Abstract
Vacuity is a leading sanity check in model-checking, applied when the system is found to satisfy the specification. The check detects situations where the specification passes in a trivial way, say when a specification that requires every request to be followed by a grant is satisfied in a system with no requests. Such situations typically reveal problems in the modelling of the system or the specification, and indeed vacuity detection is a part of most industrial model-checking tools.
Existing research and tools for vacuity concern discrete-time systems and specification formalisms. We introduce real-time vacuity, which aims to detect problems with real-time modelling. Real-time logics are used for the specification and verification of systems with a continuous-time behavior. We study vacuity for the branching real-time logic TCTL, and focus on vacuity with respect to the time constraints in the specification. Specifically, the logic TCTL includes the temporal operator \(U^J\), which specifies real-time eventualities in real-time systems: the parameter 
is an interval with integral boundaries that bounds the time in which the eventuality should hold. We define several tightenings for the \(U^J\) operator. These tightenings require the eventuality to hold within a strict subset of J. We prove that vacuity detection for TCTL is PSPACE-complete, thus it does not increase the complexity of model-checking of TCTL. Our contribution involves an extension, termed TCTL\(^+\), of TCTL, which allows the interval J not to be continuous, and for which model checking stays in PSPACE. Finally, we describe a method for ranking vacuity results according to their significance.
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Notes
- 1.
The above definition assumes that \(\psi \) appears once in \(\varPhi \), or at least that all its occurrences are of the same polarity; a definition that is independent of this assumption replaces \(\psi \) by a universally quantified proposition [5]. Alternatively, one could focus on a single occurrence of \(\psi \).
- 2.
As discussed on Sect. 1, we assumes that \(\psi \) appears once in \(\varPhi \) or focus on a single occurrence of \(\psi \) in \(\varPhi \).
- 3.
A similar construction has been used in [1] for showing the lower bound of model-checking TCTL formulas, but the proof we have here is different and more involved.
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Chockler, H., Guha, S., Kupferman, O. (2018). Timed Vacuity. In: Havelund, K., Peleska, J., Roscoe, B., de Vink, E. (eds) Formal Methods. FM 2018. Lecture Notes in Computer Science(), vol 10951. Springer, Cham. https://doi.org/10.1007/978-3-319-95582-7_26
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