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Concurrency Theorems for Non-linear Rewriting Theories

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Book cover Graph Transformation (ICGT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12741))

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Abstract

Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic matches is well-known to permit the modeling of fusing and cloning of vertices and edges, yet to date, no construction of a suitable concurrency theorem was available. The lack of such a theorem, in turn, rendered compositional reasoning for such rewriting systems largely infeasible. We develop in this paper a suitable concurrency theorem for non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming the example of adhesive categories) and with matches required to be regular monomorphisms of the given category. Our construction reveals an interesting “backpropagation effect” in computing rule compositions. We derive in addition a concurrency theorem for non-linear double pushout (DPO) rewriting in rm-adhesive categories. Our results open non-linear SqPO and DPO semantics to the rich static analysis techniques available from concurrency, rule algebra and tracelet theory.

An extended version of this paper containing additional technical appendices is available online [7].

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Notes

  1. 1.

    In this paper, we follow the conventions of compositional rewriting theory [9], i.e., we speak of “input”/“output” motifs of rules, as opposed to “left”/“right” motifs in the traditional literature [22].

  2. 2.

    Some authors prefer to not consider directly the category \(\mathbf {BRel}\), but rather define \(\mathbf {SGraph}\) as some category equivalent to \(\mathbf {BRel}\), where simple graphs are of the form \(\langle V,E\rangle \) with \(E\subseteq V\times V\). This is evidently equivalent to directly considering \(\mathbf {BRel}\), whence we chose to not make this distinction in this paper.

  3. 3.

    As demonstrated in [26, Fact 3.4], every finitary \(\mathcal {M}\)-adhesive category \(\mathbf{C}\) possesses an (extremal \(\mathcal {E}\), \(\mathcal {M}\))-factorization, so if \(\mathbf{C}\) is known to possess FPCs as required by the construction, this might allow to generalize the \(\mathcal {M}\)-FPC-PO-augmentation construction to this setting.

  4. 4.

    Note that square (1) pasted with the pullback square formed by the morphisms \(\overline{\alpha }, id_B,e,e\circ \bar{\alpha }\) yields a pullback square that is indeed of the right form to warrant the existence of a morphism n into the FPC square \((1)\,+\,(2)\).

  5. 5.

    Note that this part of the definition of general SqPO-semantics coincides precisely with the original definition of [17].

  6. 6.

    Note that we have drawn the rule from right to left so that the input, sometimes called the left-hand side, of the rule is the topmost rightmost graph. Note also that the structure of the homomorphisms may be inferred from the node positions, with the exception of the vertex clonings that are explicitly mentioned in the text.

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Authors and Affiliations

  1. Université de Paris, CNRS, IRIF, 8 Place Aurélie Nemours, 75205, Paris Cedex 13, France

    Nicolas Behr & Jean Krivine

  2. Université de Lyon, ENS de Lyon, UCBL, CNRS, LIP, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Russ Harmer

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  1. Università di Pisa, Pisa, Italy

    Fabio Gadducci

  2. Humboldt-Universität zu Berlin, Berlin, Germany

    Timo Kehrer

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Behr, N., Harmer, R., Krivine, J. (2021). Concurrency Theorems for Non-linear Rewriting Theories. In: Gadducci, F., Kehrer, T. (eds) Graph Transformation. ICGT 2021. Lecture Notes in Computer Science(), vol 12741. Springer, Cham. https://doi.org/10.1007/978-3-030-78946-6_1

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