Constrained Dynamic Tree Networks

  • Matthew HagueEmail author
  • Vincent Penelle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11123)


We generalise Constrained Dynamic Pushdown Networks, introduced by Bouajjani et al., to Constrained Dynamic Tree Networks. In this model, we have trees of processes which may monitor their children. We allow the processes to be defined by any computation model for which the alternating reachability problem is decidable. We address the problem of symbolic reachability analysis for this model. More precisely, we consider the problem of computing an effective representation of their reachability sets using finite state automata. We show that backwards reachability sets starting from regular sets of configurations are always regular. We provide an algorithm for computing backwards reachability sets using tree automata.


Model-checking Dynamic networks Concurrency Pushdown systems Alternation Higher-order Collapsible pushdown systems 



We thank the anonymous reviewers for their remarks. This work was supported by the Engineering and Physical Sciences Research Council [EP/K009907/1].


  1. 1.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997). Scholar
  2. 2.
    Bouajjani, A., Müller-Olm, M., Touili, T.: Regular symbolic analysis of dynamic networks of pushdown systems. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 473–487. Springer, Heidelberg (2005). Scholar
  3. 3.
    Brainerd, W.S.: Tree generating regular systems. Inf. Control 14(2), 217–231 (1969)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Broadbent, C.H., Carayol, A., Hague, M., Serre, O.: A saturation method for collapsible pushdown systems. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7392, pp. 165–176. Springer, Heidelberg (2012). Scholar
  5. 5.
    Broadbent, C.H., Carayol, A., Hague, M., Serre, O.: C-SHORe: a collapsible approach to higher-order verification. In: ICFP (2013)Google Scholar
  6. 6.
    Broadbent, C.H., Kobayashi, N.: Saturation-based model checking of higher-order recursion schemes. In: CSL (2013)Google Scholar
  7. 7.
    Chadha, R., Viswanathan, M.: Decidability results for well-structured transition systems with auxiliary storage. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 136–150. Springer, Heidelberg (2007). Scholar
  8. 8.
    Clemente, L., Parys, P., Salvati, S., Walukiewicz, I.: Ordered tree-pushdown systems. In: FSTTCS (2015)Google Scholar
  9. 9.
    Clemente, L., Parys, P., Salvati, S., Walukiewicz, I.: The diagonal problem for higher-order recursive schemes is decidable. In: LICS (2016)Google Scholar
  10. 10.
    Cyriac, A., Gastin, P., Kumar, K.N.: MSO decidability of multi-pushdown systems via split-width. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 547–561. Springer, Heidelberg (2012). Scholar
  11. 11.
    Gawlitza, T.M., Lammich, P., Müller-Olm, M., Seidl, H., Wenner, A.: Join-lock-sensitive forward reachability analysis for concurrent programs with dynamic process creation. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 199–213. Springer, Heidelberg (2011). Scholar
  12. 12.
    Hague, M.: Saturation of concurrent collapsible pushdown systems. In: FSTTCS (2013)Google Scholar
  13. 13.
    Hague, M., Kochems, J., Ong, C.-H.L.: Unboundedness and downward closures of higher-order pushdown automata. In: POPL (2016)Google Scholar
  14. 14.
    Hague, M., Murawski, A.S., Ong, C.-H.L., Serre, O.: Collapsible pushdown automata and recursion schemes. In: LICS (2008)Google Scholar
  15. 15.
    Hague, M., Ong, C.-H.L.: Winning regions of pushdown parity games: a saturation method. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 384–398. Springer, Heidelberg (2009). Scholar
  16. 16.
    Hague, M., Penelle, V.: Constrained dynamic tree networks (2018).,
  17. 17.
    Kobayashi, N.: Model-checking higher-order functions. In: PPDP (2009)Google Scholar
  18. 18.
    Kobayashi, N.: Higher-order model checking: from theory to practice. In: LICS (2011)Google Scholar
  19. 19.
    Kobayashi, N.: A practical linear time algorithm for trivial automata model checking of higher-order recursion schemes. In: Hofmann, M. (ed.) FoSSaCS 2011. LNCS, vol. 6604, pp. 260–274. Springer, Heidelberg (2011). Scholar
  20. 20.
    Kobayashi, N., Igarashi, A.: Model-checking higher-order programs with recursive types. In: Felleisen, M., Gardner, P. (eds.) ESOP 2013. LNCS, vol. 7792, pp. 431–450. Springer, Heidelberg (2013). Scholar
