Algorithmic Metatheorems for Decidable LTL Model Checking over Infinite Systems

  • Anthony Widjaja To
  • Leonid Libkin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6014)


By algorithmic metatheorems for a model checking problem P over infinite-state systems we mean generic results that can be used to infer decidability (possibly complexity) of P not only over a specific class of infinite systems, but over a large family of classes of infinite systems. Such results normally start with a powerful formalism F of infinite-state systems, over which P is undecidable, and assert decidability when is restricted by means of an extra “semantic condition” C. We prove various algorithmic metatheorems for the problems of model checking LTL and its two common fragments \({\text{LTL}({\text{\bf F}_{\text{s}}},{\bf G}_{\text{s}})}\) and \({\text{LTL}_{\text{det}}}\) over the expressive class of word/tree automatic transition systems, which are generated by synchronized finite-state transducers operating on finite words and trees. We present numerous applications, where we derive (in a unified manner) many known and previously unknown decidability and complexity results of model checking LTL and its fragments over specific classes of infinite-state systems including pushdown systems; prefix-recognizable systems; reversal-bounded counter systems with discrete clocks and a free counter; concurrent pushdown systems with a bounded number of context-switches; various subclasses of Petri nets; weakly extended PA-processes; and weakly extended ground-tree rewrite systems. In all cases, we are able to derive optimal (or near optimal) complexity. Finally, we pinpoint the exact locations in the arithmetic and analytic hierarchies of the problem of checking a relevant semantic condition and the LTL model checking problems over all word/tree automatic systems.


Model Check Transition System Tree Automaton Model Check Problem Tree Transducer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abdulla, P.A., Jonsson, B.: Verifying programs with unreliable channels. Inf. Comput. 127(2), 91–101 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abdulla, P.A., Jonsson, B., Nilsson, M., Saksena, M.: A survey of regular model checking. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 35–48. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Alur, R., Dill, D.: A theory of timed automata. TCS 126, 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baier, C., Bertrand, N., Schnoebelen, P.: On Computing Fixpoints in Well-Structured Regular Model Checking, with Applications to Lossy Channel Systems. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 347–361. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Barany, V.: Automatic Presentations of Infinite Structures. PhD Thesis, RWTH Aachen (2007)Google Scholar
  6. 6.
    Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: acceleration from theory to practice. STTT 10(5), 401–424 (2008)CrossRefGoogle Scholar
  7. 7.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: automata and interpretations. Theory Comput. Syst. 37(6), 641–674 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bouajjani, A., Legay, A., Wolper, P.: A Framework to Handle Linear Temporal Properties in (ω)Regular Model Checking CoRR abs/0901.4080 (2009)Google Scholar
  9. 9.
    Burkart, O., Caucal, D., Moller, F., Steffen, B.: Verification on infinite structures. In: Handbook of Process Algebra. Elsevier, Amsterdam (1999)Google Scholar
  10. 10.
    Caucal, D.: On the regular structure of prefix rewriting. TCS 106, 61–86 (1992)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Comon, H., et al.: Tree Automata: Techniques and Applications (2007),
  13. 13.
    Dang, Z., Ibarra, O., Bultan, T., Kemmerer, R., Su, J.: Binary reachability analysis of discrete pushdown timed automata. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 69–84. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Dang, Z., Ibarra, O., Pietro, P.S.: Liveness verification of reversal-bounded multicounter machines with a free counter. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 132–143. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Elgot, C., Mezei, J.: On relations defined by generalized finite automata. IBM J. Res. Develop. 9, 47–68 (1965)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Esparza, J.: Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes. Fundam. Inform. 31(1), 13–25 (1997)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theor. Comput. Sci. 256(1-2), 63–92 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Göller, S.: Reachability on prefix-recognizable graphs. Inf. Process. Lett. 108(2), 71–74 (2008)CrossRefGoogle Scholar
  19. 19.
    Grohe, M.: Logic, graphs, and algorithms. In: Logic and Automata - History and Perspectives, pp. 357–422. Amsterdam University Press (2007)Google Scholar
  20. 20.
    Hague, M., Ong, C.-H.L.: Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems. LMCS 4(4) (2008)Google Scholar
  21. 21.
    Ibarra, O., Su, J., Dang, Z., Bultan, T., Kemmerer, R.: Counter machines and verification problems. Theor. Comput. Sci. 289, 165–189 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kupferman, O., Piterman, N., Vardi, M.: Model checking linear properties of prefix-recognizable systems. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 371–385. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  23. 23.
    Lal, A., Touili, T., Kidd, N., Reps, T.: Interprocedural Analysis of Concurrent Programs Under a Context Bound. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 282–298. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. 24.
    Leroux, J., Sutre, G.: Flat Counter Automata Almost Everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Löding, C.: Infinite Graphs Generated by Tree Rewriting, PhD thesis, RWTH Aachen (2003)Google Scholar
  26. 26.
    Lugiez, D., Schnoebelen, P.: Decidable first-order transition logics for PA-processes. Inf. Comput. 203(1), 75–113 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Maidl, M.: The Common Fragment of CTL and LTL. In: FOCS 2000, pp. 643–652 (2000)Google Scholar
  28. 28.
    Mayr, R.: Decidability and Complexity of Model Checking Problems for Infinite-State Systems. PhD thesis, TU-Munich (1998)Google Scholar
  29. 29.
    Morvan, C.: On rational graphs. In: FOSSACS 2000, pp. 252–266 (2000)Google Scholar
  30. 30.
    Qadeer, S., Rehof, J.: Context-Bounded Model Checking of Concurrent Software. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 242–254. Springer, Heidelberg (2005)Google Scholar
  31. 31.
    Rehak, V.: On Extensions of Process Rewrite Systems, PhD thesis, Masaryk University (2007)Google Scholar
  32. 32.
    Sistla, A.P., Clarke, E.M.: The Complexity of Propositional Linear Temporal Logics. J. ACM 32(3), 733–749 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Thomas, W.: Constructing infinite graphs with a decidable MSO-theory. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003)Google Scholar
  34. 34.
    To, A.W., Libkin, L.: Recurrent reachability analysis in regular model checking. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 198–213. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  35. 35.
    Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. JCSS 32(2), 183–221 (1986)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Verma, K.N., Seidl, H., Schwentick, T.: On the complexity of equational Horn clauses. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 337–352. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anthony Widjaja To
    • 1
  • Leonid Libkin
    • 1
  1. 1.LFCS, School of InformaticsUniversity of Edinburgh 

Personalised recommendations