Formal Methods in System Design

, Volume 36, Issue 1, pp 65–95

Pushdown module checking



Model checking is a useful method to verify automatically the correctness of a system with respect to a desired behavior, by checking whether a mathematical model of the system satisfies a formal specification of this behavior. Many systems of interest are open, in the sense that their behavior depends on the interaction with their environment. The model checking problem for finite-state open systems (called module checking) has been intensively studied in the literature. In this paper, we focus on open pushdown systems and we study the related model-checking problem (pushdown module checking, for short) with respect to properties expressed by CTL and CTL* formulas. We show that pushdown module checking against CTL (resp., CTL*) is 2Exptime-complete (resp., 3Exptime-complete). Moreover, we prove that for a fixed CTL or CTL* formula, the problem is Exptime-complete.


Module checking Pushdown systems Branching temporal logics Tree automata 


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  1. 1.
    Aminof A, Murano A, Vardi MY (2007) Pushdown module checking with imperfect information. In: Proc 18th international conference on concurrency theory (CONCUR’07). LNCS, vol 4703. Springer, Berlin, pp 461–476 Google Scholar
  2. 2.
    Bouajjani A, Esparza J, Maler O (1997) Reachability analysis of pushdown automata: application to model-checking. In: Proc 8th international conference on concurrency theory (CONCUR’97). LNCS, vol 1243. Springer, Berlin, pp 135–150 Google Scholar
  3. 3.
    Bozzelli L (2006) Complexity results on branching-time pushdown model checking. In: Proc 7th conference on verification, model checking, and abstract interpretation (VMCAI’06). LNCS, vol 3855. Springer, Berlin, pp 65–79 CrossRefGoogle Scholar
  4. 4.
    Bozzelli L, Murano A, Peron A (2005) Pushdown module checking. In: Proc 12th int conf on logic for programming, artificial intelligence, and reasoning (LPAR’05). LNCS, vol 3835. Springer, Berlin, pp 504–518 CrossRefGoogle Scholar
  5. 5.
    Buchi JR (1962) On a decision method in restricted second order arithmetic. In: Proc internat congr logic, method and philos sci 1960, Stanford, pp 1–12 Google Scholar
  6. 6.
    Cachat T (2002) Two-way tree automata solving pushdown games. In: Automata, logics, and infinite games. LNCS, vol 2500. Springer, Berlin, pp 303–317 CrossRefGoogle Scholar
  7. 7.
    Chandra AK, Kozen DC, Stockmeyer LJ (1981) Alternation. J ACM 28(1):114–133 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Clarke EM, Emerson EA (1981) Design and verification of synchronization skeletons using branching time temporal logic. In: Proceedings of workshop on logic of programs. LNCS, vol 131. Springer, Berlin, pp 52–71 CrossRefGoogle Scholar
  9. 9.
    Emerson EA, Halpern JY (1986) Sometimes and not never revisited: on branching versus linear time. J ACM 33(1):151–178 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Emerson EA, Jutla CS (1988) The complexity of tree automata and logics of programs. In: 29th annual IEEE symposium on foundations of computer science (FOCS’88), pp 328–337 Google Scholar
  11. 11.
    Emerson EA, Jutla CS (1991) Tree automata, μ-calculus and determinacy. In: 32nd annual IEEE symposium on the foundations of computer science (FOCS’91), pp 368–377 Google Scholar
  12. 12.
    Esparza J, Kucera A, Schwoon S (2003) Model checking LTL with regular valuations for pushdown systems. Inf Comput 186(2):355–376 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ferrante A, Murano A, Parente M (2008) Enriched μ-calculi module checking. Log Methods Comput Sci 4(3):1–21 MathSciNetGoogle Scholar
  14. 14.
    Hoare CAR (1985) Communicating sequential processes. Prentice-Hall, New York MATHGoogle Scholar
  15. 15.
    Kupferman O, Grumberg O (1996) Buy one, get one free!!! J Log Comput 6(4):523–539 MATHMathSciNetGoogle Scholar
  16. 16.
    Kupferman O, Thiagarajan PS, Madhusudan P, Vardi MY (2000) Open systems in reactive environments: Control and Synthesis. In: Proc 11th international conference on concurrency theory (CONCUR’00). LNCS, vol 1877. Springer, Berlin, pp 92–107 Google Scholar
  17. 17.
    Kupferman O, Vardi MY, Wolper P (2000) An automata-theoretic approach to branching-time model checking. J ACM 47(2):312–360 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kupferman O, Vardi MY, Wolper P (2001) Module checking. Inf Comput 164(2):322–344 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kupferman O, Piterman N, Vardi MY (2002) Pushdown specifications. In: 9th int conf on logic for programming, artificial intelligence, and reasoning (LPAR’02). LNAI, vol 2514. Springer, Berlin, pp 262–277 CrossRefGoogle Scholar
  20. 20.
    Loding C, Madhusudan P, Serre O (2004) Visibly pushdown games. In: Proc 24th conference on foundations of software technology and theoretical computer science (FST&TCS’04). Springer, Berlin, pp 408–420 CrossRefGoogle Scholar
  21. 21.
    Miyano S, Hayashi T (1984) Alternating finite automata on ω-words. Theor Comput Sci 32:321–330 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Muller DE, Shupp PE (1985) The theory of ends, pushdown automata, and second-order logic. Theor Comput Sci 37:51–75 MATHCrossRefGoogle Scholar
  23. 23.
    Queille JP, Sifakis J (1981) Specification and verification of concurrent programs in Cesar. In: Proceedings of the fifth international symposium on programming. LNCS, vol 137. Springer, Berlin, pp 337–351 Google Scholar
  24. 24.
    Vardi MY (1998) Reasoning about the past with two-way automata. In: Proc 25th international colloquium on automata, languages and programming (ICALP’98). LNCS, vol 1443. Springer, Berlin, pp 628–641 CrossRefGoogle Scholar
  25. 25.
    Vardi MY, Wolper P (1986) Automata-theoretic techniques for modal logics of programs. J Comput Syst Sci 32(2):182–221 MathSciNetGoogle Scholar
  26. 26.
    Walukiewicz I (1996) Pushdown processes: games and model checking. In: Proc 8th international conference on computer aided verification (CAV’96). LNCS, vol 1102. Springer, Berlin, pp 62–74 Google Scholar
  27. 27.
    Walukiewicz I (2000) Model checking CTL properties of pushdown systems. In: Proc 20th conference on foundations of software technology and theoretical computer science (FST&TCS’00). LNCS, vol 1974. Springer, Berlin, pp 127–138 Google Scholar
  28. 28.
    Walukiewicz I (2002) Monadic second-order logic on tree-like structures. Theor Comput Sci 275:311–346 MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.IRISACampus Universitaire de BeaulieuRennes CedexFrance
  2. 2.Dipartimento di Scienze FisicheUniversità di Napoli “Federico II”NapoliItaly

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