Diagrammatic Representation and Inference

We present an algorithm for automatically laying out metro map style schematics using a force-directed approach, where we use a localized version of the standard spring embedder forces combined with an octilinear magnetic force. The two types of forces used during layout are naturally conflicting, and the existing method of simply combining these to generate a resultant force does not give satisfactory results. Hence we vary the forces, emphasizing the standard forces in the beginning to produce a well distributed graph, with the octilinear forces becoming prevalent at the end of the layout, to ensure that the key requirement of line angles at intervals of 45° is obtained. Our method is considerably faster than the more commonly used search-based approaches, and we believe the results are superior to the previous force-directed approach. We have further developed this technique to address the issues of dynamic schematic layout. We use a Delaunay triangulation to construct a schematic “frame”, which is used to retain relative node positions and permits full control of the level of mental map preservation. This technique is the first to combine mental map preservation techniques with the additional layout criteria of schematic diagrams. To conclude, we present the results of a study to investigate the relationship between the level of mental map preservation and the user response time and accuracy.

Orthogonally drawn hypergraphs have important applications, e. g. in actor-oriented data flow diagrams for modeling complex software systems. Graph drawing algorithms based on the approach by Sugiyama et al. place nodes into consecutive layers and try to minimize the number of edge crossings by finding suitable orderings of the nodes in each layer. With orthogonal hyperedges, however, the exact number of crossings is not determined until the edges are actually routed in a later phase of the algorithm, which makes it hard to evaluate the quality of a given node ordering beforehand. In this paper, we present and evaluate two cross counting algorithms that predict the number of crossings between orthogonally routed hyperedges much more accurately than the traditional straight-line method.

In this paper we introduce Tennis Plots as a novel diagram type to better understand the differently long time periods in tennis matches on different match structure granularities. We visually encode the dynamic tennis match by using a hierarchical concept similar to layered icicle representations used for visualizing information hierarchies. The time axis is represented vertically as multiple aligned scales to indicate the durations of games and points and to support comparison tasks. Color coding is used to indicate additional attributes attached to the data. The usefulness of Tennis Plots is illustrated in a case study investigating the tennis match of the women’s Wimbledon final 1988 between Steffi Graf and Martina Navratilova lasting 1 hour, 19 minutes, and 31 seconds and being played over three sets (5:7, 6:2, 6:1). Interaction techniques are described in the case study in order to explore the data for insights.

While Euler Diagrams (EDs) represent sets and their relationships, Coloured Euler Diagrams (CEDs [1]) additionally group sets into families, and sequences of CEDs enable the presentation of their dynamic evolution. Spider Diagrams (SDs) extend EDs, permitting the additional expression of elements, relationships between elements, and set membership, whilst Modelling Spider Diagrams (MSDs [2]) are used to specify the admissible states and evolutions of instances of types, enabling the verification of the conformance of configurations of instances with specifications. Transformations of MSDs generate evolutions of configurations in conformity with the specification of admissible sequences. We integrate CEDs and MSDs, proposing Coloured Modelling Spider Diagrams (CMSDs), in which underlying curves represent properties of a family of sets, whether this be state-based information or generic attributes of the domain elements and colours distinguish different families of curves. Examples of CMSDs from a visual case study of a car parking model are presented.

Regular languages can be represented by finite automata and by railroad diagrams. These two visual forms can be converted to each other. Context-free languages can also be described by (finite sets of) railroad diagrams. Based on the analogy we develop a new type of automata, the fractal automata: they accept the context-free languages. Relations between pushdown automata and fractal automata are also established.

HyperVenn is a heterogeneous logic resulting from combining the individual homogeneous logics for Euler/Venn diagrams and blocks world diagrams. We provide an example proof from the system along with a brief discussion of some of its inference rules.

We present a new ontology visualization tool, that uses Euler diagrams to represent concepts. The use of Euler diagrams as a visualisation tool allows the visual syntax to be well matched to its meaning.

The Proofscape argument mapping system for mathematical proofs is introduced. Proofscape supports argument mapping for informal proofs of the kind used by working mathematicians, and its purpose is to aid in the comprehension of existing proofs in the mathematical literature. It supports the provision of further clarification for large inference steps, which is available on demand when a proof is explored interactively through the Proofscape browser, and theory-wide exploration is possible by expanding and collapsing cited lemmas and theorems interactively. We examine how an argument map makes the structure of a proof immediately clear, and facilitates switching attention between the detailed level and the big picture. Proofscape is at http://proofscape.org.

