Abstract
We present an algorithm for verifying that a model M with timing constraints satisfies a given temporal property T. The model M is given as a composition of ω-automata P i , where each automaton P i is constrained by the bounds on delays. The property T is given as an ω-automaton as well, and the verification problem is posed as a language inclusion question \(\mathcal{L}(M) \subseteq \mathcal{L}(T)\). In constructing the composition M of the constrained automata P i , one needs to rule out the behaviors that are inconsistent with the delay bounds, and this step is (provably) computationally expensive. We propose an iterative solution which involves generating successive approximations M j to M, with containment \(\mathcal{L}(M) \subseteq \mathcal{L}(M_j )\) and monotone convergence \(\mathcal{L}(M_j ) \to \mathcal{L}(M)\) within a bounded number of steps. As the succession progresses, the M j , become more complex, but at any step of the iteration one may get a proof or a counter-example to the original language inclusion question.
We first construct M 0, the composition of the P i ignoring the delay constraints, and try to prove the language inclusion \(\mathcal{L}(M_0 ) \subseteq \mathcal{L}(T)\). If this succeeds, then \(\mathcal{L}(M) \subseteq \mathcal{L}(M_0 ) \subseteq \mathcal{L}(T)\). If this fails, we can find \(x \varepsilon \mathcal{L}(M_0 )\backslash \mathcal{L}(T)\) of the form x = σ′ σω. We give an algorithm to check for consistency of x with respect to the delay bounds of M: the time complexity of this check is linear in the length of σ′σ and cubic in the number of automata. If x is consistent with all the delay constraints of M, then x provides a counter-example to \(\mathcal{L}(M) \subseteq \mathcal{L}(T)\). Otherwise, we identify an “optimal” set of delay constraints D inconsistent with x. We generate an automaton P D which accepts only those behaviors that are consistent with the delay constraints in the set D. Then we add P d as a restriction to M 0, forming M 1, and iterate the algorithm.
In the worst case, the number of iterations needed is exponential in the number of delay constraints. Experience suggests that in typical cases, however, only a few delay constraints are material to the verification of any specific property T. Thus, resolution of the question \(\mathcal{L}(M) \subseteq \mathcal{L}(T)\) may be possible after only a few iterations of the algorithm, resulting in feasible language inclusion tests. This algorithm is being implemented into the verifier COSPAN at AT&T Bell Laboratories.
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References
R. Alur and D.L. Dill. Automata for modeling real-time systems. In Automata, Languages and Programming: Proceedings of the 17th ICALP, Lecture Notes in Computer Science 443, pages 322–335. Springer-Verlag, 1990.
Y. Choueka. Theories of automata on ω-tapes: a simplified approach. Journal of Computer and System Sciences, 8:117–141, 1974.
D.L. Dill. Timing assumptions and verification of finite-state concurrent systems. In J. Sifakis, editor, Automatic Verification Methods for Finite State Systems, Lecture Notes in Computer Science 407. Springer-Verlag, 1989.
R. P. Kurshan. Reducibility in analysis of coordination. In Lecture Notes in Computer Science, volume 103, pages 19–39. Springer-Verlag, 1987.
R. P. Kurshan. Analysis of discrete event coordination. In Lecture Notes in Computer Science, volume 430, pages 414–453. Springer-Verlag, 1990.
M. Yannakakis. Graph-theoretic methods in database theory. In Proceedings of the 9th ACM Symposium on Principles of Database Systems, pages 230–242, 1990.
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© 1993 Springer-Verlag Berlin Heidelberg
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Alur, R., Itai, A., Kurshan, R., Yannakakis, M. (1993). Timing verification by successive approximation. In: von Bochmann, G., Probst, D.K. (eds) Computer Aided Verification. CAV 1992. Lecture Notes in Computer Science, vol 663. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56496-9_12
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DOI: https://doi.org/10.1007/3-540-56496-9_12
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