5 research outputs found
Diagrammatic Polyhedral Algebra
We extend the theory of Interacting Hopf algebras with an order primitive, and give a sound and complete axiomatisation of the prop of polyhedral cones. Next, we axiomatise an affine extension and prove soundness and completeness for the prop of polyhedra
Compositional Modelling of Network Games
The analysis of games played on graph-like structures is of increasing importance due to the prevalence of social networks, both virtual and physical, in our daily life. As well as being relevant in computer science, mathematical analysis and computer simulations of such distributed games are vital methodologies in economics, politics and epidemiology, amongst other fields. Our contribution is to give compositional semantics of a family of such games as a well-behaved mapping, a strict monoidal functor, from a category of open graphs (syntax) to a category of open games (semantics). As well as introducing the theoretical framework, we identify some applications of compositionality
On Doctrines and Cartesian Bicategories
We study the relationship between cartesian bicategories and a specialisation
of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both
provide different ways of abstracting the structural properties of logical
systems: the former in algebraic terms based on a string diagrammatic calculus,
the latter in universal terms using the fundamental notion of adjoint functor.
We prove that these two approaches are related by an adjunction, which can be
strengthened to an equivalence by imposing further constraints on doctrines
Regular Monoidal Languages
We introduce regular languages of morphisms in free monoidal categories, with their associated grammars and automata. These subsume the classical theory of regular languages of words and trees, but also open up a much wider class of languages over string diagrams. We use the algebra of monoidal categories to investigate the properties of regular monoidal languages, and provide sufficient conditions for their recognizability by deterministic monoidal automata