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Symbolic Transformation of Expressions in Modular Arithmetic

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Static Analysis (SAS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14284))

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Abstract

We present symbolic methods to improve the precision of static analyses of modular integer expressions based on Abstract Interpretation. Like similar symbolic methods, the idea is to simplify on-the-fly arithmetic expressions before they are given to abstract transfer functions of underlying abstract domains. When manipulating fixed-length integer data types, casts and overflows generally act like modulo computations which hinder the use of symbolic techniques. The goal of this article is to formalize how modulo operations can be safely eliminated by abstracting arbitrary arithmetic expressions into sum, product, or division of linear forms with integer coefficients, while simplifying them. We provide some rules to simplify arithmetic expressions that are involved in the computation of linear interpolations, while ensuring the soundness of the transformation.

All these methods have been incorporated within the Astrée static analyzer that checks for the absence of run-time errors in embedded critical software, but also in an available toy abstract interpreter. The effects of our new abstract domain are then evaluated on several code excerpts from industrial code.

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Notes

  1. 1.

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Acknowledgements

We thank the anonymous referees for their constructive comments and suggestions. Furthermore, we extend our gratitude to Josselin Giet, Marc Chevalier and Antoine Miné for granting us permission to publish a toy abstract interpreter they created, which served as the foundation for our artifact.

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Authors and Affiliations

  1. École Normale Supérieure, Université PSL, Paris, France

    Jérôme Boillot & Jérôme Feret

  2. INRIA, Paris, France

    Jérôme Feret

Authors
  1. Jérôme Boillot
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  2. Jérôme Feret
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Correspondence to Jérôme Boillot .

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Editors and Affiliations

  1. U. Politécnica de Madrid (UPM) and IMDEA Software Institute, Pozuelo de Alarcón, Madrid, Spain

    Manuel V. Hermenegildo

  2. U. Politécnica de Madrid (UPM) and IMDEA Software Institute, Pozuelo de Alarcon, Spain

    José F. Morales

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Boillot, J., Feret, J. (2023). Symbolic Transformation of Expressions in Modular Arithmetic. In: Hermenegildo, M.V., Morales, J.F. (eds) Static Analysis. SAS 2023. Lecture Notes in Computer Science, vol 14284. Springer, Cham. https://doi.org/10.1007/978-3-031-44245-2_6

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  • DOI: https://doi.org/10.1007/978-3-031-44245-2_6

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