Defining star-free regular languages using diagrammatic logic

Spider diagrams are a recently developed visual logic that make statements about relationships between sets, their members and their cardinalities. By contrast, the study of regular languages is one of the oldest active branches of computer science research. The work in this thesis examines the previously unstudied relationship between spider diagrams and regular languages. In this thesis, the existing spider diagram logic and the underlying semantic theory is extended to allow direct comparison of spider diagrams and star-free regular languages. Thus it is established that each spider diagram defines a commutative star-free regular language. Moreover, we establish that every commutative star-free regular language is definable by a spider diagram. From the study of relationships between spider diagrams and commutative star-free regular languages, an extension of spider diagrams is provided. This logic, called spider diagrams of order, increases the expressiveness of spider diagrams such that the language of every spider diagram of order is star-free and regular, but not-necessarily commutative. Further results concerning the expressive power of spider diagrams of order are gained through the use of a normal form for the diagrams. Sound reasoning rules which take a spider diagram of order and produce a semantically equivalent diagram in the normal form are provided. A proof that spider diagrams of order define precisely the star-free regular languages is subsequently presented. Further insight into the structure and use of spider diagrams of order is demonstrated by restricting the syntax of the logic. Specifically, we remove spiders from spider diagrams of order. We compare the expressiveness of this restricted fragment of spider diagrams of order with the unrestricted logic.