Andrew Fish

The semantics of augmented constraint diagrams

Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They generalize Venn diagrams and Euler circles, and include syntax for quantification and navigation of relations. The notation was designed to complement the Unified Modelling Language in the development of software systems. Since symbols representing quantification in a diagrammatic language can be naturally ordered in multiple ways, some constraint diagrams have more than one intuitive meaning in first-order predicate logic. Any equally expressive notation which is based on Euler diagrams and conveys logical statements using explicit quantification will have to address this problem. We explicitly augment constraint diagrams with reading trees, which provides a partial ordering for the quantifiers (determining their scope as well as their relative ordering). Alternative approaches using spatial arrangements of components, or alphabetical ordering of symbols, for example, can be seen as implicit representations of a reading tree. Whether the reading tree accompanies the diagram explicitly (optimizing expressiveness) or implicitly (simplifying diagram syntax), we show how to construct unambiguous semantics for the augmented constraint diagram.

Automated Theorem Proving in Euler Diagram Systems

Diagrammatic reasoning has the potential to be important in numerous application areas. This paper focuses on the simple, but widely used, Euler diagrams that form the basis of many more expressive logics. We have implemented a diagrammatic theorem prover, called Edith, which has access to four sound and complete sets of reasoning rules for Euler diagrams. Furthermore, for each rule set we develop a sophisticated heuristic to guide the search for a proof. This paper is about understanding how the choice of reasoning rule set affects the time taken to find proofs. Such an understanding will influence reasoning rule design in other logics. Moreover, this work specific to Euler diagrams directly benefits the many logics based on Euler diagrams. We investigate how the time taken to find a proof depends not only on the proof task but also on the reasoning system used. Our evaluation allows us to predict the best choice of reasoning system, given a proof task, in terms of time taken, and we extract a guide for defining reasoning rules for other logics in order to minimize time requirements.

General Euler Diagram Generation

Euler diagrams are a natural method of representing set-theoretic data and have been employed in diverse areas such as visualizing statistical data, as a basis for diagrammatic logics and for displaying the results of database search queries. For effective use of Euler diagrams in practical computer based applications, the generation of a diagram as a set of curves from an abstract description is necessary. Various practical methods for Euler diagram generation have been proposed, but in all of these methods the diagrams that can be produced are only for a restricted subset of all possible abstract descriptions. We describe a method for Euler diagram generation, demonstrated by implemented software, and illustrate the advances in methodology via the production of diagrams which were difficult or impossible to draw using previous approaches. To allow the generation of all abstract descriptions we may be required to have some properties of the final diagram that are not considered nice. In particular we permit more than two curves to pass though a single point, permit some curve segments to be drawn concurrently, and permit duplication of curve labels. However, our method attempts to minimize these bad properties according to a chosen prioritization.

Euler diagram generation

Euler diagrams form the basis of many diagrammatic notations used to represent set theoretic relationships in a wide range of contexts including: file system information, statistical data representation, object-oriented modeling, logical specification and reasoning systems, and database search queries. An abstract Euler diagram is a formal abstract description of the information that is to be displayed as a concrete (or drawn) Euler diagram. If the abstract diagram can be visualized, whilst satisfying certain desirable visual properties (called well-formedness conditions), then we say the diagram is drawable. We solve the drawability problem for a given set of well-formedness conditions, identifying the properties which classify a diagram as drawable or undrawable. Furthermore, we present a high level algorithm which enables the generation of a concrete diagram from an abstract diagram, whenever it is drawable.

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.

A reading algorithm for constraint diagrams

Constraint diagrams are a visual notation 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. Their design emphasizes scalability and expressiveness while retaining intuitiveness. The formalization of constraint diagrams is non-trivial: previous attempts have exposed subtleties concerned with the ordering of symbols in the visual language. Consequently, some constraint diagrams have more than one intuitive reading. We develop the concept of the dependence graph for a constraint diagram. From the dependence graph, we obtain a set of reading trees. A reading tree provides a partial ordering for some syntactic elements of the diagram. Given a reading tree for a constraint diagram, we present an algorithm that delivers a unique semantic reading.

Embedding Wellformed Euler Diagrams

Euler diagrams are collections of labelled closed curves. They are often used to represent information about the relationship between sets and, as such, they have numerous applications including: visualizing biological data, diagrammatic logics, and visual database querying. Various methods to automatically generate Euler diagrams have been proposed. Typically, the generation process starts with an abstract description of an Euler diagram, which is then converted to a planar dual graph. Finally, the process attempts to embed the Euler diagram from the dual graph. This paper describes a method for embedding wellformed Euler diagrams from dual graphs. There are several mechanisms to generate dual graphs but, prior to the novel work described here, no general method for embedding a wellformed Euler diagram from a dual graph had been demonstrated. The method in this paper achieves an embedding of any wellformed Euler diagram. The method first triangulates the dual graph. Then, using the faces of the triangulated graph, an edge labelling technique identifies the vertices of polygons which form the closed curves of the Euler diagram. The method is demonstrated by a Java implementation. In addition, this paper discusses a number of layout improvements that can be explored for this embedding method.

User-comprehension of Euler diagrams

Euler diagrams are a diagrammatic system for representing and reasoning with set theoretic statements. Syntactic constraints called wellformedness conditions (WFCs) are often imposed with the intention of reducing comprehension errors, but there is little supporting empirical evidence that they have the desired effect. We report on experiments which support the theory that the WFCs are generally beneficial for novice user comprehension, but we discover that violating some individual WFCs, such as concurrency, can be beneficial. Furthermore, we examine a prioritisation of the WFCs, derived from the user comprehension results, which could be used to prioritise theoretical work on generation problems or to assist in the provision of a choice of a diagram to display to users, for instance. We have used similar materials to our previous ‘preference study’ for cross comparison purposes. This accumulation of work has motivated the development of a model of the user comprehension with the aim of more closely linking theoretical and empirical works examining effective notation design, general approaches to displaying notations and interacting with notations.

EulerView: a non-hierarchical visualization component

The Treeview control is the traditional way of visualizing hierarchical information in user interfaces, but in situations such as managing bookmarks in browsers, more general classifications may be more appropriate than hierarchies. We have developed a component called EulerView which is an extension of Treeview, with a similar look and feel, addressing migration, navigation and overspecificity issues amongst 2D or 3D visualisations of classifications. It has familiar characteristics, and an underlying model, based on Euler diagrams, enabling more general classifications. We found, via user testing, that users of EulerView appear to grasp a fairly accurate internal representation of Euler diagrams and that they could organise bookmarks well within in a given classification using EulerView.

Euler Diagram Decomposition

Euler diagrams are a common visual representation of set-theoretic statements, and they have been used for visualising the results of database search queries or as the basis of diagrammatic logical constraint languages for use in software specification. Such applications rely upon the ability to automatically generate diagrams from an abstract description. However, this problem is difficult and is known to be NP-complete under certain wellformedness conditions. Therefore methods to identify when and how one can decompose abstract Euler diagrams into simpler components provide a vital step in improving the efficiency of tools which implement a generation process. One such decomposition, called diagram nesting, has previously been identified and exploited. In this paper, we make substantial progress, defining the notion of a disconnecting contour and identifying the conditions on an abstract Euler diagram that allow us to identify disconnecting contours. If a diagram has a disconnecting contour, we can draw it more easily, by combining the results of drawing smaller diagrams. The drawing problem is just one context which benefits from such diagram decomposition - we can also use the disconnecting contour to provide a more natural semantic interpretation of the Euler diagram.