Abstract
To guarantee the completeness of bounded model checking (BMC) we require a completeness threshold. The diameter of the Kripke model of the transition system is a valid completeness threshold for BMC of safety properties. The recurrence diameter gives us an upper bound on the diameter for use in practice. Transition systems are usually described using (propositionally) factored representations. Bounds for such lifted representations are calculated in a compositional way, by first identifying and bounding atomic subsystems, and then composing those results according to subsystem dependencies to arrive at a bound for the concrete system. Compositional approaches are invalid when using the diameter to bound atomic subsystems, and valid when using the recurrence diameter. We provide a novel overapproximation of the diameter, called the sublist diameter, that is tighter than the recurrence diameter. We prove that compositional approaches are valid using it to bound atomic subsystems. Those proofs are mechanised in HOL4. We also describe a novel verified compositional bounding technique which provides tighter overall bounds compared to existing bottom-up approaches.
M. Abdulaziz, C. Gretton and M. Norrish—NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.
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Notes
- 1.
Allowing any action to execute in any state is a deviation from standard practice. By using total functions in our formalism, much of the resulting mathematics is relatively simple.
- 2.
With this definition the diameter will be one less than how it was defined in [2].
- 3.
- 4.
\( \rightarrow \) has a \(\Pi \) parameter, but we use it with HOL’s ad hoc overloading ability.
- 5.
top-sorted has a \(\Pi \) parameter and \({\mathcal {C}}\) has \(\Pi \) and \(G_{ vs }\) as parameters hidden with overloading.
- 6.
\(\mathsf N \) has \(\Pi \) and \(G_{ vs }\) as parameters hidden with overloading.
- 7.
\(\mathsf N _b\) has \(\Pi \) and \(G_{ vs }\) as parameters hidden with overloading.
- 8.
The point of this experiment is to show how the computed upper bound grows with different parameters, regardless of the base case function used.
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Acknowledgements
We thank Daniel Jackson for suggesting applying diameter upper bounding on the hotel key protocol verification.
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Abdulaziz, M., Gretton, C., Norrish, M. (2015). Verified Over-Approximation of the Diameter of Propositionally Factored Transition Systems. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_1
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