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Spot the Difference: A Detailed Comparison Between B and Event-B

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Logic, Computation and Rigorous Methods

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12750))

Abstract

The B landscape can be confusing to formal methods outsiders, especially due to the fact that it is partitioned into classical B for software and Event-B for systems modelling. In this article we shed light on commonalities and differences between these formalisms, based on our experience in building tools that support both of them. In particular, we examine not so well-known pitfalls. For example, despite sharing a common mathematical foundation in predicate logic, set theory and arithmetic, there are formulas that are true in Event-B and false in classical B, and vice-versa.

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Notes

  1. 1.

    See http://rodin.cs.ncl.ac.uk.

  2. 2.

    Alloy’s multiplicity annotations cannot be understood so simply in this way; see [30].

  3. 3.

    https://wiki.event-b.org/index.php/Records_Extension.

  4. 4.

    Attribute grammars with inherited and synthesised attributes (see [10]).

  5. 5.

    Without parentheses the Atelier-B parser thus interprets \(\texttt {1=2 <=> 2=3}\) as the invalid \(\texttt {((1=2) <=> 2)=3}\). In ProB the grammar specifies that = has two expressions as arguments, and expressions cannot make use of = or \(\texttt {<=>}\). Hence, even without parentheses, \(\texttt {1=2 <=> 2=3}\) is unambiguously interpreted as \(\texttt {(1=2) <=> (2=3)}\).

  6. 6.

    One reason is that classical B allows composed identifiers in the grammar (e.g., xx.xx can refer to variable xx in an included machine xx). Note that, however, \(\exists xx.2<x\) is also not accepted by Atelier-B.

  7. 7.

    Note, however, that the statement in Sect. 3.2.3 of [41]: “\(\forall x.P \implies Q\) is parsed as \((\forall x.P) \implies Q\) in classical B” is not true: without parentheses \(\forall x.P \implies Q\) cannot be parsed at all.

  8. 8.

    Private communication from Laurent Voisin, Paris, 17th September 2019.

  9. 9.

    See also https://plus.maths.org/content/pemdas-paradox for this particular example.

  10. 10.

    In other formal languages this may be different; see Sect. 5.2.

  11. 11.

    The rodin handbook requires modulo arguments to be non-negative, which is correct; [41] is in error.

  12. 12.

    ProB warns when such variable captures appear. The price to pay is that ProB does not really treat definitions as macros, every definition body has to be a valid formula; it cannot consist of a partial text of a formula.

  13. 13.

    In classical B one can of course just use the \(\varSigma \) operator for this example. Here we just wish to illustrate the various approaches to recursion on a simple example.

  14. 14.

    https://www.methode-b.com/en/download-b-tools/rodin/b2rodin/.

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Acknowledgements

Egon Börger visited my group at the University of Düsseldorf in summer of 2015. Egon was funded by a renewed Forschungspreis grant of the Humboldt Foundation. This visit was very fruitful and helped me gain a better understanding of ASMs and enabled us to write the ABZ 2016 paper on a compact encoding of ASMs in Event-B. Egon also played an important role in establishing the ABZ conference series, which was instrumental in establishing bridges between the various state-based formalisms and led to considerable cross-fertilization. I hope that this article provides an additional bridge and helps researches travel more easily between the various state-based formalisms.

I also wish to thank Jean-Raymond Abrial, Lilian Burdy, Michael Butler, Stefan Hallerstede, Luis-Fernando Mejia, Sebastian Stock, Laurent Voisin, and Fabian Vu for useful feedback, Atelier-B and rodin implementation details and pointers. Finally, an anonymous referee provided a lot of detailed feedback, for which I am grateful.

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

    Michael Leuschel

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  1. Institut für Softwaretechnik und Programmiersprachen, University of Ulm, Ulm, Germany

    Alexander Raschke

  2. Dipartimento di Informatica, Università degli Studi di Milano, Crema, Italy

    Elvinia Riccobene

  3. UIUC Institute, Zhejiang University, Haining, China

    Klaus-Dieter Schewe

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Leuschel, M. (2021). Spot the Difference: A Detailed Comparison Between B and Event-B. In: Raschke, A., Riccobene, E., Schewe, KD. (eds) Logic, Computation and Rigorous Methods. Lecture Notes in Computer Science(), vol 12750. Springer, Cham. https://doi.org/10.1007/978-3-030-76020-5_9

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