Concurrency Theorems for Non-linear Rewriting Theories

Behr, Nicolas; Harmer, Russ; Krivine, Jean

doi:10.1007/978-3-030-78946-6_1

Cited by 13 publications

(9 citation statements)

References 29 publications

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“…Quasitoposes include not only the categories of sets, directed multigraphs [4] and typed graphs [5] (all of which are toposes), but also a variety of structures such as Heyting algebras (considered as categories) [3], and the categories of simple graphs (equivalently, binary relations) [6], fuzzy sets [3], algebraic specifications [6] and safely marked Petri Nets [6]. Behr et al [15] recently proposed quasitoposes as a natural setting for non-linear rewriting.…”

Section: Pbpo + As a Unifying Theory For Quasitoposesmentioning

confidence: 99%

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Graph Rewriting and Relabeling with PBPO+: A Unifying Theory for Quasitoposes

Overbeek¹,

Endrullis²,

Rosset³

2022

Preprint

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We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO + , allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO + can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO + can define all rewrite relations definable by PBPO, AGREE and DPO, as well as additional ones. Additionally, we show that PBPO + is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.

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Section: Pbpo + As a Unifying Theory For Quasitoposesmentioning

confidence: 99%

“…The following definitions on M-partial map classification derive from work by Cockett and Lack on restriction categories [16,17], but we follow the presentations of Corradini et al [5] and Behr et al [15].…”

Section: Quasitoposesmentioning

confidence: 99%

Graph Rewriting and Relabeling with PBPO+: A Unifying Theory for Quasitoposes

Overbeek¹,

Endrullis²,

Rosset³

2022

Preprint

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“…The hypotheses that appears often together have given rise to some definitions, such as adhesive categories [17] with variations such as quasiadhesivity [18], rm-quasiadhesivity [13], M-adhesivity [11], etc. Some already existing definitions have also gained a new traction that way, such as toposes [21] and more recently a generalisation of toposes: quasitoposes [4,22]. This article looks more deeply at this latter notion of quasitoposes.…”

mentioning

confidence: 99%

Fuzzy Presheaves are Quasitoposes

Rosset¹,

Overbeek²,

Endrullis³

2023

Preprint

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Quasitoposes have proven to be an interesting framework for many graph rewriting formalisms. Since presheaves categories are toposes and fuzzy sets are quasitoposes, we prove that the combination of both, fuzzy presheaves, are also quasitoposes. This question was recently conjectured more specifically for fuzzy graphs and is now therefore proven. This entails that many fuzzy presheaves of interest can be used for graph rewriting.

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“…Quasitoposes have featured in the graph transformation literature, but not nearly as extensively as the various adhesivity notions and the notion of a topos. The rst publication in this direction was the one by Johnstone et al [JLS07], and recently the notion has received renewed attention by Behr et al [BHK21].…”

Section: Settings For Algebraic Graph Transformationmentioning

confidence: 99%

“…Quasitoposes include not only the categories of sets, directed multigraphs [LS04], typed graphs [CDE + 15], and many other graph categories (many of which are toposes), but also a variety of structures, such as Heyting algebras (considered as categories) [Wyl91], the categories of simple graphs (equivalently, binary relations) [JLS07], fuzzy sets [Wyl91], algebraic speci cations [JLS07], and safely marked Petri Nets [JLS07]. Behr et al [BHK21] moreover have recently proposed quasitoposes as a natural setting for non-linear rewriting.…”

Section: Pbpo + In Contextmentioning

confidence: 99%

A Unifying Theory for Graph Transformation

Overbeek

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The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO.

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Concurrency Theorems for Non-linear Rewriting Theories

Cited by 13 publications

References 29 publications

Graph Rewriting and Relabeling with PBPO+: A Unifying Theory for Quasitoposes

Graph Rewriting and Relabeling with PBPO+: A Unifying Theory for Quasitoposes

Fuzzy Presheaves are Quasitoposes

A Unifying Theory for Graph Transformation

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