On the Relative Expressiveness of Second-Order Spider Diagrams and Regular Expressions

This paper is about spider diagrams, an extension of Euler diagrams that includes syntax to make assertions about set cardinalities. Like many diagrammatic logics, spider diagrams are a monadic and first-order, so they are inexpressive. The limitation to first-order precludes the formalisation of many fundamental concepts such as the cardinality of a set being even. To this end, second-order spider diagrams have recently been proposed. In this paper, we establish their expressiveness relative to that of regular expressions, advancing novel research connecting diagrammatic logics with formal languages. In particular, we establish that every regular language is definable by a second-order spider diagram, demonstrating a substantial increase in expressiveness over spider diagrams.