Solving complex consistency checking tasks in natural languages is hard and requires sophisticated specialist expertise. The similar task of finding bugs in information systems can be large-scale and is often conducted with some visualisation of the data. Visualisation, therefore, could also be a useful tool when consistency checking in real world applications, such as in the case of ontology engineering. Previous experiments suggest that node-link visualisation, such as SOVA, are more effective than node-link-region visualisation, such as concept diagrams, in consistency checking tasks. In this study, we found that this tendency was not affected even in an alternative setting where multiple concept diagrams were used. Our findings have implications for the way in which information is presented visually: single (merged) visualisations are effective for these types of tasks.

Pólya and others recognised that an appropriate representation of a problem is key for enabling us to solve it. But choosing the right representation is a problem that novice problem solvers find difficult, so must turn to experts for guidance. In this paper, we present a study that examines how human experts recommend representations. We asked high school mathematics teachers to order representational systems based on their suitability generally, and with respect to a student profile. We found the teachers updated their recommendations based on the problem and student profile, but were inconsistent with each other. This inconsistency highlights a need for more training and support in representational system selection.

This paper presents an empirical study to determine whether alterations to graphical features (colour and size) of automatically generated LineSets improve task performance. LineSets are used to visualise sets and networks. The increasingly common nature of such data suggests that having effective visualisations is important. Unlike many approaches to set and network visualisation, which often use concave or convex shapes to represent sets alongside graphs, LineSets use lines overlaid on a graph. LineSets have been shown to be advantageous over shape-based approaches. However, the graphical properties of LineSets have not been fully explored. Our results suggest that automatically drawn LineSets can be significantly improved for certain tasks through the considered use of colour alongside size variations applied to their graphical elements. In particular, we show that perceptually distinguishable colours, lines of varying width, and nodes of varying diameter lead to improved task performance in automatically laid-out LineSets.

Choosing an effective representation is fundamental to the ability of the representation's user to exploit it for the intended purpose. The major contribution of this paper is to provide a novel, flexible framework, rep2rep, that can be used by AI systems to recommend effective representations. What makes an effective representation is determined by whether it expresses the necessary information, supports the execution of tasks, and reflects the user's cognitive abilities. In general, there is no single `most effective' representation for every problem and every user, which makes it difficult to choose one from the plethora of possible representations. To address this, rep2rep includes: a domain-independent language for describing representations, algorithms that compute measures of informational suitability and overall cognitive cost, and uses these measures to recommend representations. We demonstrate the application of rep2rep in the probability domain. Importantly, our framework provides the foundations for personalised interaction with AI systems in the context of representation choice.

Ontologies describe complex world knowledge in that they consist of hierarchical relations, such as is-a, which can be expressed by quantifiers or sets, and various binary relations, which can be expressed by links or networks. Should hierarchical relations be distinguished from other binary relations as essentially different ones in building cognitively accessible systems of ontologies? In this study, two kinds of ontology visualizations, a network-based visualization (SOVA) and a set-based visualization (concept diagrams), are empirically compared in the case of consistency checking. Participants were presented with one diagram and then asked to answer the question of whether the meaning of the diagram was contradictory. Our results showed that SOVA is more effective than concept diagrams, suggesting that to represent hierarchical and binary relations of ontologies in a way based on networks suits human cognition when checking ontologies' consistencies.

LineSets represent information about sets by drawing one line for each set on an existing visualization of data items. This paper addresses the following question: does manipulating the size of visual elements affect the comprehension of LineSets? We empirically evaluated two types of size treatments applied to LineSets drawn on networks: varying set-line thickness, to reflect relative set cardinality, and varying node diameter, to reflect data items' relative degree of connectivity. The evaluation required participants to perform tasks that were thought to be aided by the size variations alongside tasks where no benefit was anticipated. Viewing comprehension through accuracy and time performance, we found that varying set-line thickness and node diameter significantly improves the effectiveness of LineSets. As a consequence, this research leads to the recommendation that LineSets vary sizes of lines and nodes.

Our aim is to provide empirical evidence that diagrammatic logics are more effective than symbolic and textual logics in allowing people to better understand information. Ontologies provide an important focus for such an empirical study: people need to understand the axioms of which ontologies comprise. A between-groups study compared six frequently-used axiom types using the (textual) Manchester OWL Syntax (MOS), (symbolic) description logic (DL) and concept diagrams. Concept diagrams yielded significantly better task performance than DL for all six, and MOS for four, axiom types. MOS outperformed concept diagrams for just one axiom type and DL for only three axiom types. Thus diagrams could ensure ontologies are developed more robustly.

