String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics later formalised the context of symmetric monoidal categories. In recent years, they have found application modelling various computational structures, fields as diverse Computer Science, Physics, Control Theory, Linguistics, Biology. several these proposals, transformations systems modelled rewrite rules diagrams. These developments require mathematical foundation for string diagram rewriting: whereas theory terms is well-understood, two-dimensional nature poses quite few additional challenges. This work systematises expands series conference papers, laying down such foundation. As first step, we focus on case diagrammatic theories that feature Frobenius algebra. common structure provides more permissive notion composition than usual one available categories, has many applications areas concurrency, quantum theory, electrical circuits. Notably, this an exact correspondence between syntactic modulo combinatorial hypergraphs. Our introduces interpretation rewriting structures double-pushout hypergraph rewriting. We prove to be sound complete also show approach can generalised multiple structures. proof concept, how derive from results termination strategy Interacting Bialgebras, important study circuits signal flow graphs.