This paper presents the results from a tertiary entrance test that was delivered to two groups of candidates, one as a paper based test and the other as a computer based test. Item level differential reveals a pattern that appears related to item type: questions based on diagrammatic stimulus show a pattern of increased difficulty when delivered on computer. Differential in performance was not present in other sections of the test and it would appear unlikely to be explained by demographic differences between the groups. It is suggested this differential is due to the inability of the candidates to freely annotate on the stimulus when delivered on computer screen. More work needs to be done on considering the role of annotation as a problem solving strategy in high-stakes testing, in particular with certain kinds of stimulus, such as diagrams.

This study examined the amounts of information that students represented in diagrams compared to text when taking notes (self-directed communication) and when constructing explanations for others (others-directed communication). The participants were 98 Japanese university students who read one of two passages (differing in imageability) in Japanese (L1) and in English (L2). While reading, they could take notes, and were subsequently requested to produce an explanation of the passage using L1 or L2. The students represented more information in diagrams in notes they took from the passage of higher imageability in L1. However, in their explanation of that same passage for others – still using L1 – they represented more information in text. This finding suggests perceptual differences about the functions of diagrams in self- and others-directed communication. Results also confirmed that passage imageability and students’ language proficiency affect cognitive processing cost, which in turn influences the extent to which diagrams are used.

Although diagrams have been shown to be effective tools for promoting understanding and successful problem solving, students’ poor diagram use has been identified as a serious issue in educational practice-related reports. To enhance students’ diagram construction skills and to address problems in diagram use, creating learning situations that make it inevitable for students to use diagrams would likely be helpful. To realize this, communicative learning situations can be considered a viable option, as students would feel a greater necessity to use diagrams as a consequence of feedback they receive while explaining. Thus, this study examined the hypothesis that an interactive peer instructional learning situation would better promote students’ spontaneous diagram use compared to a non-interactive situation. Eighty-eight university students were randomly assigned to one of two conditions: interactive and non-interactive. After reading a passage relating to the science and engineering area, participants in the interactive condition were requested to explain the content of the passage to another participant next to them. In contrast, participants in the non-interactive condition were asked to record an explanation using an IC recorder by imagining that they were explaining to another person. A sheet of paper was provided to participants during the explanation, and diagram use on the paper was analyzed. The results revealed that students’ diagram use in the interactive condition was higher than in the non-interactive condition. This indicates that teachers’ provision of interactive communication situations can effectively promote students’ likelihood of using diagrams spontaneously.

Euler diagrams, used to visualize data and as a basis for visual languages, are an effective representation of information, but they can become cluttered. Previous research established a measure of Euler diagram clutter, empirically shown to correspond with how people perceive clutter. However, the impact of clutter on user understanding is unknown. An empirical study was executed with three levels of diagram clutter. We found a significant effect: increased diagram clutter leads to significantly worse performance, measured by time and error rates. In addition, we found a significant effect of zone (a region in the diagram) clutter on time and error rates. Surprisingly, the zones with a middle level of clutter had the highest error rate compared to the zones with lowest and the highest clutter. Also, the questions whose answers were placed in zones with medium clutter had the highest mean time taken to answer questions. In conclusion, both diagram clutter and zone clutter impact the interpretation of Euler diagrams. Consequently, future work will establish whether a single, but cluttered, Euler diagram is a more effective representation of information than multiple, but less cluttered, Euler diagrams.

Euler diagrams are often used for visualizing data collected into sets. However, there is a significant lack of guidance regarding graphical choices for Euler diagram layout. To address this deficiency, this paper asks the question ‘does the shape of a closed curve affect a user’s comprehension of an Euler diagram?’ By empirical study, we establish that curve shape does indeed impact on understandability. Our analysis of performance data indicates that circles perform best, followed by squares, with ellipses and rectangles jointly performing worst. We conclude that, where possible, circles should be used to draw effective Euler diagrams. Further, the ability to discriminate curves from zones and the symmetry of the curve shapes is argued to be important. We utilize perceptual theory to explain these results. As a consequence of this research, improved diagram layout decisions can be made for Euler diagrams whether they are manually or automatically drawn.