Diagrams can be an effective means of communicating complex ideas and can aid ontology engineering. Indeed, domain experts often do not have the expertise required to understand or create the complex logical statements of an ontology in description logic (DL). This paper presents a visualisation method, concept diagrams, geared toward expressing assertions and class expression axioms alongside providing support for literals, datatypes and data properties. Property diagrams are introduced, targeted at object property and data property expression axioms. We demonstrate that concept diagrams and property diagrams provide a large coverage of OWL 2 axioms and are, thus, closely aligned in expressive power.

Over the last two decades substantial advances have been made in our understanding of diagrammatic logics. Many of these logics have the expressiveness of monadic first-order logic, sometimes with equality, and are equipped with sound and complete inference rules. A particular challenge is the representation of negated statements. This paper addresses the problem of how to represent negated statements involving constants, thus asserting the absence of specific individuals, in the context of Euler-diagram-based logics. Our first contribution is to explore the potential benefits of explicitly representing absence using constants, in terms of clutter reduction, and to highlight ontological issues that arise. We go on to define a measure of clutter arising from constants. By defining a set of semantics-preserving inference rules, we are able to algorithmically minimize diagram clutter, in part made possible by the inclusion of absence. Consequently, information about individuals can be represented in a minimally cluttered way. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper addresses the problem of how best to visualize network data grouped into overlapping sets. We address it by evaluating various existing techniques alongside a new technique. Such data arise in many areas, including social network analysis, gene expression data, and crime analysis. We begin by investigating the strengths and weakness of four existing techniques, namely Bubble Sets, EulerView, KelpFusion, and LineSets, using principles from psychology and known layout guides. Using insights gained, we propose a new technique, SetNet, that may overcome limitations of earlier methods. We conducted a comparative crowdsourced user study to evaluate all five techniques based on tasks that require information from both the network and the sets. We established that EulerView and SetNet, both of which draw the sets first, yield significantly faster user responses than Bubble Sets, KelpFusion and LineSets, all of which draw the network first.

The least cognitively demanding way to create a diagram is to draw it with a pen. Yet there is also a need for more formal visualizations, that is, diagrams created using both traditional keyboard and mouse interaction. Our objective is to allow the creation of diagrams using traditional and stylus-based input. Having two diagram creation interfaces requires that changes to a diagram should be automatically rendered in the other visualization. Because sketches are imprecise, there is always the possibility that conversion between visualizations results in a lack of syntactic consistency between the two visualizations. We propose methods for converting diagrams between forms, checking them for equivalence, and rectifying inconsistencies. As a result of our theoretical contributions, we present an intelligent software system allowing users to create and edit diagrams in sketch or formal mode. Our proof-of-concept tool supports diagrams with connected and spatial syntactic elements. Two user studies show that this approach is viable and participants found the software easy to use. We conclude that supporting such diagram creation is now possible in practice.

This paper presents the first design principles that optimize the visualization of sets using linear diagrams. These principles are justified through empirical studies that evaluate the impact of graphical features on task performance. Linear diagrams represent sets using straight line segments, with line overlaps corresponding to set intersections. This study builds on recent empirical research, which establishes that linear diagrams can be superior to prominent set visualization techniques, namely Euler and Venn diagrams. We address the problem of how to best visualize overlapping sets using linear diagrams. To solve the problem, we investigate which graphical features of linear diagrams significantly impact user task performance. To this end, we conducted seven crowdsourced empirical studies involving a total of 1,760 participants. These studies allowed us to identify the following design principles, which significantly aid task performance: use a minimal number of line segments, use guidelines where overlaps start and end, and draw lines that are thin as opposed to thick bars. We also evaluated the following graphical properties that did not significantly impact task performance: color, orientation, and set order. The results are brought to life through a freely available software implementation that automatically draws linear diagrams with user-controlled graphical choices. An important consequence of our research is that users are now able to create effective visualizations of sets automatically, thus improving human--computer interaction.

This paper addresses the problem of how best to use colour in Euler diagrams. The choice of using coloured curves, rather than black curves, possibly with coloured fill is often made in tools that automatically draw Euler diagrams for information visualization as well as when they are drawn manually. We address the problem by empirically evaluating various different colour treatments: coloured or black curves combined with either no fill or coloured fill. By collecting performance data, we conclude that Euler diagrams with coloured curves and no fill significantly outperform all other colour treatments. Most automated layout algorithms adopt colour fill and are, thus, reducing the effectiveness of the Euler diagrams produced. As Euler diagrams can be used in a multitude of areas, ranging from crime control to social network analysis, our results stand to increase the ability of users to accurately and quickly extract information from their visualizations.

