Theory and Application of Diagrams

This presentation will address the representation of geospatial information in the context of group work. The focus is on visual representations that mediate between human collaborators who are participating in a joint reasoning process, within a place and/or space-based problem context. The perspective developed for addressing the challenges involved builds upon the cognitive-semiotic approach outlined in How Maps Work, extending it to consider the issues that underlie creation of maps and related diagrams that work in a group work context. This context requires representations that depict not only geospatial information but also individual perspectives on that information, the process of negotiation among those perspectives, and the behaviors (work) of individuals participating in that negotiation.

Jon Barwise was unique amongst logicians in leading engagement between logic and other disciplines, notably linguistics, computer science, and the several disciplines concerned with diagrams. My main contact with Jon was through working on cognitive analyses of the learning processes of students being taught logic using Hyperproof, the heterogeneous environment he and John Etchemendy designed. This talk will trace some of Jon’s enthusiasms for interdisciplinary interactions in this area. I hope the audience will contribute as much or more than the speaker.

Two of the most fundamental questions in visual language research are how to specify a visual language and how to recognize and understand diagrams in a particular visual language. In this tutorial we survey the many formalisms which have been suggested over the last three decades for visual language specification, discuss computational approaches to diagram understanding based on these formalisms and indicate possible applications. We shall also review recent directions in visual language theory, notably efforts to develop an analogue of the Chomsky hierarchy for visual languages, the specification of diagrammatic reasoning, and cognitive models of visual language understanding.

This session looks at some uses of diagrams in scientific discovery, particularly their role as intermediate representations which mediate between phenomena, descriptions which can be communicated and descriptions which are general. A range of examples will illustrate a variety of uses, including: the abstractive, generative role; diagrams as encoded knowledge; reasoning with diagrammatic representations in discovery; and communication (exposition and argumentation). The tutorial will encourage consideration of two issues: (a) whether, from a cognitive standpoint, diagrams are essential to reasoning about natural phenomena and processes, and (b) the relationship of diagrammatic reasoning to other types of visualisation and visual thinking in the sciences, including cognitive and computational modeling of discovery.

This tutorial covers, in brief, the road from observing behavior to the implementation of the observed behavior in a mixed rule-based and parallel network computer model. The emphasis will be on production system, or rule-based modelling. Rule-based modeling uses independently firing if-then rules to capture behavior. Why do this type of modeling, what types of data do you need, what are some advantages and limitations of the method? Simple examples of rule-based modeling will be given; these will be extended to a brief explanation of the mixed model. This model gives a theoretical explanation of the behavior of an expert using visual reasoning based on a Cartesian graph combined with verbal reasoning to teach Economics principles to students.

Many cognitive tasks, whether in everyday cognition, scientific practice, or professional life, are distributed cognitive tasks-tasks that require integrative, interactive, and dynamical processing of information retrieved from internal representations and that perceived from external representations through the interplay between perception and cognition. The representational effect is the ubiquitous phenomenon that different representations of a common structure can generate dramatically different representational efficiencies, task complexities, and behavioral outcomes. A framework of distributed representations is proposed to account for the representational effect in distributed cognitive tasks. This framework considers internal and external representations as two indispensable components of a single system and suggests that the relative distribution of information across internal and external representations is the major factor of the representational effect in distributed cognitive tasks. A representational determinism is also proposed-the form of a representation determines what information can be perceived, what processes can be activated, and what structures can be learned and discovered from the specific representation. Applications of the framework of distributed representations will be described for three domains: problem solving, relational information displays, and numberation systems.

Venn diagrams and Euler circles have long been used as a means of expressing relationships among sets using visual metaphors such as “disjointness” and “containment” of topological contours. Although the notation is effective in delivering a clear visual modeling of set theoretical relationships, it does not scale well. In this work we study “projection contours”, a new means for presenting sets intersections, which is designed to reduce the clutter in such diagrams. Informally, a projected contour is a contour which describes a set of elements limited to a certain context. The challenge in introducing this notation is in producing precise and consistent semantics for the general case, including a diagram comprising several, possibly interacting, projections, which might even be of the same base set. The semantics investigated here assigns a “positive” meaning to a projection, i.e., based on the list of contours with which it interacts, where contours disjoint to it do not change its semantics. This semantics is produced by a novel Gaussian-like elimination process for solving set equations. In dealing with multiple projections of the same base set, we introduce yet another extension to Venn-Euler diagrams in which the same set can be described by multiple contours.

Spider diagram systems provide a visual language that extends the popular and intuitive Venn diagrams and Euler circles. Designed to complement object-oriented modelling notations in the specification of large software systems they can be used to reason diagrammatically about sets, their cardinalities and their relationships with other sets. A set of reasoning rules for a spider diagram system is shown to be sound and complete. We discuss the extension of this result to diagrammatically richer notations and also consider their expressiveness. Finally, we show that for a rich enough system we can diagrammatically express the negation of any diagram.

