# 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} }

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…

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## 20 Citations

Processes and unfoldings: concurrent computations in adhesive categories

- Computer Science, MathematicsMathematical Structures in Computer Science
- 2014

This work generalises both the notion of a non-sequential process and the unfolding construction to the abstract setting of (single pushout) rewriting of objects in adhesive categories and shows that processes are in one-to-one correspondence with switch-equivalent classes of derivations.

Double-pushout-rewriting in S-Cartesian functor categories: Rewriting theory and application to partial triple graphs

- Computer Science, MathematicsJ. Log. Algebraic Methods Program.
- 2020

This work shows the comprehensive theory of double-pushout-rewriting for S-cartesian functor categories which fulfill additional sufficient conditions and obtains all the classical results for double- Pushout- Rewriting in these categories by construction.

Analysis and Abstraction of Graph Transformation Systems via Type Graphs

- Computer Science
- 2019

This work introduces a general framework which can be suitably instantiated, in order to obtain methods usable in practice, and shows how three different refinements of the basic framework influence decidability, expressiveness and closure properties of type graph specification languages.

Adhesivity with Partial Maps instead of Spans

- Mathematics, Computer ScienceFundam. Informaticae
- 2012

Adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi- category of spans.

Recognizable languages of arrows and cospans

- Computer ScienceMathematical Structures in Computer Science
- 2018

This article gives a category-theoretical characterization of recognizability, and shows that a recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets as a straightforward generalization of finite automata.

Stochastic mechanics of graph rewriting

- Computer Science, Mathematics2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016

An algebraic approach to stochastic graph-rewriting which extends the classical construction of the Heisenberg-Weyl algebra and its canonical representation on the Fock space and finds that natural variants of the evaluation morphism map give rise to concepts of graph transformations hitherto not considered.

Single Pushout Rewriting in Comprehensive Systems

- Mathematics, Computer ScienceICGT
- 2020

Heindel’s characterisation yields cocompleteness of the category of comprehensive systems equipped with closed partial morphisms and thus enables computing by SPO graph transformation.

Robustness and closure properties of recognizable languages in adhesive categories

- Computer ScienceSci. Comput. Program.
- 2015

This paper shows that the notion of recognizable languages is robust in the sense that also semi-functors, i.e., functors that do not necessarily preserve identities, characterize recognizable languages, and shows that recognizable languages are closed under concatenation by constructing an automaton accepting the Concatenation.

A Logic on Subobjects and Recognizability

- Computer ScienceIFIP TCS
- 2010

A simple logic that allows to quantify over the subobjects of a categorical object is introduced and it is shown that, for the category of graphs, this logic is equally expressive as second-order monadic graph logic (msogl).

Hereditary Pushouts Reconsidered

- Mathematics, Computer ScienceICGT
- 2010

It is shown that the same lemmas already hold for pushouts that are hereditary, i.e. those pushout that remain pushouts when they are embedded into the associated category of partial maps.

## References

SHOWING 1-10 OF 89 REFERENCES

Modelling Concurrent Computations: from Contextual Petri Nets to Graph Grammars

- Computer Science
- 2000

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.

Processes for Adhesive Rewriting Systems

- Computer ScienceFoSSaCS
- 2006

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.

Algebraic Approaches to Graph Transformation - Part I: Basic Concepts and Double Pushout Approach

- Mathematics, Computer ScienceHandbook of Graph Grammars
- 1997

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.

Fundamental Theory for Typed Attributed Graph Transformation

- Computer ScienceICGT
- 2004

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.

Unfolding Grammars in Adhesive Categories

- Computer ScienceCALCO
- 2009

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.

Unfolding semantics of graph transformation

- Computer Science, MathematicsInf. Comput.
- 2007

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.

Graph Rewriting in Some Categories of Partial Morphisms

- Computer ScienceGraph-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 standard…

Grammar Morphisms and Weakly Adhesive Categories

- Mathematics, Computer ScienceICGT
- 2008

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.

Single-pushout rewriting in categories of spans I: the general setting

- Mathematics
- 1997

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 are…

On the composition of processes

- Computer SciencePOPL '82
- 1982

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.