Corpus ID: 12495064

A category theoretical approach to the concurrent semantics of rewriting: adhesive categories and related concepts

@inproceedings{Heindel2009ACT,
  title={A category theoretical approach to the concurrent semantics of rewriting: adhesive categories and related concepts},
  author={Tobias Heindel},
  year={2009}
}
  • T. Heindel
  • Published 30 April 2010
  • Mathematics, Computer Science
This thesis studies formal semantics for a family of rewriting formalisms that have arisen as category theoretical abstractions of the so-called algebraic approaches to graph rewriting. The latter in turn generalize and combine features of term rewriting and Petri nets. Two salient features of (the abstract versions of) graph rewriting are a suitable class of categories which captures the structure of the objects of rewriting, and a notion of independence or concurrency of rewriting steps – as… Expand
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References

SHOWING 1-10 OF 89 REFERENCES
Modelling Concurrent Computations: from Contextual Petri Nets to Graph Grammars
TLDR
This thesis provides graph transformation systems with truly concurrent semantics based on (concatenable) processes and on a Winskel’s style unfolding construction, as well as with more abstract semanticsbased on event structures and domains. Expand
Processes for Adhesive Rewriting Systems
TLDR
The main result of the paper shows that processes capture the notion of true concurrency—there is a one-to-one correspondence between concurrent derivations, where the sequential order of independent steps is immaterial, and (isomorphism classes of) processes. Expand
Algebraic Approaches to Graph Transformation - Part I: Basic Concepts and Double Pushout Approach
TLDR
This chapter starts with an overwiev of the basic notions common to the two algebraic approaches, the "double-pushout (DPO) approach) and the "single-push out (SPO) approaches"; next it is presented the classical theory and some recent development of the double- pushout approach. Expand
Fundamental Theory for Typed Attributed Graph Transformation
TLDR
A rigorous approach to typed attributed graph transformation is obtained, providing as fundamental results the Local Church-Rosser, Parallelism, Concurrency, Embedding and Extension Theorem and a Local Confluence Theorem known as Critical Pair Lemma in the literature. Expand
Unfolding Grammars in Adhesive Categories
TLDR
The unfolding semantics is generalized to the abstract setting of (single pushout) rewriting over adhesive categories, which applies to a wider class of systems, which is due to the use of a refined notion of grammar morphism. Expand
Unfolding semantics of graph transformation
TLDR
This paper fully extend Winskel's approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of (semi-weighted) graph grammar and the categoryof prime algebraic domains, which factorises through the categoryOf occurrence grammARS and the categories of asymmetric event structures. Expand
Graph Rewriting in Some Categories of Partial Morphisms
  • R. Kennaway
  • Computer Science
  • Graph-Grammars and Their Application to Computer Science
  • 1990
We present a definition of term graph rewriting as the taking of a pushout in a category of partial morphisms, adapting the rather ad hoc definitions we gave in [Ken87] so as to use a standardExpand
Grammar Morphisms and Weakly Adhesive Categories
TLDR
In this paper, the proposed notion of weakly adhesive categories is presented, which has emerged during work on the generalization of the co-reflective semantics of Petri nets to the realm of graph transformation and adhesive rewriting systems. Expand
Single-pushout rewriting in categories of spans I: the general setting
A unifying view of all constructions of pushouts of partial morphisms considered so far in the literature of single-pushout transformation is given in this paper. Pushouts of partial morphisms areExpand
On the composition of processes
TLDR
A model of net-connected processes is described that amounts to a reformulation of a model derived by Brock and Ackerman from the Kahn-McQueen model of processes as relations on streams of data, which leads directly to a straightforward definition of process composition. Expand
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