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Graph-Based Shape Analysis Beyond Context-Freeness

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Software Engineering and Formal Methods (SEFM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10886))

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Abstract

We develop a shape analysis for reasoning about relational properties of data structures. Both the concrete and the abstract domain are represented by hypergraphs. The analysis is parameterized by user-supplied indexed graph grammars to guide concretization and abstraction. This novel extension of context-free graph grammars is powerful enough to model complex data structures such as balanced binary trees with parent pointers, while preserving most desirable properties of context-free graph grammars.

One strength of our analysis is that no artifacts apart from grammars are required from the user; it thus offers a high degree of automation. We implemented our analysis and successfully applied it to various programs manipulating AVL trees, (doubly-linked) lists, and combinations of both.

Matheja, C.—Supported by Deutsche Forschungsgemeinschaft (DFG) Grant NO 401/2-1.

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Notes

  1. 1.

    We often draw multiple black circles, but they all correspond to the same location.

  2. 2.

    Again, note that we consider a single execution path in this example. The full analysis also explores the cases in which X is substituted by \(z\) and \(sz\).

  3. 3.

    External nodes are needed to define the semantics of nonterminal edges.

  4. 4.

    I.e., \(v_{\mathtt {null}}= \textit{ext}(1)\) and for each \(e \in E\) with \(\textit{lab}(e) \in N\), we have \(\textit{att}(e)(1) = v_{\mathtt {null}}\).

  5. 5.

    \(f \upharpoonright M\) denotes the restriction of function f to domain M and . Moreover, function \(\textit{mod} = \{ \textit{ext}_{K}(k) \mapsto \textit{att}_{H}(e)(k) ~|~ 1 \le k \le |\textit{ext}_{K}| \} \cup \{ v \mapsto v ~|~ v \in V\setminus \textit{ext}_{K} \}\) is lifted to sequences of nodes by pointwise application.

  6. 6.

    \(H\left[ \nu \mapsto \rho \right] = (V_{H},E_{H},\textit{att}_{H},\textit{lab}_{H},\textit{ind},\textit{ext}_{H})\) with \(\textit{ind}= \{ \textit{ind}_{H}(e)\left[ \nu \mapsto \rho \right] ~|~ e \in E_{H} \}\).

  7. 7.

    denotes sequential composition of f and g, i.e. .

  8. 8.

    https://github.com/moves-rwth/attestor-examples/releases/tag/v0.3.5-SEFM2018. Also confer the extended version [4].

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Hannah Arndt, Christina Jansen, Christoph Matheja & Thomas Noll

Authors
  1. Hannah Arndt
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  2. Christina Jansen
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  3. Christoph Matheja
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  4. Thomas Noll
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Correspondence to Christoph Matheja .

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Editors and Affiliations

  1. University of Oslo, Oslo, Norway

    Einar Broch Johnsen

  2. Braunschweig University of Technology, Braunschweig, Germany

    Ina Schaefer

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Arndt, H., Jansen, C., Matheja, C., Noll, T. (2018). Graph-Based Shape Analysis Beyond Context-Freeness. In: Johnsen, E., Schaefer, I. (eds) Software Engineering and Formal Methods. SEFM 2018. Lecture Notes in Computer Science(), vol 10886. Springer, Cham. https://doi.org/10.1007/978-3-319-92970-5_17

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  • DOI: https://doi.org/10.1007/978-3-319-92970-5_17

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