Characterisation of Parallel Independence in AGREE-Rewriting

Lowe, M. J. S.

doi:10.1007/978-3-319-92991-0_8

Cited by 8 publications

(6 citation statements)

References 10 publications

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“…This property is satisfied by adhesive categories (cf. Lemma 3 of Appendix A), yet to the best of our knowledge the question of which more general types of categories possess this property has not yet been investigated to quite the level of generality as analogous classification problems in the case of DPO rewriting, even though there does exist a large body of work on classes of categories that admit SqPO constructions [16,39,36,14,37]. Within these classes, according to [13,14] guarantees for the existence of FPCs may be provided for categories that possess a so-called M -partial map classifier.…”

Section: Background: Adhesive Categories and Final Pullback Complementsmentioning

confidence: 99%

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Behr

2019

Electron. Proc. Theor. Comput. Sci.

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Sesqui-pushout (SqPO) rewriting is a variant of transformations of graph-like and other types of structures that fit into the framework of adhesive categories where deletion in unknown context may be implemented. We provide the first account of a concurrency theorem for this important type of rewriting, and we demonstrate the additional mathematical property of a form of associativity for these theories. Associativity may then be exploited to construct so-called rule algebras (of SqPO type), based upon which in particular a universal framework of continuous-time Markov chains for stochastic SqPO rewriting systems may be realized. * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 753750.1 While non-linear rules in SqPO rewriting have interesting applications in their own right (permitting e.g. the cloning and fusing of vertices in graphs), this most general case is left for future work.SqPO Rewriting: Concurrency, Associativity and Rule Algebra Framework adhesive categories (see Assumption 1) in order to ensure certain technical properties necessary for our concurrency and associativity theorems to hold. To the best of our knowledge, apart from some partial results in the direction of developing a concurrency theorem for SqPO-type rewriting in [16,36,15], prior to this work neither of the aforementioned theorems had been available in the SqPO framework.Associativity of SqPO rewriting theories plays a pivotal role in our development of a novel form of concurrent semantics for these theories, the so-called SqPO-type rule algebras. Previous work on associative DPO-type rewriting theories [3, 5, 7] (see also [8]) has led to a category-theoretical understanding of associativity that may be suitably extended to the SqPO setting. In contrast to the traditional and well-established formalisms of concurrency theory for rewriting systems (see e.g. [42,25,23,15] for DPO-type semantics and [16,15] for a notion of parallel independence and a Local Church-Rosser theorem for SqPO-rewriting of graphs), wherein the focus of the analysis is mostly on derivation traces and their sequential independence and parallelism properties, the focus of our rule-algebraic approach differs significantly: we propose instead to put sequential compositions of linear rules at the center of the analysis (rather than the derivation traces), and moreover to employ a vector-space based semantics in order to encode the non-determinism of such rule compositions. It is for this reason that the concurrency theorem plays a quintessential role in our rule algebra framework, in that it encodes the relationship between sequential compositions of linear rules and derivation traces, which in turn gives rise to the socalled canonical representations of the rule algebras (see Section 4). This approach in particular permits to uncover certain combinatorial properties of rewriting systems that would otherwise not be accessible. While undoubtedly not a stand...

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Section: Background: Adhesive Categories and Final Pullback Complementsmentioning

confidence: 99%

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Behr

2019

Electron. Proc. Theor. Comput. Sci.

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“…Kreowski [17] tackled the problem of parallel graph transformations and introduced the notion of parallel independence. This pioneering work has been considered for several algebraic graph transformation approaches, see [15] as well as the more recent contributions [7,26,25]. At almost the same period, parallel graph transformations has been used as an extension of L-systems [29,31] as was proposed in, e.g., [21].…”

Section: Conclusion and Related Workmentioning

confidence: 99%

Parallel Coherent Graph Transformations

Tour

Echahed

2021

Recent Trends in Algebraic Development Techniques

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Cellular automata as well as simultaneous assignments in Python can be understood as the parallel application of local rules to a grid or an environment that can be easily represented as an attributed graph. Since the result of such transformations cannot generally be obtained by a sequential application of the involved rules, this situation infringes the standard notion of parallel independence. An algebraic approach with production rules of the form L Ð K Ð I Ñ R is adopted and a condition of parallel coherence more general than parallel independence is formulated, that enable the definition of the Parallel Coherent Transformation (PCT). This transformation supports a generalisation of the Parallelism Theorem in the theory of adhesive HLR categories, showing that the PCT yields the expected result of sequential rewriting steps when parallel independence holds. Categories of finitely attributed structures are proposed, in which PCTs are guaranteed to exist. These notions are introduced and illustrated on several detailed examples.

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“…Kreowski [14] tackled the problem of parallel graph transformations and proposed conditions under which parallel graph transformations could be sequentialized and how sequential independent graph transformations could be parallelized. This pioneering work has been considered for several algebraic graph transformation approaches, see, e.g., the most recent contributions [9,22,21] or Volume 3 of the Handbook of Graph Grammars and Computing by Graph Transformation [12]. However, this stream of work departs drastically from our goal where parallel graph transformations are not aimed to be sequentialized.…”

Section: Examplementioning

confidence: 99%

Combining Parallel Graph Rewriting and Quotient Graphs

Tour

Echahed

2020

Rewriting Logic and Its Applications

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We define two graph transformations, one by parallelizing graph rewrite rules, the other by taking quotients of graphs. The former consists in the exhaustive application of local transformations defined by graph rewrite rules expressed in a set-theoretic framework. Compared with other approaches to parallel rewriting, we allow a substantial amount of overlapping only restricted by a condition called the effective deletion property. This transformation can be reduced by factoring out possibly many equivalent matchings by the automorphism groups of the rules. The second transformation is based on the use of equivalence relations over graph items and offers a new way of performing simultaneous merging operations. The relevance of combining the two transformations is illustrated on a running example.

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Characterisation of Parallel Independence in AGREE-Rewriting

Cited by 8 publications

References 10 publications

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Parallel Coherent Graph Transformations

Combining Parallel Graph Rewriting and Quotient Graphs

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