076

  23. Emerson, E.A., Namjoshi, K.S.: Reasoning about rings. In: Cytron, R.K., Lee, P. (eds.) Conference Record of POPL’95: 22nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, San Francisco, California, USA, 23–25 January 1995, pp. 85–94. ACM Press (1995). https://doi.org/10.1145/199448.199468

  24. Emerson, E.A., Namjoshi, K.S.: On model checking for non-deterministic infinite-state systems. In: Thirteenth Annual IEEE Symposium on Logic in Computer Science, Indianapolis, Indiana, USA, 21–24 June 1998, pp. 70–80. IEEE Computer Society (1998). https://doi.org/10.1109/LICS.1998.705644

  25. Esparza, J.: Keeping a crowd safe: on the complexity of parameterized verification (invited talk). In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, 5–8 March 2014, Lyon, France. LIPIcs, vol. 25, pp. 1–10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014). https://doi.org/10.4230/LIPICS.STACS.2014.1

  26. Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, 2–5 July 1999, pp. 352–359. IEEE Computer Society (1999). https://doi.org/10.1109/LICS.1999.782630

  27. Esparza, J., Ganty, P., Majumdar, R.: Parameterized verification of asynchronous shared-memory systems. J. ACM 63(1), 10:1–10:48 (2016). https://doi.org/10.1145/2842603

  28. Esparza, J., Raskin, M., Weil-Kennedy, C.: Parameterized analysis of immediate observation petri nets. In: Donatelli, S., Haar, S. (eds.) PETRI NETS 2019. LNCS, vol. 11522, pp. 365–385. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21571-2_20

    Chapter  MATH  Google Scholar 

  29. Finkel, A.: Reduction and covering of infinite reachability trees. Inf. Comput. 89(2), 144–179 (1990). https://doi.org/10.1016/0890-5401(90)90009-7

  30. Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theor. Comput. Sci. 256(1–2), 63–92 (2001). https://doi.org/10.1016/S0304-3975(00)00102-X

    Article  MathSciNet  MATH  Google Scholar 

  31. German, S.M., Sistla, A.P.: Reasoning about systems with many processes. J. ACM 39(3), 675–735 (1992). https://doi.org/10.1145/146637.146681

    Article  MathSciNet  MATH  Google Scholar 

  32. Jacobs, S., Sakr, M.: Analyzing guarded protocols: better cutoffs, more systems, more expressivity. In: VMCAI 2018. LNCS, vol. 10747, pp. 247–268. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73721-8_12

    Chapter  MATH  Google Scholar 

  33. Jacobs, S., Sakr, M., Völp, M.: Automatic repair and deadlock detection for parameterized systems. In: Griggio, A., Rungta, N. (eds.) 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, 17–21 October 2022, pp. 225–234. IEEE (2022). https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_29

  34. Kozen, D.: Lower bounds for natural proof systems. In: 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October - 1 November 1977, pp. 254–266. IEEE Computer Society (1977). https://doi.org/10.1109/SFCS.1977.16

  35. Kruskal, J.B.: The theory of well-quasi-ordering: a frequently discovered concept. J. Comb. Theory, Ser. A 13(3), 297–305 (1972). https://doi.org/10.1016/0097-3165(72)90063-5

  36. de Luca, A., Varricchio, S.: Well quasi-orders and regular languages. Acta Informatica 31(6), 539–557 (1994). https://doi.org/10.1007/BF01213206

    Article  MathSciNet  MATH  Google Scholar 

  37. Pnueli, A., Xu, J., Zuck, L.: Liveness with (0,1, \(\infty \))- counter abstraction. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 107–122. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45657-0_9

    Chapter  MATH  Google Scholar 

  38. Schmitz, S., Schnoebelen, P.: The power of well-structured systems. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013. LNCS, vol. 8052, pp. 5–24. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40184-8_2

    Chapter  MATH  Google Scholar 

  39. Suzuki, I.: Proving properties of a ring of finite-state machines. Inf. Process. Lett. 28(4), 213–214 (1988). https://doi.org/10.1016/0020-0190(88)90211-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank Javier Esparza and Pierre Ganty for many helpful discussions at the start of this paper. P. Eichler carried out this work as a member of the Saarbrücken Graduate School of Computer Science. This research was funded in part by the German Research Foundation (DFG) grant GSP&Co (No. 513487900). C. Weil-Kennedy’s work was supported by the grant PID2022-138072OB-I00, funded by MCIN, FEDER, UE and partially supported by PRODIGY Project (TED2021-132464B-I00) funded by MCIN and the European Union NextGeneration.

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Authors and Affiliations

  1. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Paul Eichler & Swen Jacobs

  2. IMDEA Software Institute, Madrid, Spain

    Chana Weil-Kennedy

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  1. Paul Eichler
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Correspondence to Paul Eichler .

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  1. Indian Institute of Technology Bombay, Mumbai, Maharashtra, India

    Krishna Shankaranarayanan

  2. University of Colorado Boulder, Boulder, CO, USA

    Sriram Sankaranarayanan

  3. University of Colorado Boulder, Boulder, CO, USA

    Ashutosh Trivedi

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Eichler, P., Jacobs, S., Weil-Kennedy, C. (2025). Parameterized Verification of Systems with Precise (0,1)-Counter Abstraction. In: Shankaranarayanan, K., Sankaranarayanan, S., Trivedi, A. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2025. Lecture Notes in Computer Science, vol 15529. Springer, Cham. https://doi.org/10.1007/978-3-031-82700-6_5

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