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Polymorphic Sesqui-Pushout Graph Rewriting

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Book cover Graph Transformation (ICGT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9151))

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Abstract

The paper extends Sesqui-Pushout Graph Rewriting (SqPO) by polymorphism, a key concept in object-oriented design. For this purpose, the necessary theory for rule composition and decomposition is elaborated on an abstract categorical level. The results are applied to model rule extension and type dependent rule application. This extension mechanism qualifies SqPO – with its very useful copy mechanism for unknown contexts – as a modelling technique for extendable frameworks. Therefore, it contributes to the applicability of SqPO in software engineering. A version management example demonstrates the practical applicability of the combination of context-copying and polymorphism.

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Notes

  1. 1.

    Composition and prefix closure: for all \(f\in \mathcal {L}\), \(f\circ g\in \mathcal {L}\iff g\in \mathcal {L}\).

  2. 2.

    If \(g\circ f'=f\circ g'\), \((g',f')\) is pullback in \(\mathcal {C}\) of (f, g), and \(g\in \mathcal {L}\), then \(g'\in \mathcal {L}\).

  3. 3.

    If \(g'\circ f'=f\circ g\), \((g',f')\) is final pullback complement of (f, g), \(g\in \mathcal {L}\), then \(g'\in \mathcal {L}\).

  4. 4.

    Note that (C5) also implies \(i_{0},i_{2},i_{3}\in \mathcal {L}\) due to (C3) and (C4).

  5. 5.

    In SPO, \(\mathcal {L}\) is required to be a suitable subset of the monomorphisms in \(\mathcal {C}\).

  6. 6.

    The statement “\(t'\left\langle m'\right\rangle \circ m\) is match” requires that \(t'\left\langle m'\right\rangle \circ m\in \mathcal {C}\), which – being more precise – means that \(t'\left\langle m'\right\rangle \circ m=[\mathrm {id},f]\) for some \(f\in \mathcal {C}\).

  7. 7.

    Note that this notion of parallel independence is a conservative generalisation of the corresponding notions in [2, 5]. If the rules \(t=(l:K\rightarrow L,r:K\rightarrow R)\) and \(t'=(l':K'\rightarrow L',r':K'\rightarrow R')\) have monic left-hand sides l and \(l'\), we obtain monic morphisms \(l\left\langle m\right\rangle \) and \(l'\left\langle m'\right\rangle \) in the traces for matches m and \(m'\). In this case, the existence of morphisms n and \(n'\) with \(l'\left\langle m'\right\rangle \circ n=m\) and \(l\left\langle m\right\rangle \circ n'=m'\) implies that \((n,\mathrm {id}_{L})\) and \((n',\mathrm {id}_{L'})\) are pullbacks of \((m,l'\left\langle m'\right\rangle )\) and \((m',l\left\langle m\right\rangle )\) resp.

  8. 8.

    Compare Fact 2.

  9. 9.

    These maximal rules model object-oriented operations.

  10. 10.

    Compare “Negative Application Conditions” in [12].

  11. 11.

    Vertex or edge.

  12. 12.

    Compare also [19].

  13. 13.

    For an arbitrary hierarchy H, the Dedekind/MacNeille-completion [23] provides the smallest order closed under least upper and greatest lower bounds containing H.

  14. 14.

    If \(f,g:X\rightarrow G\) are two mappings into a partially ordered set \(G=(G,\le )\), we write \(f\le g\) if \(f(x)\le g(x)\) for all \(x\in X\).

  15. 15.

    The comparison operator \(\le \) in (11) is replaced by \(=\).

  16. 16.

    The notation \(o\in D\) stands here and in the following five occurrences for \(o\in Object_{D}\).

  17. 17.

    The composition \(c\circ a\) exists.

  18. 18.

    Version management systems typically store a successor version of an atomic component by some delta-information or text differences wrt. to its direct predecessor.

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Authors and Affiliations

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

    Michael Löwe

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  1. Michael Löwe
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Correspondence to Michael Löwe .

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  1. Dipartimento di Informatica, Sapienza Università di Roma, Rome, Italy

    Francesco Parisi-Presicce

  2. Angewandte Informatik 1, Universität Bayreuth, Bayreuth, Germany

    Bernhard Westfechtel

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Löwe, M. (2015). Polymorphic Sesqui-Pushout Graph Rewriting . In: Parisi-Presicce, F., Westfechtel, B. (eds) Graph Transformation. ICGT 2015. Lecture Notes in Computer Science(), vol 9151. Springer, Cham. https://doi.org/10.1007/978-3-319-21145-9_1

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  • DOI: https://doi.org/10.1007/978-3-319-21145-9_1

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