A key component of computational diagrammatic reasoning is the automated interpretation of diagram notations. One common and successful approach to this is based on attributed multiset grammars. The disadvantages of grammars are, however, that they do not allow ready integration of semantic information and that the underlying theory is not strongly developed. Therefore, embeddings of grammars into first-order logic have been investigated. Unfortunately, these are unsatisfactory: Either they are complex and unnatural or else, because of the monotonicity of classical first-order logic, cannot handle diagrammatic reasoning. We investigate the use of two non-standard logics, namely linear logic and situation theory, for the formalization of diagram interpretation and reasoning. The chief advantage of linear logic is that it is a resource-oriented logic, which renders the embedding of grammars straightforward. Situation theory, on the other hand, has been designed for capturing the semantics of natural language and offers powerful methods for modelling more complex aspects of language, such as incomplete views of the world. The paper illustrates embeddings of grammar-based interpretation into both formalisms and also discusses their integration.

By devising a new reading method for Peirce’s Existential Graphs (EG), this paper moves away from the traditional method of evaluating diagrammatic systems against the criteria appropriate to symbolic systems. As is well-known, symbolic systems have long been preferred to diagrammatic systems and the distinction between the two types of systems has not been well defined. This state of affairs has resulted in a vicious circle: because the unique strengths of visual systems have not been discovered, diagrammatic systems have been criticized for lacking the properties of a symbolic system, which, in turn, reinforces the existing prejudice against non-symbolic systems. Peirce’s EG is a classic example of this vicious circle.Logicians commonly complain that EG is too complicated to put to actual use. This paper locates a main source of this criticism in the traditional reading methods of EG, none of which fully exploits the visual features of the system. By taking full advantage of the iconicity of EG, I present a much more transparent and useful reading of the Beta graphs. I pursue this project by (i) uncovering important visual features of EG, and (ii) implementing Peirce’s original intuitions for the system.

This paper examines diagrams which exploit qualitative spatial relations (QSRs) for representation. Our point of departure is the theory that such diagram systems are most effective when their formal properties match those of the domains that they represent (e.g. [1,2,3]). We argue that this is true in certain cases (e.g. when a user is constructing diagrammatic representations of a certain kind) but that formal properties cannot be studied in isolation from an account of the cognitive capacities of diagram users to detect and categorize diagram objects and relations.We discuss a cognitively salient repertoire of elements in qualitative visual languages, which is different from the set of primitives in mathematical topology, and explore how this repertoire affects the expressivity of the languages in terms of their vocabulary and the possible spatial relations between diagram elements.We then give a detailed analysis of the formal properties of relations between the diagram elements. It is shown that the analysis can be exploited systematically for the purposes of designing a diagram system and analysing expressivity. We demonstrate this methodology with reference to several domains, e.g. diagrams for file systems and set theory (see e.g. [4]).

Interlacing knotwork forms a significant part of celtic art. From the perspective of computer science, it is a visual language following mathematically precise rules of construction. In this paper, we study the syntactic generation of celtic knots using collage grammars. Several syntactic regulation mechanisms are employed in order to ensure that only consistent designs are generated.

Philosophers and practitioners commonly distinguish between descriptions, depictions and diagrams as visual representations. But how are we to understand the differences between these representational types? Many suggestions have been made, all of them unsatisfactory. A common assumption has been that these representational types must be evaluated in terms of the presence or absence of a single property. I argue that this assumption is both questionable and overly restrictive, and advance a two-property analysis in terms of what I call Assimilability and Discretion. I argue that this analysis allows us give a general differentiation of the various types and to understand better what factors could affect changes in classification. This suggests an outline framework for empirical research. Philosophically, it can also be used to capture a core idea of perspicuousness, and to ground an argument for the general perspicuousness of diagrams as a representational type.

The question is posed: in which respects and to what extent are logical systems which employ diagrammatic representations “formal”? I propose to characterize “formal” rules to be those which are reducible to simple constructive operations on the representations themselves. Formal systems, then, are those which employ such formal rules. It is argued that “formality” thus characterized underlies a particular strategy for meeting a certain epistemological challenge. Some diagrammatic and heterogeneous logical systems are tested for formality, and it is suggested that any robust heterogeneous system is unlikely to be formal. The analysis of this paper, then, provides a principled account of how some diagrammatic systems differ significantly from linguistic ones.

