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Characterisation of Parallel Independence in AGREE-Rewriting

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Graph Transformation (ICGT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10887))

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Abstract

AGREE is a new approach to algebraic graph transformation. It provides sophisticated mechanisms for object cloning and non-local manipulations. It is for example possible to delete all edges in a graph by a single rule application. Due to these far-reaching effects of rule application, Church-Rosser results are difficult to obtain. Currently, there are only sufficient conditions for parallel independence, i.e. there are independent situations which are not classified independent by the existing conditions. In this paper, we characterise all independent situations.

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Notes

  1. 1.

    For details compare Definition 2 below.

  2. 2.

    This is mainly due to a pullback construction that is part of definition of rewrite in AGREE. For details compare Sect. 3.

  3. 3.

    As in [2], we consider right-linear rules only.

  4. 4.

    This is only half of the usually required adhesivity, compare [4].

  5. 5.

    Compare [6, 7]. Facts F1 and F2 are consequences of the fact that all pushouts in a category with partial arrow classifiers are hereditary. A pushout \((f',g')\) of (gf) as depicted in sub-diagram (1) of Fig. 2 is hereditary, if for all commutative situations as presented in Fig. 2 in which sub-diagrams (2) and (3) are pullbacks, and \(m,m_{P},\mathrm {\,and\,}m_{Q}\) are monomorphisms the following property is valid: the co-span \((f'_{m},g'_{m})\) is pushout of the span \((g_{m},f_{m})\), if and only if sub-diagrams (4) and (5) are pullbacks and \(m_{U}\) is monomorphism. Hereditariness implies stability of pushouts under pullbacks along monomorphisms. For some of the following results, however, we need requirement P3 which does not require monic \(m,m_{P},m_{Q},\mathrm {\,and\,}m_{U}\).

  6. 6.

    Special case of F3 for \(d=\mathrm {id}_{B}\) and \(b=c\circ e\).

  7. 7.

    Consequence of F3 for \(d=\mathrm {id}_{B}\).

  8. 8.

    For the proof, see [11].

  9. 9.

    Compare top part of Fig. 9.

  10. 10.

    Compare left part of Fig. 9.

  11. 11.

    Compare Fig. 9 where \((i\circ m')^{\bullet }\) is called \(m_{i}\)!.

  12. 12.

    Equal up to isomorphism.

  13. 13.

    These two lemmata demonstrate that gluing constructions in categories of abstract (right linear) spans possess the same composition and decomposition properties as simple pushouts in arbitrary categories.

  14. 14.

    \((\pi _{2},\beta _{1})\) is pullback of \((\alpha _{1},t_{1})\), compare proof of Proposition 22.

References

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Löwe, M. (2018). Characterisation of Parallel Independence in AGREE-Rewriting. In: Lambers, L., Weber, J. (eds) Graph Transformation. ICGT 2018. Lecture Notes in Computer Science(), vol 10887. Springer, Cham. https://doi.org/10.1007/978-3-319-92991-0_8

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  • DOI: https://doi.org/10.1007/978-3-319-92991-0_8

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