Diagrammatic reasoning for hierarchical systems

What is it about?

Natural sciences, as well as many areas of informatics and engineering, organise into hierarchical layers of abstraction. For instance, chemical reactions may be described abstractly as networks or more finely as molecular interactions; similarly, in designing computer architecture, high-level process descriptions must ultimately be implemented as microelectronic circuits. In recent years, many diagrammatic languages have been developed for describing such hierarchies, which allow reasoning by drawing diagrams – all while retaining full mathematical precision. Examples include diagrammatic languages for computation, scalable graphical notation for quantum computing and hierarchical diagrams for logical reasoning and database queries. In this work, we systematise existing developments by defining a novel diagrammatic language that is able to describe a system at different levels of abstraction. In doing so, we retain the key features of ordinary diagrammatic languages: a visually intuitive notion of information flow, and built-in notions of parallel and sequential composition, i.e. ability to say that two processes occur either simultaneously or one after another. Our language further allows mixing several diagrammatic languages (together with translations between them) within the same diagram. To illustrate the motivation and potential use cases, we provide extended case studies – including digital circuits, electrical circuits, probabilistic channels, quantum circuits, concurrency theory and chemical reactions – of notions appearing in the literature that are instances of our diagrammatic language.

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Photo by Steve A Johnson on Unsplash

Photo by Steve A Johnson on Unsplash

Why is it important?

The study of the levels of abstraction is motivated by hierarchies of scientific theories. For instance, chemistry provides the vocabulary to talk about interactions between molecules, whereas particle physics lacks the language to describe interactions of such complexity. Similarly, biology can describe the function of a cell within an organism, while chemistry is restricted to describing reactions that are necessary to fulfil this function. Choosing the right level of abstraction is important for successfully interpreting one theory in another one. Our aim is to provide a mathematical formalism for doing so.