Euler diagrams and graphs are used as visualisations individually in a large variety of application areas such as network analysis, medicine and engineering. Existing methods which combine both Euler diagrams and graphs such as Bubble Sets and Euler View provide somewhat limited results with suboptimal layout. In particular, they do not produce diagrams that are known to be most effective for performing user-driven tasks. That said, our knowledge is rather limited about what constitutes an effective layout for Euler diagrams and graphs in combination. Our ultimate aim is to automatically visualise large networks in an effective manner. To produce effective layouts, we need to identify properties that may correlate with effective layouts of Euler diagrams combined with graphs. Such properties are considered in this paper. In future, empirical studies will be conducted to inform and validate the combined properties.

Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where nonconvex curves are necessary, our method draws an appropriate diagram using nonconvex polygons. Thus, we are now always able to automatically visualize data for up to three sets.

The graphical choices made when laying out Euler diagrams impact upon both aesthetic quality and comprehensiveness. Graphical choices include the shape, size and colour of closed curves which are commonly described as retinal variables to which we are known to be perceptually sensitive. There is copious literature providing guidance as to how best use retinal variables to visualise both quantitative and qualitative information for a wide range range of diagram types. Further, this guidance is explicitly defined to optimise the users' comprehension of such information. However, there exists little, if any, literature affording guidance as to how best to use retinal variables when laying out Euler diagrams. Here we present a novel insight as to where retinal variables manifest in Euler diagrams, how they might influence our perception of Euler diagrams and, as a consequence, provide motivation and guidance for establishing layout guidelines.

Ensuring the privacy of end-user data is of paramount importance. Allowing data analysts access to personally identifiable information can lead to legal problems, especially when the end-user has not given permission for such data to be used. This paper is concerned with the specification and processing of data collected from consumers as they interact with services such as those presented as `smart-phone' applications. Consumers are (should be) asked whether they want to opt-out of various kinds of their data being collected and/or used for marketing and other data analysis purposes. This impacts on the manner in which their data can be used by analysts. The process to determine the transformations that data must undergo in order to be used lawfully involves a wide range of stakeholders: software engineers, lawyers, analysts and marketing personnel, to name some. Each of these stakeholders has different expertise and prior knowledge that must be taken into account when devising methods of communication between them. We argue that visual (diagrammatic) notations developed for this purpose aid effective communication between these diverse groups. To date, there is no general method for visualising the transformations that raw data must undergo. We address this by utilising the concept diagram notation to rigorously depict, abstractly, how the data needs to be processed and the effects upon that data under various privacy related transformations.

Diagrammatic modes of communication have long been recognized for their accessible representations of information. One area in which they have been developed is that of logical reasoning, where symbolic notations are perceived by many as difficult to use. Significant progress has been made on formalizing diagrammatic logics and proving formal properties of their inference rules. To-date, most inference rules for diagrammatic logics have been designed from the perspective of a logician, aiming for the essential and, thus, desirable properties of soundness and completeness. However, this approach overlooks a fundamental goal of providing diagrammatic logics: to overcome barriers posed by symbolic logics to non-mathematicians. Even if the diagrams themselves are accessible, having inference rules that result in unwieldy proofs will fail to fulfil this fundamental goal. Thus, the time is ripe to fully address this goal and show how to design inference rules that give rise to more natural proofs. In this paper we take significant steps towards this ambitious target by devising new inference rules for spider diagrams. We demonstrate that they allow substantially shorter proofs to be written and, we argue, the resulting proofs are more natural.

Constraint diagrams were proposed as a means of modelling software systems, with generalized constraint diagrams being a recent refinement. It is known that generalized constraint diagrams are more expressive than constraint diagrams and can express any first-order logic statement that uses monadic or dyadic predicates. Thus, the generalized constraint diagram logic is undecidable. In this article, we develop a decision procedure for the so-called existential fragment of generalized constraint diagrams, building on previous work for a simpler, less expressive, fragment of the logic.

Codecharts are expressively lightweight whilst sufficiently rich to develop insight into program design. This paper contributes an informal description of the semantics of codecharts using the abstract syntax defined in [4]. Future work is to further formalize the semantics, define an inference system in which to prove soundness and completeness, and to investigate applications. In particular, we believe that some interesting program metrics, ascertaining program complexity or the level of dependencies and coupling present, can be readily defined and visualized using codecharts. For instance, we can make the area of an ellipse proportional the level of coupling, with relatively large ellipses indicating the need to consider refactoring. Such applications of codecharts could lead to improved software design and understanding.

The process of designing and modelling an ontology can be difficult, especially if the user finds the syntax to be relatively inaccessible. Providing users with graphical syntax with which they can model and visualise their ontology has the potential to be helpful. Previously, we informally introduced concept diagrams for ontology visualisation and modelling. We present a case study comprising: (a) a set of axioms for an ontology, and (b) a set of theorems that follow from the axioms, together with their proofs. The proofs have been constructed so that they are, in our opinion, of an intuitive style. From these proofs, we derive a set of sound inference rules that can be used to formally reason about ontologies following the same intuitive style. This approach to designing inference rules differs from previous efforts where the primary focus has been on obtaining a set of sound and complete inference rules, rather than on intuitiveness.

