PaL Diagrams: a linear diagram-based visual language

Linear diagrams have recently been shown to be more effective than Euler diagrams when used for set-based rea- soning. However, unlike the growing corpus of knowledge about formal aspects of Euler and Venn diagrams, there has been no formalisation of linear diagrams. To fill this knowl- edge gap, we present and formalise Point and Line (PaL) di- agrams, an extension of simple linear diagrams containing points, thus providing a formal foundation for an effective visual language. We prove that PaL diagrams are exactly as expressive as monadic first-order logic with equality, gain- ing, as a corollary, an equivalence with the Euler diagram extension called spider diagrams. The method of proof pro- vides translations between PaL diagrams and sentences of monadic first-order logic.