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International Static Analysis Symposium

SAS 2016: Static Analysis pp 338–360Cite as

A Parametric Abstract Domain for Lattice-Valued Regular Expressions

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9837))

Abstract

We present a lattice-valued generalization of regular expressions as an abstract domain for static analysis. The parametric abstract domain rests on a generalization of Brzozowski derivatives and works for both finite and infinite lattices. We develop both a co-inductive, simulation algorithm for deciding ordering between two domain elements and a widening operator for the domain. Finally we illustrate the domain with a static analysis that analyses a communicating process against a lattice-valued regular expression expressing the environment’s network communication.

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Notes

  1. 1.

    It has later been pointed out [14, 26, 27] that Brzozowski’s proof had a minor flaw, that could be fixed by patching the statement of the theorem [27] or by patching the definition of derivatives to avoid the syntactic occurrence of \(\delta \) [26]. We have followed the latter approach in our generalization.

  2. 2.

    Solving the equations for such error states before step 5 (collapsing) has the same effect: their collective solution is \(\emptyset \) in the matrix, and substituting the solution in removes any transitions to and from them, and thereby any observable effect of grouping an error state and a non-error state in the same equivalence class.

  3. 3.

    The result is also \(\epsilon + \emptyset \) for [20; 20], [21; 21], [22; 22] up to ACI of +, but that just constitutes a refinement identifying even more equivalent atoms.

  4. 4.

    Generally the solution to this equation is an over-approximation but so is the result of widening.

  5. 5.

    https://github.com/jmid/regexpanalyser.

  6. 6.

    http://janmidtgaard.dk/papers/Midtgaard-Nielson-Nielson:SAS16-full.pdf.

  7. 7.

    These are however referred to as ‘atomic lattices’ contradicting standard terminology [10, 15].

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Authors and Affiliations

  1. DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark

    Jan Midtgaard, Flemming Nielson & Hanne Riis Nielson

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  1. Jan Midtgaard
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  2. Flemming Nielson
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Correspondence to Jan Midtgaard .

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  1. Ecole Normale Supérieure , Paris, France

    Xavier Rival

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Midtgaard, J., Nielson, F., Nielson, H.R. (2016). A Parametric Abstract Domain for Lattice-Valued Regular Expressions . In: Rival, X. (eds) Static Analysis. SAS 2016. Lecture Notes in Computer Science(), vol 9837. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53413-7_17

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  • DOI: https://doi.org/10.1007/978-3-662-53413-7_17

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