Abstract
The model checking problem for open systems (called module checking) has been intensively studied in the literature, both for finite–state and infinite–state systems. In this paper, we focus on pushdown module checking with respect to μ–calculus enriched with graded and nominals (hybrid graded μ -calulus). We show that this problem is decidable and solvable in double–exponential time in the size of the formula and in exponential time in the size of the system. This result is obtained by exploiting a classical automata–theoretic approach via pushdown nondeterministic parity tree automata. In particular, we reduce in exponential time our problem to the emptiness problem for these automata, which is known to be decidable in Exptime. As a key step of our algorithm, we show an exponential improvement of the construction of a nondeterministic parity tree automaton accepting all models of a formula of the considered logic. This result, not only allows our algorithm to match the known lower bound, but it is also interesting by itself, since it allows investigating decision problems related to enriched μ-calculus formulas in a greatly simplified manner. We conclude the paper with a discussion on the model checking w.r.t. μ-calculus formulas enriched with backward modalities as well.
Work partially supported by MIUR FIRB Project no. RBAU1P5SS and the grant Metodi per la Verifica di Sistemi Software e Real-Time from University of Salerno.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bonatti, P.A., Lutz, C., Murano, A., Vardi, M.Y.: The complexity of enriched μ-calculi. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 540–551. Springer, Heidelberg (2006)
Bozzelli, L., Murano, A., Peron, A.: Pushdown module checking. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 504–518. Springer, Heidelberg (2005)
Bonatti, P.A., Peron, A.: On the undecidability of logics with converse, nominals, recursion and counting. Artif. Intelligence 158(1), 75–96 (2004)
Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Logics of Programs. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)
Clarke, E.M., Grumberg, O., Peled, D.A.: Model checking. MIT Press, Cambridge (1999)
Ferrante, A., Murano, A.: Enriched μ-calculus module checking. In: FOSSACS 2007. LNCS, vol. 4423, pp. 183–197 (2007)
Hoare, C.A.R.: Communicating sequential processes. Prentice-Hall International, Upper Saddle River (1985)
Harel, D., Pnueli, A.: On the development of reactive systems. In: Logics and Models of Concurrent Systems. NATO Advanced Summer Institutes, vol. F-13, pp. 477–498. Springer, Heidelberg (1985)
Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27, 333–354 (1983)
Kupferman, O., Piterman, N., Vardi, M.Y.: Pushdown specifications. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 262–277. Springer, Heidelberg (2002)
Kupferman, O., Sattler, U., Vardi, M.Y.: The complexity of the graded μ-calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002)
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. of ACM 47(2), 312–360 (2000)
Kupferman, O., Vardi, M.Y., Wolper, P.: Module checking. Information & Computation 164, 322–344 (2001)
Queille, J.P., Sifakis, J.: Specification and verification of concurrent systems in cesar. In: Dezani-Ciancaglini, M., Montanari, U. (eds.) International Symposium on Programming. LNCS, vol. 137, pp. 337–351. Springer, Heidelberg (1982)
Sattler, U., Vardi, M.Y.: The hybrid mu-calculus. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 76–91. Springer, Heidelberg (2001)
Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ferrante, A., Murano, A., Parente, M. (2007). Enriched μ–Calculus Pushdown Module Checking. In: Dershowitz, N., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2007. Lecture Notes in Computer Science(), vol 4790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75560-9_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-75560-9_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75558-6
Online ISBN: 978-3-540-75560-9
eBook Packages: Computer ScienceComputer Science (R0)