On the expressiveness of spider diagrams and commutative star-free regular languages

Spider diagrams provide a visual logic to express relations between sets and their elements, extending the expressiveness of Venn diagrams. Sound and complete inference systems for spider diagrams have been developed and it is known that they are equivalent in expressive power to monadic first-order logic with equality, MFOL[=]. In this paper, we further characterize their expressiveness by articulating a link between them and formal languages. First, we establish that spider diagrams define precisely the languages that are finite unions of languages of the form K {black small square} ?*, where K is a finite commutative language and ? is a finite set of letters. We note that it was previously established that spider diagrams define commutative star-free languages. As a corollary, all languages of the form K {black small square} ?* are commutative star-free languages. We further demonstrate that every commutative star-free language is also such a finite union. In summary, we establish that spider diagrams define precisely: (a) languages definable in MFOL[=], (b) the commutative star-free regular languages, and (c) finite unions of the form K {black small square} ?*, as just described.