Abstract
We show that the model checking problem for μ-calculus on graphs of bounded tree-width can be solved in time linear in the size of the system. The result is presented by first showing a related result: the winner in a parity game on a graph of bounded tree-width can be decided in polynomial time. The given algorithm is then modified to obtain a new algorithm for μ-calculus model checking. One possible use of this algorithm may be software verification, since control flow graphs of programs written in high-level languages are usually of bounded tree-width. Finally, we discuss some implications and future work.
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References
Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)
Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science: Volume B: Formal Models and Semantics, pp. 193–242. Elsevier, Amsterdam (1990)
Emerson, E., Jutla, C.: Tree automata, mu-calculus and determinancy. In: Proc. 5th IEE Foundations of Computer Science, pp. 368–377 (1991)
Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model-checking for fragments of mu-calculus. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 385–396. Springer, Heidelberg (1993)
Emerson, E.A., Lei, C.L.: Efficient model checking in fragments of the propositional μ–calculus. In: Symposion on Logic in Computer Science, pp. 267–278. IEEE Computer Society Press, Los Alamitos (1986)
Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: LICS 2002, pp. 215–224. IEEE Computer Society, Los Alamitos (2002)
Gustedt, J., Mæhle, O., Telle, J.A.: The treewidth of Java programs. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, p. 86. Springer, Heidelberg (2002)
Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)
Kloks, T. (ed.): Treewidth – computations and approximations. LNCS, vol. 842. Springer, Heidelberg (1994)
Kozen, D.: Results on the propositional μ-calculus. Theoretical Computer Science (TCS) 27, 333–354 (1983)
Long, D.E., Browne, A., Clarke, E.M., Jha, S., Marrero, W.R.: An improved algorithm for the evaluation of fixpoint expressions. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 338–350. Springer, Heidelberg (1994)
Mader, A.: Verification of Modal Properties Using Boolean Equation Systems, 8th edn. Bertz Verlag, Berlin (1997)
Obdržálek, J.: Fast mu-calculus model checking when tree-width is bounded. Technical report, LFCS (July 2003), http://www.dcs.ed.ac.uk/home/s0128832
Robertson, N., Seymour, P.D.: Graph Minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B 36, 49–63 (1984)
Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)
Stirling, C.: Modal and Temporal Properties of Processes. Texts in Computer Science. Springer, Heidelberg (2001)
Thorup, M.: All structured programs have small tree-width and good register allocation. Information and Computation 142(2), 159–181 (1998)
Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)
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Obdržálek, J. (2003). Fast Mu-Calculus Model Checking when Tree-Width Is Bounded. In: Hunt, W.A., Somenzi, F. (eds) Computer Aided Verification. CAV 2003. Lecture Notes in Computer Science, vol 2725. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45069-6_7
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DOI: https://doi.org/10.1007/978-3-540-45069-6_7
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