Investigating reasoning with constraint diagrams

Constraint diagrams are a visual notation designed to express logical constraints. Augmenting the diagrams with a reading tree (effectively a partial ordering of quantifiers) ensures that each diagram has a unique semantic interpretation.In this paper, we discuss examples of reasoning rules for augmented constraint diagrams which exhibit interesting properties or difficulties that can arise when developing rules for such a diagrammatic system. We do not present a complete set of rules, but investigate the generic problems arising, providing solutions. One problem corresponds to the nesting of quantifiers and another relates to the domain of universal quantification. These issues may be an important consideration in the definition of other logical reasoning systems which explicitly represent quantification diagrammatically.