Jean Flower

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.

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.

Generating Readable Proofs: A Heuristic Approach to Theorem Proving With Spider Diagrams

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.

Nesting in Euler Diagrams: syntax, semantics and construction

This paper considers the notion of nesting in Euler diagrams, and how nesting affects the interpretation and construction of such diagrams. After setting up the necessary definitions for concrete Euler diagrams (drawn in the plane) and abstract diagrams (having just formal structure), the notion of nestedness is defined at both concrete and abstract levels. The concept of a dual graph is used to give an alternative condition for a drawable abstract Euler diagram to be nested. The natural progression to the diagram semantics is explored and we present a “nested form” for diagram semantics. We describe how this work supports tool-building for diagrams, and how effective we might expect this support to be in terms of the proportion of nested diagrams.

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.

Drawing graphs in Euler diagrams

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.

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.

Dynamic Euler Diagram Drawing

In this paper we describe a method to lay out a graph enhanced Euler diagram so that it looks similar to a previously drawn graph enhanced Euler diagram. This task is nontrivial when the underlying structures of the diagrams differ. In particular, if a structural change is made to an existing drawn diagram, our work enables the presentation of the new diagram with minor disruption to the users mental map. As the new diagram can be generated from an abstract representation, its initial embedding may be very different from that of the original. We have developed comparison measures for Euler diagrams, integrated into a multicriteria optimizer, and applied a force model for associated graphs that attempts to move nodes towards their positions in the original layout. To further enhance the usability of the system, the transition between diagrams can be animated

Automated Theorem Proving with Spider Diagrams

Spider diagrams are a visual notation for expressing logical statements. In this paper we describe a tool that supports reasoning with a sound and complete spider diagram system. The tool allows the construction of diagrams and proofs by users. We present an algorithm which the tool uses to determine whether one diagram semantically entails another. If the premise diagram does semantically entail the conclusion diagram then a proof is presented to the user. Otherwise it gives a counterexample: a model for the premise that is not a model for the conclusion. The proof of completeness given in [Howse, J., G. Stapleton and J. Taylor, Spider diagrams, In preparation, to appear: www.cmis.brighton.ac.uk/research/vmg] can be used to create an alternative proof writing algorithm. The algorithm described here improves upon this by providing counterexamples and significantly shorter proofs.