The use of Euler diagrams as a basis for visual languages is commonplace and they are often used for visualizing information. The ability to automatically draw these diagrams is, therefore, likely to be of widespread practical use. The Euler diagram drawing problem is recognized as challenging, but the potential pay-off from the derivation of a comprehensive solution, that produces usable and effective diagrams, is significant. Previous research on automated Euler diagram drawing has used various different approaches, each of which had their own problems, including: (a) failure to draw a diagram in all cases, (b) poor diagram layout, and (c) inability to ensure that certain wellformedness properties of the drawn diagrams hold. In this paper, we present a novel approach to Euler diagram drawing that draws diagrams with circles, ellipses and curves in general. This new approach will draw a diagram in all cases, avoiding bad layout where possible (by the use of `nice' geometric shapes) and can enforce wellformedness properties as chosen by the user.

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.

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.

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.

There has been significant research effort focussed on the study of regular languages, since they play a vital role in our understanding of computation. This existing research draws a large number of connections with other areas, such as algebra and symbolic logic. Recently, research has begun into how diagrammatic logics can define regular languages, providing another mechanism through which we can understand regular languages. However, the formalised diagrammatic logics are first-order, so they cannot define non-starfree regular languages. The primary contributions of this paper are: (a) to develop and formalise a second-order diagrammatic logic, extending spider diagrams of order, and (b) to establish a class of regular languages that this logic can define. This lays the essential foundations for providing an exact classification of the regular languages that are definable using this new second-order logic.

Euler diagrams are frequently used for visualizing information about collections of objects and form an important component of various visual languages. Properties possessed by Euler diagrams correlate with their usability, such as whether the diagram has only simple curves or possesses concurrency. Sometimes, every diagram that represents some given information possesses some undesirable properties, and reducing the number of violations of undesirable properties is beneficial. In this paper we show how to count the number of violations from the reduced Euler graph. We then define various transformations on the Euler graph which can reduce the number of violations of a given property, but sometimes at the expense of increasing the number of violations of another property. These transformations can be used to improve the quality of the drawn diagram, which is important for effective information visualization.

Many data sets are visualized effectively with area proportional Venn diagrams, where the area of the regions is in proportion to a defined specification. In particular, Venn diagrams with three intersecting curves are considered useful for visualizing data in many applications, including bioscience, ecology and medicine. To ease the understanding of such diagrams,using restricted dasianicepsila shapes for the curves is considered beneficial. Many research questions on the use of such diagrams are still open. For instance, a general solution to the question of when given area specifications can be represented by Venn3 using convex curves is still unknown.In this paper we study symmetric Venn3 drawn with convex curves and show that there is a symmetric area specification that cannot be represented with such a diagram. In addition, by using symmetric diagrams drawn with polygons, we show that, if area specifications are restricted so that the double intersection areas are no greater than the triple intersection area then the specification can be drawn with convex curves. We also propose a construction that allows the representation of some area specifications when the double intersection areas are greater than the triple intersection area.Finally, we present some open questions on the topic.

Euler diagrams form the basis of several visual modelling notations, including statecharts and constraint diagrams. Recently, various techniques for automated Euler diagram drawing have been proposed, contributing to the Euler diagram generation problem: given an abstract description, draw an Euler diagram with that description and which possesses certain properties. A common generation method is to find a dual graph from which an Euler diagram is subsequently created. In this paper we define transformations of the dual graph that allow us to alter the properties that the generated diagram possesses. In addition, because the dual graph of a previously generated diagram can be found, our transformations can be used to take such a diagram and produce a new diagram with the same abstract description, but with different properties. As a result, we can produce a variety of different diagrams for any given abstract description, allowing us to choose an Euler diagram that conforms to the properties that a user prefers.

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.

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 in that they can be used for specifying system invariants and operation contracts in the context of a UML model. Constraint diagrams can also be used independently of the UML. In this paper, we illustrate a range of counter-intuitive features of constraint diagrams and highlight some (potential) expressiveness limitations. We propose a generalized version of the constraint diagram language that overcomes the illustrated counter-intuitive features and limitations.

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.

Spider diagrams are a visual language for expressing logical statements. Spiders represent the existence of elements and contours denote sets. Several sound and complete spider diagram systems have been developed and it is known that the spider diagram language is equivalent in expressive power to monadic first order logic with equality. However, these sound and complete spider diagram systems do not contain syntactic elements analogous to constants in first order predicate logic. We extend the spider diagram language to include constant spiders which represent specific individuals and give formal semantics for the extended diagram language. We then prove that this extended system is equivalent in expressive power to the language of spider diagrams without constants.

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.