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Fast and Effective Well-Definedness Checking

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Integrated Formal Methods (IFM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12546))

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Abstract

Well-Definedness is important for many formal methods. In B and Event-B it ensures that certain kinds of errors (e.g., division by 0) cannot appear and that proof rules based on two-valued logic are sound. For validation tools such as ProB, well-definedness is important for constraint solving. B and Event-B establish well-definedness by generating dedicated proof obligations (POs). Unfortunately, the standard provers are not always very good at discharging them. In this paper, we present a new integrated technique to simultaneously generate and discharge well-definedness POs. The implementation contains a dedicated rule-based prover written in Prolog supporting B, Event-B and extensions thereof for data validation. We show that the generation and discharging is significantly faster than existing implementations in rodin and Atelier-B and that a large number of POs are automatically discharged. The POs are fine-grained enough to provide precise source code feedback, and allow inspection of problematic POs within various editors.

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Notes

  1. 1.

    Taken from https://en.wikipedia.org/wiki/Brouwer?Hilbert_controversy.

  2. 2.

    [12] discusses combining power of \(\mathcal{D}\) with the efficiency of \(\mathcal{L}\), but is not used in practice as far as we know. It seems to require one to establish the truth or falsity of individual formulas, which may not be easily feasible in practice.

  3. 3.

    For modulo -3 mod 2 = −1 is well-defined and true in Event-B, but is not well-defined in classical B. But this is not due to a difference in the WD condition, but due to the fact that -3 mod 2 is parsed as -(3 mod 2) in rodin and (-3) mod 2 in Atelier-B. The rodin handbook requires modulo arguments to be non-negative, which is correct; [26] is in error.

  4. 4.

    Atelier-B now uses full WP calculus (private communication from Lilian Burdy).

  5. 5.

    See FiniteRan.java in org.eventb.internal.core.seqprover.eventbExtensions at https://sourceforge.net/p/rodin-b-sharp/rodincore. A lot of other proof rules are more compact, though.

  6. 6.

    The missing occurs check in Prolog is not an issue, because we use the ground representation for the B formulas, and hence any variable in a proof rule is always instantiated to a ground term.

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Acknowledgements

Big thanks go to Philipp Körner for scripts for extracting benchmark specification list, Thierry Lecomte for writing the sum Logic Solver example, Sebastian Stock for the VSCode integration and David Geleßus for the ProB2-UI integration. I wish to thank Jean-Raymond Abrial, Lilian Burdy, Michael Butler, Stefan Hallerstede and Laurent Voisin for useful feedback, Atelier-B and rodin implementation details and pointers to related research. In particular Laurent Voisin provided many useful hints and corrections. Finally, the anonymous referees of iFM provided very useful feedback.

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  1. Institut für Informatik, Universität Düsseldorf Universitätsstr. 1, 40225, Düsseldorf, Germany

    Michael Leuschel

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  1. Michael Leuschel
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Correspondence to Michael Leuschel .

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  1. University of Surrey, Guildford, UK

    Brijesh Dongol

  2. Royal Institute of Technology - KTH, Stockholm, Sweden

    Elena Troubitsyna

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Leuschel, M. (2020). Fast and Effective Well-Definedness Checking. In: Dongol, B., Troubitsyna, E. (eds) Integrated Formal Methods. IFM 2020. Lecture Notes in Computer Science(), vol 12546. Springer, Cham. https://doi.org/10.1007/978-3-030-63461-2_4

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  • DOI: https://doi.org/10.1007/978-3-030-63461-2_4

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