On the Descriptional Complexity of a Diagrammatic Notation

Spider diagrams are a widely studied, visual logic that are able to make statements about relationships between sets and their cardinalities. Various meta-level results for spider diagrams have been established, including their soundness, completeness and expressiveness. In order to further enhance our understanding of spider diagrams, we can compare them with other languages; in the case of this paper we consider star-free regular languages. We establish relationships between various fragments of the spider diagram language and certain well-known subclasses of the star-free regular class. Utilising these relationships, given any spider diagram, we provide an upper-bound on the state complexity of minimal deterministic finite automata corresponding to that spider diagram. We further demonstrate cases where this bound is tight.