Abstract. We consider the problem of drawing Venn diagrams for which each region’s area is proportional to some weight (e.g., population or percentage) assigned to that region. These area-proportional Venn diagrams have an enhanced ability over traditional Venn diagrams to visually convey information about data sets with interacting characteristics. We develop algorithms for drawing area-proportional Venn diagrams for any population distribution over two characteristics using circles and over three characteristics using rectangles and near-rectangular polygons; modifications of these algorithms are then presented for drawing the more general Euler diagrams. We present results concerning which population distributions can be drawn using specific shapes. A program to aid further investigation of area-proportional Venn diagrams is also described. 1

We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles the long-standing open problem of the decidability of string graph recognition (Sinden [Sin66], Graham [Gra76]). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Grigni, Papadias, Papadimitriou [GPP95]). The general theory of Euler diagrams turns out to be as hard as second-order arithmetic.

Abstract. An important aim of diagrammatic reasoning is to make it easier for people to create and understand logical arguments. We have worked on spider diagrams, which visually express logical statements. Ideally, automatically generated proofs should be short and easy to understand. An existing proof generator for spider diagrams successfully writes proofs, but they can be long and unwieldy. In this paper, we present a new approach to proof writing in diagrammatic systems, which is guaranteed to find shortest proofs and can be extended to incorporate other readability criteria. We apply the A ∗ algorithm and develop an admissible heuristic function to guide automatic proof construction. We demonstrate the effectiveness of the heuristic used. The work has been implemented as part of a spider diagram reasoning tool. 1

This paper shows by a constructive method the existence of a diagrammatic representation called extended Euler diagrams for any collection of sets X1 , ..., Xn , n 9. These diagrams are adapted for representing sets inclusions and intersections: each set X i and each non empty intersection of a subcollection of X1 , ..., Xn is represented by a unique connected region of the plane. Starting with an abstract description of the diagram, we define the dual graph G and reason with the properties of this graph to build a planar representation of the X1 , ..., Xn . These diagrams will be used to visualize the results of a complex request on any indexed video databases. In fact, such a representation allows the user to perceive simultaneously the results of his query and the relevance of the database according to the query.

Abstract. We describe a method for drawing graph-enhanced Euler diagrams using a three stage method. The first stage is to lay out the underlying Euler diagram using a multicriteria optimizing system. The second stage is to find suitable locations for nodes in the zones of the Euler diagram using a force based method. The third stage is to minimize edge crossings and total edge length by swapping the location of nodes that are in the same zone with a multicriteria hill climbing method. We show a working version of the software that draws spider diagrams. Spider diagrams represent logical expressions by superimposing graphs upon an Euler diagram. This application requires an extra step in the drawing process because the embedded graphs only convey information about the connectedness of nodes and so a spanning tree must be chosen for each maximally connected component. Similar notations to Euler diagrams enhanced with graphs are common in many applications and our method is generalizable to drawing Hypergraphs represented in the subset standard, or to drawing Higraphs where edges are restricted to connecting with only atomic nodes. 1

Abstract — We apply the A ∗ algorithm to guide a diagrammatic theorem proving tool. The algorithm requires a heuristic function, which provides a metric on the search space. In this paper we present a collection of metrics between two spider diagrams. We combine these metrics to give a heuristic function that provides a lower bound on the length of a shortest proof from one spider diagram to another, using a collection of sound reasoning rules. We compare the effectiveness of our approach with a breadthfirst search for proofs. I.

Diagrammatic notations, such as the Unified Modeling Language (UML), are in common use in software development. They allow many aspects of software systems to be described diagrammatically, but typically they rely on textual notations for logical constraints. In contrast, spider diagrams provide a visual notation for expressing a natural class of settheoretic statements in a diagrammatic form. In this paper we present a tableau system for spider diagrams, and describe an implementation of the system. In a software development context, the system allows users to explore the implications of design choices, and thus to validate specifications; beyond this, the tableau algorithm and system are of general interest to visual reasoners. 1.

We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Our work is intended to inform automatic Euler diagram layout research by confirming the importance of various Euler diagram aesthetic criteria. The three criteria under investigation were: smoothness, zone area equality and edge closeness. Subjects were asked to interpret diagrams with different combinations of levels for each of the criteria. Results for this investigation indicate that, within the parameters of the study, all three criteria are important for understanding Euler diagrams and we have a preliminary indication of the ordering of importance for the criteria. 1.

Abstract. Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They were designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Due to the lack of a linear ordering of symbols inherent in a diagrammatic language which expresses logical statements, some constraint diagrams have more than one intuitive meaning. We generalize, from an example based approach, to suggest a default reading for constraint diagrams. This reading is usually unique, but may require a small number of simple user choices.

In this paper, we present a new graphical interface for traditional library environments, which allows the user to elaborate easily and efficiently new strategies in search processes. This tool is based on two linked interactive Euler diagram representations. The first one is an interactive representation of the structures composing the documentary kernel of the library. The user may navigate and select items, making their own understanding of the database content, structure and access. The second one is a set based visualization of the results of a composed query. This allows the user to validate his search context and to elaborate strategies to go through the results. The association of both interfaces generates a tool that allows the user to elaborate the main search strategies through graphical manipulations.