We introduce a graphical language for relational data called the probabilistic entityrelationship (PER) model. The model is an extension of the entity-relationship model, a common model for the abstract representation of database structure. We concentrate on the directed version of this model—the directed acyclic probabilistic entity-relationship (DAPER) model. The DAPER model is closely related to the plate model and the probabilistic relational model (PRM), existing models for relational data. The DAPER model is more expressive than either existing model, and also helps to demonstrate their similarity. In addition to describing the new language, we discuss important facets of modeling relational data, including the use of restricted relationships, self relationships, and probabilistic relationships. Many examples are provided.

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

Abstract — The Unified Modeling Language (UML) is a collection of notations which are mainly diagrammatic. These notations are used by software engineers in the process of object oriented modelling. The only textual notation in the UML is the Object Constraint Language (OCL). The OCL is used to express logical constraints such as system invariants. Constraint diagrams are designed to provide a diagrammatic alternative to the OCL. Since constraint diagrams are visual they complement existing notations in the UML. Spider diagrams form the basis of constraint diagrams and sound and complete reasoning systems have been developed. Spider diagrams allow subset relations between sets and cardinality constraints on sets to be expressed. In addition to this, constraint diagrams allow universal quantification and relational navigation and hence are vastly more expressive. In this paper we present the first constraint diagram reasoning system. We give syntax and semantics for constraint diagrams we call CD1 diagrams. We identify syntactic criteria that allow us to determine whether a CD1 diagram is satisfiable. We give descriptions of a set of sound and complete reasoning rules for CD1 diagrams. I.

In this chapter, we introduce a graphical language for relational data called the probabilistic entity-relationship (PER) model. The model is an extension of the entity-relationship model, a common model for the abstract representation of database structure. We concentrate on the directed version of this model—the directed acyclic probabilistic entity-relationship (DAPER) model. The DAPER model is closely related to the plate model and the probabilistic relational model (PRM), existing models for relational data. The DAPER model is more expressive than either existing model, and also helps to demonstrate their similarity. In addition to describing the new language, we discuss important facets of modeling relational data, including the use of restricted relationships, self relationships, and probabilistic relationships. Many examples are provided.

Abstract. Projected contours enable Euler diagrams to scale better. They enable the representation of information using less syntax and can therefore increase visual clarity. Here informal reasoning rules are given that allow the transformation of spider diagrams with respect to projected contours. 1 Spider diagrams and projected contours A spider diagram [3] is an Euler diagram with plane trees (spiders) that represent the existence of elements and shading in regions that indicate upper bounds on the cardinalities of sets they denote. Projected contours [1, 2] here are dashed and non-projected contours are called given contours. The semantics of projected contours are given in [1]: a projected contour represents the intersection of the set denoted by its label with the set denoted by its context (the smallest region, defined in terms of given contours, that it intersects).

The uptake in the software industry of notations for designing systems visually has been accelerated with the standardization of the Unified Modeling Language (UML). The formalization of diagrammatic notations is important for the development of essential tool support and to allow reasoning to take place at the diagrammatic level. Focusing on an extended version of Venn and Euler diagrams (which was developed to complement UML in the specification of software systems), this paper presents two levels of syntax for this system: type-syntax and token-syntax. Token-syntax is about particular diagrams instantiated on some physical medium, and type-syntax provides a formal definition with which a concrete representation of a diagram must comply. While these two levels of syntax are closely related to each other, the domains of type-syntax and token-syntax are ontologically and the other concrete. We discuss the roles of typesyntax and token-syntax in diagrammatic systems and show that it is important to consider both levels of syntax in diagrammatic reasoning systems and in developing software tools to support such systems.