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Left-Linear Rewriting in Adhesive Categories

Authors Paolo Baldan , Davide Castelnovo , Andrea Corradini , Fabio Gadducci

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Author Details

Paolo Baldan
  • Department of Mathematics, University of Padua, Italy
Davide Castelnovo
  • Department of Mathematics, University of Padua, Italy
Andrea Corradini
  • Department of Computer Science, University of Pisa, Italy
Fabio Gadducci
  • Department of Computer Science, University of Pisa, Italy

Funding

The research has been partially supported by the EuropeanUnion - NextGenerationEU under the National Recovery and Resilience Plan (NRRP) - Call PRIN 2022 PNRR - Project P2022HXNSC "Resource Awareness in Programming: Algebra, Rewriting, and Analysis", by the Italian MUR - Call PRIN 2022 - Project 20228KXFN2 "Spatio-Temporal Enhancement of Neural nets for Deeply Hierarchical Automatised Logic" and by the University of Pisa - Call PRA 2022 - Project 2022_99 "Formal Methods for the Healthcare Domain based on Spatial Information".

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Paolo Baldan, Davide Castelnovo, Andrea Corradini, and Fabio Gadducci. Left-Linear Rewriting in Adhesive Categories. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CONCUR.2024.11

Abstract

When can two sequential steps performed by a computing device be considered (causally) independent? This is a relevant question for concurrent and distributed systems, since independence means that they could be executed in any order, and potentially in parallel. Equivalences identifying rewriting sequences which differ only for independent steps are at the core of the theory of concurrency of many formalisms. We investigate the issue in the context of the double pushout approach to rewriting in the general setting of adhesive categories. While a consolidated theory exists for linear rules, which can consume, preserve and generate entities, this paper focuses on left-linear rules which may also "merge" parts of the state. This is an apparently minimal, yet technically hard enhancement, since a standard characterisation of independence that - in the linear case - allows one to derive a number of properties, essential in the development of a theory of concurrency, no longer holds. The paper performs an in-depth study of the notion of independence for left-linear rules: it introduces a novel characterisation of independence, identifies well-behaved classes of left-linear rewriting systems, and provides some fundamental results including a Church-Rosser property and the existence of canonical equivalence proofs for concurrent computations. These results properly extends the class of formalisms that can be modelled in the adhesive framework.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Semantics and reasoning
Keywords
  • Adhesive categories
  • double-pushout rewriting
  • left-linear rules
  • switch equivalence
  • local Church-Rosser property

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