  21. 21.
    Kobayashi, N.: GTRecS2: a model checker for recursion schemes based on games and types (2012).
  22. 22.
    La Torre, S., Muscholl, A., Walukiewicz, I.: Safety of parametrized asynchronous shared-memory systems is almost always decidable. In: CONCUR (2015)Google Scholar
  23. 23.
    Lammich, P., Müller-Olm, M., Wenner, A.: Predecessor sets of dynamic pushdown networks with tree-regular constraints. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 525–539. Springer, Heidelberg (2009). Scholar
  24. 24.
    Lammich, P., Müller-Olm, M., Seidl, H., Wenner, A.: Contextual locking for dynamic pushdown networks. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 477–498. Springer, Heidelberg (2013). Scholar
  25. 25.
    Löding, C.: Infinite graphs generated by tree rewriting. Ph.D. thesis, RWTH Aachen (2003)Google Scholar
  26. 26.
    Lugiez, D.: Forward analysis of dynamic network of pushdown systems is easier without order. Int. J. Found. Comput. Sci. 22(4), 843–862 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lugiez, D., Schnoebelen, P.: The regular viewpoint on PA-processes. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 50–66. Springer, Heidelberg (1998). Scholar
  28. 28.
    Madhusudan, P., Parlato, G.: The tree width of auxiliary storage. In: POPL (2011)Google Scholar
  29. 29.
    Muscholl, A., Seidl, H., Walukiewicz, I.: Reachability for dynamic parametric processes. In: Bouajjani, A., Monniaux, D. (eds.) VMCAI 2017. LNCS, vol. 10145, pp. 424–441. Springer, Cham (2017). Scholar
  30. 30.
    Neatherway, R.P., Ramsay, S.J., Ong, C.-H.L.: A traversal-based algorithm for higher-order model checking. In: ICFP (2012)Google Scholar
  31. 31.
    Nordhoff, B., Müller-Olm, M., Lammich, P.: Iterable forward reachability analysis of monitor-DPNs. In: Semantics, Abstract Interpretation, and Reasoning About Programs: Essays Dedicated to David A. Schmidt on the Occasion of his Sixtieth Birthday (2013)Google Scholar
  32. 32.
    Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: LICS (2006)Google Scholar
  33. 33.
    Parys, P.: The complexity of the diagonal problem for recursion schemes. In: FSTTCS (2018)Google Scholar
  34. 34.
    Penelle, V.: Rewriting higher-order stack trees. In: Beklemishev, L.D., Musatov, D.V. (eds.) CSR 2015. LNCS, vol. 9139, pp. 364–397. Springer, Cham (2015). Scholar
  35. 35.
    Qadeer, S., Rehof, J.: Context-bounded model checking of concurrent software. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 93–107. Springer, Heidelberg (2005). Scholar
  36. 36.
    Ramsay, S.J., Neatherway, R.P., Ong, C.-H.L.: A type-directed abstraction refinement approach to higher-order model checking. In: POPL (2014)Google Scholar
  37. 37.
    Schwoon, S.: Model-checking pushdown systems. Ph.D. thesis, Technical University of Munich (2002)Google Scholar
  38. 38.
    Seth, A.: Games on higher order multi-stack pushdown systems. In: Bournez, O., Potapov, I. (eds.) RP 2009. LNCS, vol. 5797, pp. 203–216. Springer, Heidelberg (2009). Scholar
  39. 39.
    Song, F., Touili, T.: Model checking dynamic pushdown networks. Form. Asp. Comput. 27(2), 397–421 (2015)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Suwimonteerabuth, D., Berger, F., Schwoon, S., Esparza, J.: jMoped: a test environment for java programs. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 164–167. Springer, Heidelberg (2007). Scholar
  41. 41.
    Touili, T., Atig, M.F.: Verifying parallel programs with dynamic communication structures. Theor. Comput. Sci. 411(38–39), 3460–3468 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yasukata, K., Kobayashi, N., Matsuda, K.: Pairwise reachability analysis for higher order concurrent programs by higher-order model checking. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 312–326. Springer, Heidelberg (2014). Scholar
  43. 43.
    Yasukata, K., Tsukada, T., Kobayashi, N.: Verification of higher-order concurrent programs with dynamic resource creation. In: Igarashi, A. (ed.) APLAS 2016. LNCS, vol. 10017, pp. 335–353. Springer, Cham (2016). Scholar
  44. 44.
    Zetzsche, G.: An approach to computing downward closures. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 440–451. Springer, Heidelberg (2015). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Royal Holloway, University of LondonEghamUK
  2. 2.Université de Bordeaux, LaBRI, UMR 5800TalenceFrance

Personalised recommendations