Spatial abilities involved in reasoning with diagrams have been assessed using tests supposed to require mental rotation (cube figures of the Vandenberg & Kruse). However, Hegarty (2010) described alternative strategies: Mental rotation is not always used; analytical strategies can be used instead. In this study, we compared three groups of participants in three external formats of presentation of the referent figure in the Vandenberg & Kruse test: static, animated, interactive. During the test, participants were eye tracked. After the test, they were interrogated on their strategies for each item during the viewing of the replay of their own eye movement in a cued retrospective verbal protocol session. Results showed participants used varied strategies, part of them similar to those shown by Hegarty. Presentation format influenced significantly the strategy. Participants with high performance at the test used more mental rotation. Participants with lower performance tended to use more analytical strategies than mental rotation.

There are a range of diagram types that can be used to visualize sets. However, there is a significant lack of insight into which is the most effective visualization. To address this knowledge gap, this paper empirically evaluates four diagram types: Venn diagrams, Euler diagrams with shading, Euler diagrams without shading, and the less well-known linear diagrams. By collecting performance data (time to complete tasks and error rate), through crowdsourcing, we establish that linear diagrams outperform the other three diagram types in terms of both task completion time and number of errors. Venn diagrams perform worst from both perspectives. Thus, we provide evidence that linear diagrams are the most effective of these four diagram types for representing sets.

Colour is one of the primary aesthetic elements of a visualization. It is often used successfully to encode information such as the importance of a particular part of the diagram or the relationship between two parts. Even so, there are few investigations into the human reading of colour coding on diagrams from the scientific community. In this paper we report on an experiment with graph diagrams comparing a black and white composition with two other colour treatments. We drew our subjects from engineering, art, visual design, physical education, tourism, psychology and social science disciplines. We found that colouring the nodes of interest reduced the time taken to find the shortest path between the two nodes for all subjects. Engineers, tourism and social scientists proved significantly faster with artist/designers just below the overall average speed. From this study, we contribute that adding particular colour treatments to diagrams increases legibility. In addition, preliminary work investigating colour treatments and schemes indicates potential for future gains.

The neural mechanisms underlying global reading on tabular representations were investigated using the task-switching paradigm in event-related functional magnetic resonance imaging. Participants were required to make an appropriate response based on the latest task cue using stimuli tabulated in five rows and five columns with labels. The task was either local or global, and critical events included both cue and target events, which enabled separate analyses of the preparation and execution stages of each type of reading process. Neuroimaging results revealed differential activations between local and global tasks in both preparation and execution stages. For the preparation stage, global cues led to larger activation in the extrastriate cortex, which has been shown as the neural basis of selective attention in the literature. For the execution stage, the left middle temporal gyrus and inferior parietal lobule were more activated in the local task. These areas comprise an object-based attentional selection network, which serves to attend to a particular element in the table that changed with each event. For the global task, the left inferior frontal junction showed high activation, suggesting that the task demanded more cognitive control. The implications of these findings are discussed with respect to the characteristics of global reading.

The aim of this paper is to study the relationship between two important families of diagrams that are used in logic, viz. Aristotelian diagrams (such as the well-known ‘square of oppositions’) and Hasse diagrams. We discuss some obvious similarities and dissimilarities between both types of diagrams, and argue that they are in line with general cognitive principles of diagram design. Next, we show that a much deeper connection can be established for Aristotelian/Hasse diagrams that are closed under the Boolean operators. We consider the Boolean algebra \(\mathbb{B}_n\) with 2ⁿ elements, whose Hasse diagram can be drawn as an n-dimensional hypercube. Both the Aristotelian and the Hasse diagram for \(\mathbb{B}_n\) can be seen as (n − 1)-dimensional vertex-first projections of this hypercube; whether the diagram is Aristotelian or Hasse depends on the projection axis. We show how this account provides a unified explanation of the (dis)similarities between both types of diagrams, and illustrate it with some well-known Aristotelian/Hasse diagrams for \(\mathbb{B}_3\) and \(\mathbb{B}_4\).

