2019/06/04 by Filippo Bonchi, Bonchi, Filippo, Robin Piedeleu +5 · 1 voice
Computer Science · Mathematics · #03G99 #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO) #Programming Languages (cs.PL) #cs.LO #cs.PL #math.CT
paper · pdf · doi:10.48550/arxiv.1906.01519
arxiv published 2019/06/04 · arxiv updated 2019/07/02
Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law over that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, à la Hoare and à la Milner.