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Double-Pushout Rewriting in Context

Rule Composition and Parallel Independence

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Graph Transformation (ICGT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11629))

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Abstract

Recently, we introduced double-pushout rewriting in context (DPO-C) as a conservative extension of the classical double-pushout approach (DPO) at monic matches. DPO-C allows non-monic rules such that the split and merge of items can be specified together with deterministic context distribution and joining. First results showed that DPO-C is practically applicable, for example in the area of model refactoring, and that the theory of the DPO-approach is very likely to carry over to DPO-C. In this paper, we extend the DPO-C-theory. We investigate rule composition and characterise parallel independence.

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Notes

  1. 1.

    For more examples, see [11].

  2. 2.

    Every well-structured object-oriented model shall be hierarchical.

  3. 3.

    Two-headed arrows stand for a pair of arrows one in each direction.

  4. 4.

    The examples in the introduction also use this underlying category.

  5. 5.

    Again, the associations and inheritance relations the classifier adds and the associations and inheritance relations which are mapped to them by the totalisation are painted as dotted arrows.

  6. 6.

    The pushout morphisms \(c_{l}\) and \(c_{r}\) are monic by Fact 2 (3).

  7. 7.

    If (m, g) and (p, h) are pushouts of (n, l) resp. (n, r) in a DPO-derivation with rule \(\varrho =(l,r)\) at match m, we denote the trace (g, h) by \(\varrho \left\langle m\right\rangle \) and co-match p by \(m\left\langle \varrho \right\rangle \).

  8. 8.

    See [13] for composition of partial maps and especially composition of monic spans.

  9. 9.

    To my knowledge, rule and trace composition has never been investigated in isolation in the DPO-approach. Some aspects are handled within the concurrency theorem [4].

  10. 10.

    The latter by pullback decomposition of \((c_{12},l_{2}')\).

  11. 11.

    See Rewrite Property 9(1).

  12. 12.

    This condition is identical to the one in [3].

  13. 13.

    This notion of residual is a conservative extension of the notion for DPO-rewriting with monic rules at monic matches, since traces become monic in this case as well and it is easy to see that a triangle \(g\circ m_{D}=m\) with monic g is a pullback diagram.

  14. 14.

    Compare Fig. 12.

  15. 15.

    More precisely, pushouts preserve isomorphisms.

  16. 16.

    Pushout \((p_{1},h_{1})\) of \((r_{1},n_{1})\) is also pullback by Fact 2(4) and pullbacks compose.

References

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Authors and Affiliations

  1. FHDW Hannover, Freundallee 15, 30173, Hanover, Germany

    Michael Löwe

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  1. Michael Löwe
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Correspondence to Michael Löwe .

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Editors and Affiliations

  1. Universidad Autónoma de Madrid, Madrid, Spain

    Esther Guerra

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Fernando Orejas

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Löwe, M. (2019). Double-Pushout Rewriting in Context. In: Guerra, E., Orejas, F. (eds) Graph Transformation. ICGT 2019. Lecture Notes in Computer Science(), vol 11629. Springer, Cham. https://doi.org/10.1007/978-3-030-23611-3_2

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  • DOI: https://doi.org/10.1007/978-3-030-23611-3_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23610-6

  • Online ISBN: 978-3-030-23611-3

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