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Moving from interface theories to assembly theories

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Abstract

We show how interface theories supporting pairwise component analysis can be extended in a generic way to a multi-component environment. This leads to the abstract framework of an assembly theory which captures notions of assembly refinement and communication-safety in assemblies of interacting components. An assembly theory supports also encapsulation of assemblies into interfaces and hence hierarchical constructions. We propose general rules that should be satisfied by any concrete assembly theory, like compositional construction and refinement of communication-safe assemblies. We discuss general procedures how to construct an assembly theory on top of a given interface theory such that (some of) the laws of an assembly theory are automatically guaranteed by the properties of an underlying interface theory. As a proof of concept we consider two instances of our approach. The first one starts from the (optimistic) interface theory of interface automata proposed by de Alfaro and Henzinger, and the second one from the (pessimistic) interface theory of modal I/O-interfaces. In the latter case, we propose a new notion of modal assembly refinement which has all the required properties, in particular it preserves modal communication-safety of assemblies. A small case-study illustrates how our concepts can be methodologically applied.

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Notes

  1. It is assumed that the composition of compatible interfaces is defined.

  2. Commutativity means that for all \(F, G \in {\mathcal {F}}\), if \(F, G\) are composable then \(G, F\) are composable and \(F \mathbin {\otimes }G = G \mathbin {\otimes }F\), i.e., \(F \mathbin {\otimes }G\) and \(G \mathbin {\otimes }F\) are set-theoretically equal.

  3. Associativity means that for all \(F, G, H \in {\mathcal {F}}\), if \(F, G\) and \(H\) are pairwise composable then \((F \mathbin {\otimes }G) \mathbin {\otimes }H\) and \(F \mathbin {\otimes }(G \mathbin {\otimes }H)\) are defined and \((F \mathbin {\otimes }G) \mathbin {\otimes }H = F \mathbin {\otimes }(G \mathbin {\otimes }H)\).

  4. In our considerations an environment for an interface \(F\) is just another interface, say \(E\), which is composable with \(F\). We do not impose any closedness assumption on \(F \mathbin {\otimes }E\), since this is not possible in the abstract framework of an interface theory. This could be done, however, in the framework of “labelled interface theories” considered in [5].

  5. More precisely, we consider interface automata as representatives of their isomorphism classes w.r.t. bijections on states.

  6. We refer to the different parts of modal interfaces by subscripting, such that, e.g., \(S_{M_1}\) denotes the states of the modal interface \(M_1\).

  7. More precisely, we consider modal interfaces as representatives of their isomorphism classes w.r.t. bijections on states.

  8. Hence for each \(A \in {\mathcal {A}}\), any non-empty sub-family of \(A\) is also in \({\mathcal {A}}\).

  9. The modal interface \(M\) is considered as a MIO with no communication labels and the same holds for \(M_j\) in the subsequent case.

  10. Recall that \(C_{E_j}\) are the communication labels of \(E_j\) and \((O_{E_j} {\setminus } I_{M_j})\) the output labels of \(E_j\) unshared with the input labels of \(M_j\), i.e., not used for communication between \(E_j\) and \(M_j\). The silent must-transitions of \(E_j\) are anyway subsumed in the notation \({\mathrel {{\overset{\widehat{X_j}}{\rightarrow }\!\!\!{\genfrac{}{}{0.0pt}1{}{E_j}}}}}\); see Sect. 3.2.

  11. In order to prove the equivalence one has to check that \( pack ^{{\mathrm {mi}}}(\mathsf{CashDeskAssembly}) \mathrel {\preccurlyeq ^{\mathrm {mi}}} min ( pack ^{{\mathrm {mi}}}(\mathsf{CashDeskAssembly}))\) and vice versa. Both directions have been verified with the MIO-Workbench [6]. We believe that the interface \( min ( pack ^{{\mathrm {mi}}}(\mathsf{CashDeskAssembly}))\) is indeed minimal. Whether minimal behaviours for modal interfaces always exist and how they can be computed is an open question.

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Authors and Affiliations

  1. Ludwig-Maximilians-Universität München, Munich, Germany

    Rolf Hennicker

  2. Universität Augsburg, Augsburg, Germany

    Alexander Knapp

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  1. Rolf Hennicker
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  2. Alexander Knapp
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Correspondence to Alexander Knapp.

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Dedicated to Walter Vogler on the occasion of his 60th birthday.

This work has been partially sponsored by the European Union under the FP7-project ASCENS, 257414.

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Hennicker, R., Knapp, A. Moving from interface theories to assembly theories. Acta Informatica 52, 235–268 (2015). https://doi.org/10.1007/s00236-015-0220-7

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