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Inferring physical units in formal models

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Abstract

Most state-based formal methods, like B, Event-B or Z, provide support for static typing. However, these methods and the associated tools lack support for annotating variables with (physical) units of measurement. There is thus no obvious way to reason about correct or incorrect usage of such units. We present a technique that analyzes the usage of physical units throughout B and Event-B machines infers missing units and notifies the user of incorrectly handled units. The technique combines abstract interpretation with classical animation, constraint solving and model checking and has been integrated into the ProB validation tool, both for classical B and for Event-B. It provides source-level feedback about errors detected in the models. We also describe how to extend our approach to TLA \(^+\), an untyped formal language. We provide an in-depth empirical evaluation and demonstrate that our technique scales up to real-life industrial models.

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Notes

  1. See, however, [24].

  2. We return to this example in Sect. 7.1 and provide further details there.

  3. Note that this set includes not just functions but all relations between integers. In B and ProB  functions and relations have the same type; operators for relations can be applied to functions and vice versa. The fact that a relation is indeed a function is encoded as an invariant and verified for each state, i.e., it is a safety property of the system.

  4. Note that for Event-B, the Rodin tool will produce an error message if a variable or expression’s type contains \(\top \).

  5. For convenience, some SI derived units and units accepted for use with the SI standard (see [35]) are stored on their own rather than converting them.

  6. Basically, which means adding the exponents of the leading 10 of each SI unit.

  7. In contrast to TLA \(^+\), B does not support \(n < 0\) as the first argument to the modulo operator. This is taken into account by the translation from TLA \(^+\) to B.

  8. Note that there is only experimental support for floating point numbers in B and Event-B. Hence, we currently do not support conversions like the one from seconds to minutes using \(\frac{1}{60}\) as a conversion factor. It would, however, be possible to implement this as an extension of our approach, and we will do so once real/float support stabilizes. TLA \(^+\) supports real numbers. However, at the moment, our translation does not.

  9. http://www.stups.uni-duesseldorf.de/models/sosym_units/.

  10. Both were counted on the internal representation of the machines. Thus, the metrics include code from imported machines. Comments are not counted, as they are not in the internal representation. However, new lines used for pretty printing are counted.

  11. The tutorial including the machines can be found at http://www.tools.clearsy.com/wp1/?page_id=161.

  12. The exception being variables and sets belonging to the system’s status. Here, no unit of measurement applies, and no unit was inferred.

  13. Upon request of our industrial customers, we since added several custom units to the units included by default. Conversion rules were added as well.

  14. A relation constructed as the set union of the two lambda expressions. Depending on the condition of the if expression, one of the lambda expressions is empty, while the other one contains exactly one element: the result of the if expression. We apply the relation to 1 to extract this element.

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Acknowledgments

We are grateful to reviewers of SEFM and SoSyM for their useful feedback, which helped to improve the paper. Our thanks also go to Luis-Fernando Mejia for providing us with interesting industrial case studies.

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

    Sebastian Krings & Michael Leuschel

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  1. Sebastian Krings
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Correspondence to Sebastian Krings.

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Communicated by Prof. Hierons, Prof. Merayo, and Prof. Bravetti.

Part of this research has been sponsored by the EU funded FP7 project 287563 (ADVANCE) and the DFG funded research project GEPAVAS II.

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Krings, S., Leuschel, M. Inferring physical units in formal models. Softw Syst Model 16, 25–47 (2017). https://doi.org/10.1007/s10270-015-0458-0

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