Formalizing the essence of diagrammatic syntax

摘要

Diagrams have a long history as visual aids which assist in structuring and simplifying potentially complex reasoning tasks. Recent years have witnessed a rapid, ongoing popularisation of diagrammatic notations in the specification, modelling and programming of computing systems, leading to diagrammatic languages requiring increasingly complex semantic interpretations. A general theory of diagrammatic languages, following those of more typical text-based languages, requires as a minimum an account of diagram syntax, of semantics, and of interpretations: the relationships between syntax and semantics. A satisfactory theory of diagrams must also account for more cognitive aspects, notably the ways in which pragmatic features of diagrams contribute to their effectiveness for human users, and how individual cognitive differences affect human interpretation of diagrams. We present preliminary work and ideas in the direction of providing a mathematical understanding of the essential features and idiosyncrasies of diagrammatic syntax. In particular, we are investigating the applicability of methods and devices drawn from the theory of categories [5], from which we adapt the notions of sketch [1] and morphism in order to capture the constituent elements of diagrams as well as relations among diagrams in the same class.