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.

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.

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.

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.

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.

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.

Diagrams of one sort or another have always been used as aids to abstract reasoning. Although many are informal mnemonics, reminding their authors about structures and relationships they have observed or deduced, considerable research effort has been expended on formalising graphical notations so that they may play a more central role in the application of logic to problems.

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.

With the adoption of UML by the OMG and industry as the linguae-francae of visual systems modelling, one begins to ponder what will come next in this field? This tutorial brings a vision for visual modelling beyond UML. We present and consolidate radical new notations, proposed in a series of research papers and with quickly increasing adoption by industry, for the specification of complex systems in an intuitive visual, yet precise manner. The recurring theme of these notations is the upgrading of familiar diagrams into a powerful visual language. Spider diagrams considerably extend Venn-diagrams to the specification of OO-systems. Most familiar OO-concepts are translated to set theoretical terms: class into set of objects, inheritance corresponding to subset, and even Harel's statecharts interpreted as the set of objects in that state. Constraint diagrams enhance the arrow notation to describe static system invariants which cannot be described by UML class-object diagram. Reasoning rules are developed for the notation and strong completeness results are given. Finally, 3D-diagrams show how the third dimension and VRML modelling can be used for a conceptual modelling of dynamic system behaviour. Much of the tutorial will be based on a case study developed in industry, illustrating how the new notations are combined with those of UML, including OCL.Highlights include:• A crash critical overview in UML, stressing its weaknesses and strengths,• A rich visual constraint language and an insight into subtle issues that arise when defining a visual language, for applying the popular design-by-contract using a visual formalism• A discussion of diagrammatic reasoning with the notation, including completeness results• A case study• A demonstration of a graphical editor for the constraint-diagrams language• A look to the future of visual modelling, including ideas about 3D modelling notations and visual modelling tools.

While it is crucial to understand the formal structure of the semantic domain of an information system, in this paper we raise an ontological issue about the syntactic aspect of a representation system through a case study on a diagrammatic system. 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 diagram(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, the domains of type-syntax and token-syntax are ontologically independent, that is, one is abstract and the other concrete. We discuss the roles of type-syntax 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.

The tutorial is example driven and illustrates how the new notations are combined with those of UML, including OCL. Some of the examples are drawn from industrial contexts, in particular the telecomms sector. Highlights include:

The Object Constraint Language (OCL), which forms part of the UML 1.1. set of modelling notations is a precise, textual language for expressing constraints that cannot be shown in the standard diagrammatic notation used in UML. A semantics for OCL lays the foundation for building CASE tools that support integrity checking of whole UML models, not just the component expressed using OCL. This paper provides a semantics for OCL, at the same time providing a semantics for classes, associations, attributes and states.