{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T19:59:45Z","timestamp":1762459185290},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,6,26]],"date-time":"2014-06-26T00:00:00Z","timestamp":1403740800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2014,8]]},"abstract":"<jats:p>Amalgamation is a well-known concept for graph transformations that is used to model synchronised parallelism of rules with shared subrules and corresponding transformations. This concept is especially important for an adequate formalisation of the operational semantics of statecharts and other visual modelling languages, where typed attributed graphs are used for multiple rules with nested application conditions. However, the theory of amalgamation for the double-pushout approach has so far only been developed on a set-theoretical basis for pairs of standard graph rules without any application conditions.<\/jats:p><jats:p>For this reason, in the current paper we present the theory of amalgamation for<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0960129512000345_inline2\" \/><jats:tex-math>$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories, which form a slightly more general framework than (weak) adhesive HLR categories, for a bundle of rules with (nested) application conditions. The two main results are the Complement Rule Theorem, which shows how to construct a minimal complement rule for each subrule, and the Multi-Amalgamation Theorem, which generalises the well-known Parallelism and Amalgamation Theorems to the case of multiple synchronised parallelism. In order to apply the largest amalgamated rule, we use maximal matchings, which are computed according to the actual instance graph. The constructions are illustrated by a small but meaningful running example, while a more complex case study concerning the firing semantics of Petri nets is presented as an introductory example and to provide motivation.<\/jats:p>","DOI":"10.1017\/s0960129512000345","type":"journal-article","created":{"date-parts":[[2014,6,26]],"date-time":"2014-06-26T14:22:13Z","timestamp":1403792533000},"source":"Crossref","is-referenced-by-count":16,"title":["Multi-amalgamation of rules with application conditions in -adhesive categories"],"prefix":"10.1017","volume":"24","author":[{"given":"ULRIKE","family":"GOLAS","sequence":"first","affiliation":[]},{"given":"ANNEGRET","family":"HABEL","sequence":"additional","affiliation":[]},{"given":"HARTMUT","family":"EHRIG","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,6,26]]},"reference":[{"key":"S0960129512000345_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-17322-6_7"},{"key":"S0960129512000345_ref1","first-page":"1","article-title":"A Subgraph Operator for Graph Transformation Languages.","volume":"6","author":"Balasubramanian","year":"2007","journal-title":"Electronic Communications of the EASST"},{"key":"S0960129512000345_ref11","first-page":"111","article-title":"Categorical Frameworks for Graph Transformation and HLR Systems based on the DPO Approach.","volume":"102","author":"Ehrig","year":"2010","journal-title":"Bulletin of the EATCS"},{"key":"S0960129512000345_ref23","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90068-5"},{"key":"S0960129512000345_ref25","first-page":"1","article-title":"Repotting the Geraniums: On Nested Graph Transformation Rules.","volume":"18","author":"Rensink","year":"2009","journal-title":"Electronic Communications of the EASST"},{"key":"S0960129512000345_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-47884-1_2"},{"key":"S0960129512000345_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-46464-8_21"},{"key":"S0960129512000345_ref4","unstructured":"Biermann E. , Ermel C. , Schmidt J. and Warning A. (2010c) Visual Modeling of Controlled EMF Model Transformation using HENSHIN. In: Proceedings of the Fourth International Workshop on Graph-Based Tools (GraBaTs 2010). 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(2014) M-Adhesive Transformation Systems with Nested Application Conditions. Part 1: Parallelism, Concurrency and Amalgamation. Mathematical Structures in Computer Science (this volume).","DOI":"10.1017\/S0960129512000357"},{"key":"S0960129512000345_ref29","doi-asserted-by":"crossref","unstructured":"Taentzer G. (1996) Parallel and Distributed Graph Transformation: Formal Description and Application to Communication Based Systems, Ph.D. thesis, Technische Universit\u00e4t Berlin.","DOI":"10.1007\/3-540-61228-9_95"},{"key":"S0960129512000345_ref13","unstructured":"Ermel C. (2006) Simulation and Animation of Visual Languages based on Typed Algebraic Graph Transformation, Ph.D. thesis, Technische Universit\u00e4t Berlin."},{"key":"S0960129512000345_ref16","first-page":"1","article-title":"A Visual Interpreter Semantics for Statecharts Based on Amalgamated Graph Transformation.","volume":"39","author":"Golas","year":"2011","journal-title":"Electronic Communications of the EASST"},{"key":"S0960129512000345_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02408-5_6"},{"key":"S0960129512000345_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129508007202"},{"key":"S0960129512000345_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-65306-6"},{"key":"S0960129512000345_ref28","doi-asserted-by":"publisher","DOI":"10.1142\/9789812815149_0013"},{"key":"S0960129512000345_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-25959-6_35"},{"key":"S0960129512000345_ref26","doi-asserted-by":"publisher","DOI":"10.1142\/3303"},{"key":"S0960129512000345_ref32","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45832-8_28"},{"key":"S0960129512000345_ref15","doi-asserted-by":"crossref","unstructured":"Golas U. 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