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Evaluation Diversity for Graph Conditions

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Graph Transformation (ICGT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12741))

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Abstract

Graphs are used as a universal data structure in various domains. Sets of graphs (and likewise graph morphisms) can be specified using the graph logic of Graph Conditions (GCs). The evaluation of a graph against a GC results in a satisfaction judgement on whether the graph is specified by the GC. is as expressive as first-order logic on graphs and infinitely many graphs may be evaluated against a given GC. Therefore, a complete compact overview of how a given GC may be evaluated for varying graphs can support GC validation, testing, debugging, and repair.

As a main contribution, we generate such an overview for a given GC in the form of a complete finite set of diverse evaluations for varying associated graphs formally given by so called Evaluation Trees (ETs). Each of these ETs concretely describes how its associated graph is evaluated against the given GC by presenting each evaluation step. The returned ETs are complete since each possible ET subsumes one of the returned ETs and diverse by not containing superfluous ETs subsuming smaller ETs. We apply an implementation of our ET generation procedure in the tool to a running example.

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Notes

  1. 1.

    Type graphs and typed graphs can be understood as formalizations of UML class diagrams and UML object diagrams.

  2. 2.

    The operation used here is a simplification of the operation presented in [23, 24], which returned symbolic models representing all graphs satisfying a given GC. Technically, we apply the operation from [23, 24], remove additional information with which the returned graphs are equipped and remove all graphs from that set for which the set contains a subgraph already.

  3. 3.

    Note that the runtime of is usually insignificant when it is applied offline (i.e., at design time) to small hand-written GCs only (despite the need to solve the NP-hard subgraph isomorphism problem when computing the possibly exponential number of graph overlappings). If one or both of these assumptions are not satisfied in an application context, further evaluations are required to ensure a practical runtime in that setting and the property (b) from Fact 1 on partial results may become relevant.

  4. 4.

    Intuitively, compactness means that all graph elements of the host graph G have been matched by some considered mono during the evaluation.

  5. 5.

    Intuitively, the operations of this three-valued logic can be computed based on numerical values using the bijective homomorphism \(\langle \cdot \rangle \), which maps the elements of the base set as follows: , , and . Then, negation is translated into the multiplication with \(-1\) and disjunction is translated into taking the maximum of its arguments. E.g. because \(-1\times \mathsf {max}(0,1)=-1\).

  6. 6.

    The operation is an adaptation of the predicate \(\models _{\mathrm {ST}}\) from [22] accommodating for the additional operator .

  7. 7.

    The returned satisfaction judgement for is either or .

  8. 8.

    Since is either or , this also has to hold for .

  9. 9.

    This ES is given by the operation in [22].

  10. 10.

    To simplify our presentation, we also employ the regular expression based specification via for existential quantification. We thereby restrict ourselves to finite sets of monos , which is guaranteed e.g. for finite host graphs. However, the used rules can easily be relaxed to handle also infinite match sets

  11. 11.

    Steps in Fig. 2 are i.a. labeled with the satisfaction judgement for the derived ET.

  12. 12.

    Note that for (c), there may be in general multiple ways as to why a sub-GC is not satisfied and fixing one particular reason for all matches would not allow for a set of ETs that is complete in the sense of Definition 10.

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Authors and Affiliations

  1. University of Potsdam, Hasso Plattner Institute, Potsdam, Germany

    Sven Schneider & Leen Lambers

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  1. Sven Schneider
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Correspondence to Sven Schneider .

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  1. Università di Pisa, Pisa, Italy

    Fabio Gadducci

  2. Humboldt-Universität zu Berlin, Berlin, Germany

    Timo Kehrer

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Schneider, S., Lambers, L. (2021). Evaluation Diversity for Graph Conditions. In: Gadducci, F., Kehrer, T. (eds) Graph Transformation. ICGT 2021. Lecture Notes in Computer Science(), vol 12741. Springer, Cham. https://doi.org/10.1007/978-3-030-78946-6_7

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  • DOI: https://doi.org/10.1007/978-3-030-78946-6_7

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