Equivalences in Euler-based diagram systems through normal forms

Abstract

The form of information presented can influence its utility for the conveying of knowledge byaffecting an interpreter’s ability to reason with the information. There are distinct types ofrepresentational systems (e.g. symbolic versus diagrammatic logics), various sub-systems (e.g.propositional versus predicate logics), and even within a single representational system theremay be different means of expressing the same piece of information content. Thus to displayinformation, choices must be made between its different representations, depending upon manyfactors such as: the context, the reasoning tasks to be considered, user preferences or desires (e.g.for short symbolic sentences or minimal clutter within diagrammatic systems). The identificationof all equivalent representations with the same information content is a sensible precursor toattempts to minimize a metric over this class. We posit that defining notions of semantic-redundancy and identifying the syntactic properties that encapsulate redundancy can help inachieving the goal of completely identifying equivalences within a single notational system oracross multiple systems, but that care must be taken when extending systems, since refinementsof redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler diagram basednotations. Such notations can be used to represent logical information and have applicationsincluding visualization of database queries, social network visualisation, statistical data visuali-sation, and as the basis of more expressive diagrammatic logics such as constraint languages usedin software specification and reasoning. The development of the new associated machinery andconcepts required is important in its own right since it increases the growing body of knowledgeon diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow a much greater degree offlexibility of representation. We give syntactic properties that encapsulate semantic equivalencein both systems, whilst observing that the same semantic concept of redundancy is significantlymore difficult to realize as syntactic properties in the extended system with projections.