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Automata-based verification of programs with tree updates

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Abstract

This paper describes a verification framework for Hoare-style pre- and post-conditions of programs manipulating balanced tree-like data structures. Since the considered verification problem is undecidable, we appeal to the standard semi-algorithmic approach in which the user has to provide loop invariants, which are then automatically checked, together with the program pre- and post-conditions. We specify sets of program states, representing tree-like memory configurations, using Tree Automata with Size Constraints (TASC). The main advantage of this new class of tree automata is that they recognise tree languages based on arithmetic reasoning about the lengths of various (possibly all) paths in trees, like, e.g., in AVL trees or red–black trees. TASCs are closed under union, intersection, and complement, and their emptiness problem is decidable. Thus we obtain a class of automata which are an interesting theoretical contribution by itself. Further, we show that, under few restrictions, one can automatically compute the effect of tree-updating program statements on the set of configurations represented by a TASC, which makes TASC a practical verification tool. We tried out our approach on the insertion procedure for red–black trees, for which we verified that the output on an arbitrary balanced red–black tree is also a balanced red–black tree.

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Authors and Affiliations

  1. LIAFA, Université Paris Diderot—Paris 7/CNRS, Case 7014, 75205, Paris 13, France

    Peter Habermehl

  2. VERIMAG, Université Joseph Fourier/CNRS/INPG, 2 av. de Vignate, 38610, Gières, France

    Radu Iosif

  3. FIT, Brno University of Technology, Božetěchova 2, 61266, Brno, Czech Republic

    Tomáš Vojnar

Authors
  1. Peter Habermehl
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  2. Radu Iosif
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  3. Tomáš Vojnar
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Corresponding author

Correspondence to Peter Habermehl.

Additional information

A short version of this paper appeared in the Proceedings of TACAS 2006. The work was supported by the French Ministry of Research (RNTL project AVERILES), the Czech Science Foundation within the project 102/07/0322, the Czech-French Barrande project MEB 020840, and the Czech Ministry of Education, Youth, and Sports by the project MSM 0021630528.

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Habermehl, P., Iosif, R. & Vojnar, T. Automata-based verification of programs with tree updates. Acta Informatica 47, 1–31 (2010). https://doi.org/10.1007/s00236-009-0108-5

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  • DOI: https://doi.org/10.1007/s00236-009-0108-5

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