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A Formal Framework to Measure the Incompleteness of Abstract Interpretations

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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

In program analysis by abstract interpretation, backward-completeness represents no loss of precision between the result of the analysis and the abstraction of the concrete execution, while forward-completeness stands for no imprecision between the concretization of the analysis result and the concrete execution. Program analyzers satisfying one of the two properties (or both) are considered precise. Regrettably, as for all approximation methods, the presence of false-alarms is most of the time unavoidable and therefore we need to deal somehow with incompleteness of both. To this end, a new property called partial completeness has recently been formalized as a relaxation of backward-completeness allowing a limited amount of imprecision measured by quasi-metrics. However, the use of quasi-metrics enforces distance functions to adhere precisely the abstract domain ordering, thus not suitable to be used to weaken the forward-completeness property which considers also abstract domains that are not necessarily based on Galois Connections. In this paper, we formalize a weaker form of quasi-metric, called pre-metric, which can be defined on all domains equipped with a pre-order relation. We show how this newly defined notion of pre-metric allows us to derive other pre-metrics on other domains by exploiting the concretization and, when available, the abstraction maps, according to the information and the corresponding level of approximation that we want to measure. Finally, by exploiting pre-metrics as our imprecision meter, we introduce the partial forward/backward-completeness properties.

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Notes

  1. 1.

    The term “forward-completeness” was introduced in [32] in order to distinguish it from the well known backward-completeness property requiring an abstraction function.

  2. 2.

    In the standard abstract interpretation framework [20, 21] dealing with GCs, the backward-completeness property is simply called completeness or exactness.

  3. 3.

    This is not a standard term in the literature: sometimes it is used to refer to other generalizations of metrics such as pseudosemi-metrics [8] or pseudo-metrics [34]; it sometimes appears as pra-metric [3]. This definition is taken from Wikipedia [1].

  4. 4.

    We assume the following results when the \(\infty \) symbol is involved: \(Av(k - \infty ) = Av(\infty - k) = \infty \) with \(k \in \mathbb {R}\), while \(\infty - \infty = 0\).

  5. 5.

    Interproc is freely available at http://pop-art.inrialpes.fr/interproc/interprocweb.cgi.

  6. 6.

    For the convex polyhedra analysis, we activated the option of 2 descending steps.

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Acknowledgements

We wish to thank the anonymous reviewers of SAS 2023 for their detailed comments. This work has been partially supported by the grant PRIN2017 (code: 201784YSZ5) “AnalysiS of PRogram Analyses (ASPRA)”.

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Authors and Affiliations

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

    Marco Campion & Caterina Urban

  2. Department of Computer Science, University of Verona, Verona, Italy

    Mila Dalla Preda

  3. Department of Computer Science, University of Arizona, Tucson, AZ, USA

    Roberto Giacobazzi

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  1. Marco Campion
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  4. Roberto Giacobazzi
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Correspondence to Marco Campion .

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  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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Campion, M., Urban, C., Dalla Preda, M., Giacobazzi, R. (2023). A Formal Framework to Measure the Incompleteness of Abstract Interpretations. 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_7

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