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On submodels and submetamodels with their relation

A uniform formalization through inclusion properties

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Abstract

Model-driven engineering (MDE) recognized software models as first-class objects with their own relationships and operations, up to constitute full structured model spaces. We focus on inclusion capacities through the concepts of submodel and submetamodel which contribute a lot to the structuring effort. Submodels and submetamodels underlie many MDE practices which require their precise characterization for plain control. A typical application is model management as offered by model repositories. On the basis of results on submodel inclusion we stated in a preceding paper, we concentrate on the special form of submodels which are submetamodels and their specific role in model space structuring. Pointing out that relating submodels and submetamodels is two ways, their respective inclusion hierarchies will be systematically characterized and symmetrically compared under the logical relationships of metamodel membership and model well-formedness. As a major result, it will be shown that submodel well-formedness w.r.t submetamodels closely relates to submodel invariance (a property which guarantees transitive structure preservation) applied at both levels. The uniform formalization offers algebraic grounding to better comprehension and control of model spaces which underlie MDE activities. At a much more practical level, reusable technology which takes advantage of established results will be offered.

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Notes

  1. http://www.cristal.univ-lille.fr/caramel/submodel.

  2. For sake of continuity, examples of the present paper will be directly inspired from those of [12].

  3. \(\sqsubseteq _{m}\) being an order, it is reflective so that we systematically have: \(\forall x\in \widetilde{m}, x\sqsubseteq _{m}x\). For sake of simplicity, we will not list these reflective dependencies. In graph representation, this corresponds to “Hasse Diagram” simplification which ignores them but also systematic transitive links. It will be used in figures.

  4. Remind that they were introduced as submodels which do not preserve structure to motivate the problem. Backwardly, due to the formalism, structure preserving submodels were formulated as invariant ones.

  5. As far as possible, a figure will be associated with each property in order to pictorially schematize its content. Bold double arrows (which evoke implication) will be used to represent deduced relationships.

  6. More theoretically, in that case function meta must be monotonic when applied to model elements of m with the structure that this model asserts. A monotonic function is a special case of the more general concept of (homo)morphism of set theory when used in the field of order (and also graph) theory. In the general case (set theory), morphisms are structure preserving functions between structured sets. Within the field of order theory, structuring is ordering. It is the case here, comparing structured sets (\(\widetilde{m},\sqsubseteq _{m})\) and \((\widetilde{M},\sqsubseteq _{M})\) under the meta function where \(\sqsubseteq _{m}\) and \(\sqsubseteq _{M}\) are ordering relationships.

  7. A poset is a set of things equipped with a partial order, here (meta) models arranged by submodel inclusion partial ordering. Depending on the partial order properties, a poset may have bound elements: “greatest” one(s) and/or “smallest” one(s).

  8. See quantitative evaluation of the results in Sect. 6.3.

  9. See quantitative evaluation of the results in Sect. 6.3.

  10. Thanks to the formalism, query model can be full well-formed models but also unspecified model fragments such as simple sets of model ingredients or partial models resulting from incremental or intermediate design.

  11. Concerning proper metamodels, they are stored and indexed at the same time than their related model. An additional bidirectional mapping table is also maintained between model and their proper metamodel for accelerating checking required by several searches.

  12. on Intel I7, 2.10 GHz \(\times \) 4 core.

  13. http://www.genmymodel.com.

  14. More precisely, \(O(e_m + c_m)\) and \(O(c_{\widehat{m}} \times c_M)\), with \(e_x=\) number of model elements of x, \(c_x=\) number of constraints of x.

  15. More precisely, \(O(e_m^2 \times c_M)\) for method 1 compared to \(O(e_m^2 \times c_{\widehat{m}})\) for method 2.

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Acknowledgments

The authors would like to thank Prof. J. Gray and the anonymous reviewers for their work and comments in order to improve the paper.

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Authors and Affiliations

  1. CRIStAL, UMR CNRS 9189, University of Lille, 59655, Villeneuve d’Ascq Cedex, France

    Bernard Carré, Gilles Vanwormhoudt & Olivier Caron

  2. Institut Mines-Télécom, 75634, Paris Cedex 13, France

    Gilles Vanwormhoudt

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  1. Bernard Carré
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Correspondence to Bernard Carré.

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Communicated by Dr. Kevin Lano.

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Carré, B., Vanwormhoudt, G. & Caron, O. On submodels and submetamodels with their relation. Softw Syst Model 17, 1105–1137 (2018). https://doi.org/10.1007/s10270-016-0540-2

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