We present transitive-closure-based model checking (TCMC): a symbolic representation of the semantics of computational tree logic with fairness constraints (CTLFC) for finite models in first-order logic with transitive closure (FOLTC). TCMC is an expression of the complete model checking problem for CTLFC as a set of constraints in FOLTC without induction, iteration, or invariants. We implement TCMC in the Alloy Analyzer, showing how a transition system can be expressed declaratively and concisely in the Alloy language. Since the total state space is rarely representable due to the state-space explosion problem, we present scoped TCMC where the property is checked for state spaces of a size smaller than the total state space. We address the problem of spurious instances and carefully describe the meaning of results from scoped TCMC with respect to the complete model checking problem. Using case studies, we demonstrate scoped TCMC and compare it with bounded model checking, highlighting how TCMC can check infinite paths.
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This translation increases the size of a transition system.
A full subgraph of a graph is a subset of the nodes with all edges between these nodes that are found in the original graph.
Infinite liveness, also described in this figure, is explained in a later subsection.
The use of id[X] in EG (from which AF and AU are derived) in the TCMC implementation in Fig. 2 requires there to be a looping path from a state back to itself to make an infinite path.
Existential TCMC requires the satisfying TS instance to have some path from all initial states of the TS instance; however, unless the model requires there to be multiple initial states, usually there is a TS instance with only one initial state meaning there is some path from some initial state.
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We thank Amin Bandali, George Gao, Eunsuk Kang, Mitchell Kember, Joseph Poremba, Jose Serna, Khadija Tariq and Lynn Tran for their help in discussions regarding Alloy. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Communicated by Dr. Antonio Cerone.
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Farheen, S., Day, N.A., Vakili, A. et al. Transitive-closure-based model checking (TCMC) in Alloy. Softw Syst Model (2020). https://doi.org/10.1007/s10270-019-00763-8
- Symbolic model checking
- Declarative models