Gem Stapleton

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.

The Expressiveness of Spider Diagrams

Spider diagrams are a visual language for expressing logical statements. In this paper we identify a well-known fragment of first-order predicate logic that we call MFOL=, equivalent in expressive power to the spider diagram language. The language MFOL= is monadic and includes equality but has no constants or function symbols. To show this equivalence, in one direction, for each diagram we construct a sentence in MFOL= that expresses the same information. For the more challenging converse we prove that there exists a finite set of models for a sentence S that can be used to classify all the models for S. Using these classifying models we show that there is a diagram expressing the same information as S.

Visualizing ontologies: a case study

Concept diagrams were introduced for precisely specifying ontologies in a manner more readily accessible to developers and other stakeholders than symbolic notations. In this paper, we present a case study on the use of concept diagrams in visually specifying the Semantic Sensor Networks (SSN) ontology. The SSN ontology was originally developed by an Incubator Group of the W3C. In the ontology, a sensor is a physical object that implements sensing and an observation is observed by a single sensor. These, and other, roles and concepts are captured visually, but precisely, by concept diagrams. We consider the lessons learnt from developing this visual model and show how to convert description logic axioms into concept diagrams. We also demonstrate how to merge simple concept diagram axioms into more complex axioms, whilst ensuring that diagrams remain relatively uncluttered.

A Survey of Reasoning Systems Based on Euler Diagrams

Euler diagrams have been used for centuries as a means for conveying ideas in an intuitive, informal way. Recently much research has been conducted to develop formal, diagrammatic reasoning systems based on Euler diagrams. Most of these systems extend Euler diagrams by adding further syntax to increase expressiveness. In this paper we survey such systems and draw comparisons between them.

Inductively Generating Euler Diagrams

Euler diagrams have a wide variety of uses, from information visualization to logical reasoning. In all of their application areas, the ability to automatically layout Euler diagrams brings considerable benefits. In this paper, we present a novel approach to Euler diagram generation. We develop certain graphs associated with Euler diagrams in order to allow curves to be added by finding cycles in these graphs. This permits us to build Euler diagrams inductively, adding one curve at a time. Our technique is adaptable, allowing the easy specification, and enforcement, of sets of well-formedness conditions; we present a series of results that identify properties of cycles that correspond to the well-formedness conditions. This improves upon other contributions toward the automated generation of Euler diagrams which implicitly assume some fixed set of well-formedness conditions must hold. In addition, unlike most of these other generation methods, our technique allows any abstract description to be drawn as an Euler diagram. To establish the utility of the approach, a prototype implementation has been developed.

A Decidable Constraint Diagram Reasoning System

Constraint diagrams are a visual notation designed for use by software engineers to formally specify information systems. In this paper we formalize a fragment of the constraint diagram language. A set of reasoning rules are defined and we prove that this set is both sound and complete. Given constraint diagrams D1 and D2 such that D2 is a semantic consequence of D1, to prove completeness we construct a proof of D2 from D1. A decision procedure can be extracted from this proof construction process and it follows that the system is decidable.

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.

Evaluating and generalizing constraint diagrams

The constraint diagram language was designed to be used in conjunction with the unified modelling language (UML), primarily for placing formal constraints on software models. In particular, constraint diagrams play a similar role to the textual object constraint language (OCL) in that they can be used for specifying system invariants and operation contracts in the context of a UML model. Unlike the OCL, however, constraint diagrams can be used independently of the UML. In this paper, we illustrate a range of intuitive and counter-intuitive features of constraint diagrams and highlight some (potential) expressiveness limitations. The counter-intuitive features are related to how the individual pieces of syntax interact. A generalized version of the constraint diagram language that overcomes the illustrated counter-intuitive features and limitations is proposed. In order to discourage specification readers and writers from overlooking certain semantic information, the generalized notation allows this information to be expressed more explicitly than in the non-generalized case. The design of the generalized notation takes into account five language design principles which are discussed in the paper. We provide a formalization of the syntax and semantics for generalized constraint diagrams. Moreover, we establish a lower bound on the expressiveness of the generalized notation and show that they are at least as expressive as constraint diagrams.

Drawing Euler Diagrams with Circles: The Theory of Piercings

Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Euler diagram, it is computationally expensive to generate the diagram. Moreover, the generated diagrams represent sets by polygons, sometimes with quite irregular shapes that make the diagrams less comprehensible. In this paper, we address these two issues by developing the theory of piercings, where we define single piercing curves and double piercing curves. We prove that if a diagram can be built inductively by successively adding piercing curves under certain constraints, then it can be drawn with circles, which are more esthetically pleasing than arbitrary polygons. The theory of piercings is developed at the abstract level. In addition, we present a Java implementation that, given an inductively pierced abstract description, generates an Euler diagram consisting only of circles within polynomial time.

SketchSet: Creating Euler diagrams using pen or mouse

Euler diagrams form the basis of various visual languages but tool support for creating them is generally limited to generic diagram editing software using mouse and keyboard interaction. A more natural and convenient mode of entry is via a sketching interface which facilitates greater cognitive focus on the task of diagram creation. Previous work has developed sketching interfaces for Euler diagrams drawn with ellipses. This paper presents SketchSet, the first sketch tool for Euler diagrams whose curves can be circles, ellipses, or arbitrary shapes. SketchSet allows the creation of formal diagrams via point and click interaction. The user drawn diagram, in sketched or formal format, is automatically converted to a diagram in the other format, thus maintaining both views. We provide a mechanism that allows semantic differences between the sketch and the formal diagram to be rectified automatically. Finally, we present a user study that evaluates the effectiveness of the tool.