Elena Tulchinsky

Research thesis

Positive Semantics of Projections in Venn-Euler Diagrams

Supervisor: Dr. Joseph (Yossi) Gil

Table of contents:

Abstract

    Venn diagrams and Euler circles have long been used to express relationship among sets using visual metaphors such as disjointness and containment of topological contours. Although the notation is effective in delivering a clear visual modeling of set theoretical relationship, it does not scale well. The topology of Venn diagrams of four contours or more is so cluttered that it becomes impractical to use it for visualization of set relationships.
In this work we study "projection contours", a new means for presenting sets intersections, which is designed to reduce the clutter in such diagrams. Informally, a projection is a contour which describes a set of elements limited to a certain context. The challenge in introducing this notation is in producing precise and consistent semantics for the general case, including a diagram comprising several, possibly interacting, projections, which might even be of the same base set. The semantics investigated here assigns a "positive" meaning to a projection, i.e., based on the list of contours with which it interacts, where contours disjoint to it do not change its semantics. Another approach, calling in literature a negative semantics, assigns to a projection the semantics based on the contours interacting with the projection as well on the contours disjoint to the projection.
Positive semantics for a projection is produced by a novel Gaussian-like elimination process for solving set equations.
In dealing with multiple projections of the same base set, we introduce yet another extension to Venn-Euler diagrams in which the same set can be described by multiple contours. This extension is of independent interest as a powerful means for reducing the clutter in Venn-Euler diagrams.

 

Seminar lecture slides

Final thesis paper

This work was presented at the International Conference on the Theory and Application of Diagrams (DAIGRAMS 2000) in September 2000. The final paper, submitted to the Senate of the Technion, is availabled All Venn-Euler diagrams for the thesis were prepared using visual editor, named CD-Editor, developped as a part of  Yan Sorkin thesis.