A new graphical representation/model of Boolean logic is presented. We use horizontal and vertical lines to represent the logical constants, serial and parallel circuits for conjunction and disjunction. For negation of a (sub)formula, we use a turn of the circuit by \(\frac{\pi}{2}\). A web-based system that is built on the model is also presented.

As an application of their channel theory, Barwise & Seligman sketched a set-theoretic model of representation systems. Their model has the attraction of capturing many important logical properties of diagrams, but few attempts have been made to apply it to actual diagrammatic systems. We attribute this to a lack of precision in their explanation of what their model is about—what a “representation system” is. In this paper, we propose a concept of representation system on the basis of Barwise & Seligman’s original ideas, supplemented by Millikan’s theory of reproduction. On this conception, a representation system is a family of individual representational acts formed through a repetitive reproduction process that preserves a set of syntactic and semantic constraints. We will show that this concept lets us identify a piece of reality that the Barwise-Seligman model is concerned with, making the model ready for use in the logical analysis of real-world representation systems.

This paper concerns the Aristotelian relations of contradiction, contrariety, subcontrariety and subalternation between 14 contingent formulae, which can get a 2D or 3D visual representation by means of Aristotelian diagrams. The overall 3D diagram representing these Aristotelian relations is the rhombic dodecahedron (RDH), a polyhedron consisting of 14 vertices and 12 rhombic faces (Section 2). The ultimate aim is to study the various complementarities between Aristotelian diagrams inside the RDH. The crucial notions are therefore those of subdiagram and of nesting or embedding smaller diagrams into bigger ones. Three types of Aristotelian squares are characterised in terms of which types of contradictory diagonals they contain (Section 3). Secondly, any Aristotelian hexagon contains 3 squares (Section 4), and any Aristotelian octagon contains 4 hexagons (Section 5), so that different types of bigger diagrams can be distinguished in terms of which types of subdiagrams they contain. In a final part, the logical complementarities between 6 and 8 formulae are related to the geometrical complementarities between the 3D embeddings of hexagons and octagons inside the RDH (Section 6).

In graphical or diagrammatic representations, not only the basic component of a diagram, but also a collection of multiple components can form a unit with semantic significance. We call such a collection a “global object”, and we consider how this can assist in reasoning using diagrammatic representation. In this paper, we investigate reasoning with correspondence tables as a case study. Correspondence tables are a basic, yet widely applied graphical/diagrammatical representation system. Although there may be various types of global objects in a table, here we concentrate on global objects consisting of rows or columns taken as a whole. We investigate reasoning with tables by exploiting not only local conditions, specifying the values in individual table entries, but also global conditions, which specify constraints on rows and columns in the table. This type of reasoning with tables would typically be employed in a task solving simple scheduling problems, such as assigning workers to work on different days of the week, given global conditions such as the number of people to be assigned to each day, as well as local conditions such as the days of the week on which certain people cannot work. We investigate logical properties of reasoning with tables, and conclude, from the perspective of free ride, that the application of global objects makes such reasoning more efficient.

Heterogeneous reasoning refers to theorem proving with mixed diagrammatic and sentential languages and inference steps. We introduce a heterogeneous logic that enables a simple and flexible way to extend logics of existing general-purpose theorem provers with representations from entirely different and possibly not formalised domains. We use our heterogeneous logic in a framework that enables integrating different reasoning tools into new heterogeneous reasoning systems. Our implementation of this framework is MixR – we demonstrate its flexibility and extensibility with a few examples.

We present syntax and semantics of a diagrammatic language based on Venn diagrams in which a diagram is not read as a statement about sets, but as a set itself. We prove that our set of rules is sound and complete with respect to the intended semantics. Our system has two slight advantages in relation to the systems we usually encounter in the literature. First, the drawing of diagrams for terms is made inside the system, i.e., by a completely mechanical process based just on the rules of the system. Second, as a consequence, the validity of an inclusion is also verified inside the system and does not depend on any other means than those afforded by our set of rules. These characteristics are absent in the majority of the Venn diagrammatic systems.

By using Euler diagrams, early Peircean abduction is explained as an inference based on the shrinkage of a class of properties; this renders it dual to inductive inference, which is based on the enlargement of a class of subjects. In fact, at a very general level these inferences can be interpreted as (category-theoretic) dual constructions, by representing them as commutative diagrams.