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A Generalized Concurrent Rule Construction for Double-Pushout Rewriting

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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

Double-pushout rewriting is an established categorical approach to the rule-based transformation of graphs and graph-like objects. One of its standard results is the construction of concurrent rules and the Concurrency Theorem pertaining to it: The sequential application of two rules can equivalently be replaced by the application of a concurrent rule and vice versa. We extend and generalize this result by introducing generalized concurrent rules (GCRs). Their distinguishing property is that they allow identifying and preserving elements that are deleted by their first underlying rule and created by the second one. We position this new kind of composition of rules among the existing ones and obtain a Generalized Concurrency Theorem for it. We conduct our work in the same generic framework in which the Concurrency Theorem has been presented, namely double-pushout rewriting in \(\mathcal {M}\)-adhesive categories via rules equipped with application conditions.

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Notes

  1. 1.

    This means, every pair of morphisms with the same codomain can be factored as a pair of morphisms belonging to \(\mathcal {E}^{\prime }\) followed by an \(\mathcal {M}\)-morphism. We do not directly need this property in any of our proofs but it is assumed for the computation of application conditions of concurrent and, hence, also generalized concurrent rules. Moreover, it guarantees the existence of E-related transformations [6, Fact 5.29].

    Since we restrict ourselves to the case of \(\mathcal {M}\)-matching, decomposition of \(\mathcal {M}\)-morphisms then ensures that all occurring pairs \((e_1,e_2) \in \mathcal {E}^{\prime }\) are in fact even pairs of \(\mathcal {M}\)-morphisms. This in turn (by closedness of \(\mathcal {M}\) under pullbacks) implies that in any common kernel k compatible to a given E-dependency relation, the embedding morphisms \(u_1,u_2\) are necessarily \(\mathcal {M}\)-morphisms.

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Acknowledgments

This work was partially funded by the German Research Foundation (DFG), project TA294/17-1.

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  1. Philipps-Universität Marburg, Marburg, Germany

    Jens Kosiol & Gabriele Taentzer

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  1. Jens Kosiol
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  2. Gabriele Taentzer
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Correspondence to Jens Kosiol .

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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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Kosiol, J., Taentzer, G. (2021). A Generalized Concurrent Rule Construction for Double-Pushout Rewriting. 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_2

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  • DOI: https://doi.org/10.1007/978-3-030-78946-6_2

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