Abstract
We study languages of nested trees—structures obtained by augmenting trees with sets of nested jump-edges. These graphs can naturally model branching behaviors of pushdown programs, so that the problem of branching-time software model checking may be phrased as a membership question for such languages. We define finite-state automata accepting such languages—these automata can pass states along jump-edges as well as tree edges. We find that the model-checking problem for these automata on pushdown systems is EXPTIME-complete, and that their alternating versions are expressively equivalent to NT-μ, a recently proposed temporal logic for nested trees that can express a variety of branching-time, “context-free” requirements. We also show that monadic second order logic (MSO) cannot exploit the structure: MSO on nested trees is too strong in the sense that it has an undecidable model checking problem, and seems too weak to capture NT-μ.
This research was partially supported by ARO URI award DAAD19-01-1-0473 and NSF award CCR-0306382.
Chapter PDF
References
Alur, R., Chaudhuri, S., Madhusudan, P.: A fixpoint calculus for local and global program flows. In: Proc. of POPL 2006, pp. 153–165 (2006)
Alur, R., Chaudhuri, S., Madhusudan, P.: Languages of nested trees. University of Pennsylvania Technical Report MS-CIS-06-10 (2006)
Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. of STOC 2004, pp. 202–211 (2004)
Alur, R., Madhusudan, P.: Adding nesting structure to words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)
Ball, T., Rajamani, S.: The SLAM project: debugging system software via static analysis. In: Proc. of POPL 2002, pp. 1–3 (2002)
Burkart, O., Steffen, B.: Model checking the full modal mu-calculus for infinite sequential processes. Theoretical Computer Science 221, 251–270 (1999)
Caucal, D.: On infinite transition graphs having a decidable monadic theory. Theor. Comput. Sci. 290(1), 79–115 (2003)
Comon, H., Dauchet, M., Gilleron, R., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications. Draft (2003)
Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus, and determinacy. In: Proc. of FOCS 1991, pp. 368–377 (1991)
Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)
Henzinger, T.A., Jhala, R., Majumdar, R., Necula, G.C., Sutre, G., Weimer, W.: Temporal-safety proofs for systems code. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 526–538. Springer, Heidelberg (2002)
Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Reading (1979)
Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996)
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)
Löding, C.: Private communication
Löding, C., Madhusudan, P., Serre, O.: Visibly pushdown games. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 408–420. Springer, Heidelberg (2004)
McMillan, K.L.: Symbolic model checking: an approach to the state explosion problem. Kluwer Academic Publishers, Dordrecht (1993)
Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theor. Comput. Sci. 54(2-3), 267–276 (1987)
Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science 37, 51–75 (1985)
Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Transactions of the AMS 141, 1–35 (1969)
Reps, T., Horwitz, S., Sagiv, S.: Precise interprocedural dataflow analysis via graph reachability. In: Proc. of POPL 1995, pp. 49–61 (1995)
Walukiewicz, I.: Pushdown processes: Games and model-checking. Information and Computation 164(2), 234–263 (2001)
Walukiewicz, I.: Monadic second-order logic on tree-like structures. Theor. Comput. Sci. 275(1-2), 311–346 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alur, R., Chaudhuri, S., Madhusudan, P. (2006). Languages of Nested Trees. In: Ball, T., Jones, R.B. (eds) Computer Aided Verification. CAV 2006. Lecture Notes in Computer Science, vol 4144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11817963_31
Download citation
DOI: https://doi.org/10.1007/11817963_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37406-0
Online ISBN: 978-3-540-37411-4
eBook Packages: Computer ScienceComputer Science (R0)