A number of criteria for discriminating diagrammatic from sentential systems of representation by their manner of semantic interpretation have been proposed. Often some sort of spatial homomorphism between diagram and its referent is said to distinguish diagrammatic from sentential systems (e.g. Barwise & Etchemendy 1990). Or the distinction is analysed in terms of Peirce’s distinctions between symbol, icon and index (see Shin (forthcoming)). Shimojima (1999) has proposed that the sharing of ‘nomic’ constraints between representing and represented relations is what distinguishes diagrams. We have proposed that the fundamental distinction is between direct and indirect systems of representation, where indirect systems have an abstract syntax interposed between representation and represented entities (Stenning & Inder 1994; Gurr, Lee & Stenning 1999; Stenning & Lemon (in press).The purpose of the present paper is to relate the distinction between directness and indirectness to the other criteria, and to further develop the approach through a comparison Peirce’s Existential Graphs both with sentential logics and with diagrammatics ones. Peirce’s system is a particularly interesting case because its semantics can be viewed as either direct or indirect according to the level of interpretation. The paper concludes with some remarks on the consequences of sentential vs. diagrammatic modalities for the conduct of proof.

Graph comprehension is constrained by the goals of the cognitive system that processes the graph and by the context in which the graph appears. In this paper we report the results of a study using a sentence-graph verification paradigm. We recorded participants’ reaction times to indicate whether the information contained in a simple bar graph matched a written description of the graph. Aside from the consistency of visual and verbal information, we manipulated whether the graph was ascending or descending, the relational term in the verbal description, and the labels of the bars of the graph. Our results showed that the biggest source of variance in people’s reaction times is whether the order in which the referents appear in the graph is the same as the order in which they appear in the sentence. The implications of this finding for contemporary theories of graph comprehension are discussed.

Eye-tracking equipment has proven useful in examining the cognitive processes people use when understanding and reasoning with diagrams. However, eye-tracking has several drawbacks: accurate eye-tracking equipment is expensive, often awkward for participants, requires frequent re-calibration and the data can be difficult to interpret. We introduce an alternative tool for diagram research: the Restricted Focus Viewer (RFV). This is a computer program which takes an image, blurs it and displays it on a computer monitor, allowing the participant to see only a small region of the image in focus at any time. The region in focus can be moved using the computer mouse. The RFV records what the participant is focusing on at any point in time. It is cheap, non-intrusive, does not require calibration and provides accurate data about which region is being focused upon. We describe this tool, and also provide an experimental comparison with eye-tracking. We show that the RFV gives similar results to those obtained by Hegarty (1992) when using eye-tracking equipment to investigate reasoning about mechanical diagrams.

Static mixed-mode presentations consisting of verbal explanations illustrated with diagrams have long been used to communicate information. With the advent of multimedia, such presentations have become dynamic, by migrating from paper to the computer and by adding interactivity and animation. The conventional wisdom is that computer-based multimedia presentations are better than printed presentations. However, does the communicative power of mixed-mode representations stem from their careful design to match cognitive processes involved in comprehension or from their interactive and animated nature? This is an important issue that has never been investigated. This paper first presents a cognitive model of comprehension of mixed-mode representations. We describe how this model generates design guidelines for mixed-mode representations that present expository material in two domains - the concrete domain of mechanical systems and the abstract domain of computer algorithms. We then report on a series of studies that compared computer-based interactive multimedia presentations and their paper-based counterparts. Both were designed in accordance with the comprehension model and were compared against each other and against competing representational forms such as books, CD-ROMs, and animations. These studies indicate that the effectiveness of mixed-mode presentations has more to do with their match with comprehension processes than the medium of presentation. In other words, benefits of interactivity and animation are likely being overstated in the current milieu of fascination with multimedia.

This paper examines capacity limits in mental animation of static diagrams of mechanical systems and interprets these limits within current theories of working memory. I review empirical studies of mental animation that examined (1) the relation of spatial ability to mental animation (2) the effects of working memory loads on mental animation, (3) use of external memory in mental animation and (4) strategies for task decomposition that enable complex mental animation problems to be accomplished within the limited capacity of working memory. The effects of capacity limits on mental animation are explored by implementing a simple production system model of mental animation in the 3CAPS production system architecture, limiting the working memory resources available to the model, and implementing strategies for managing scarce working memory resources. It is proposed that mental animation involves the visual-spatial and executive components of working memory and that individual differences in mental animation reflect the operation of these working memory components.

Management of time and commitments is a central problem for high-discretion employees in the information society. A variety of conventions have evolved for the representation of time in calendars, diaries, and project management packages. Yet current time management products remain very close to paper-based conventions with respect to their support for visualisation of scheduling problems; indeed their displays may be even more restrictive than the paper diary. We report an exploratory study aiming at “thinking outside the box” of current computerised diaries by an empirical investigation in which a heterogeneous sample of white-collar workers generated diagrammatical representations of their time and commitments. Design issues are raised for diagrammatic representations that can empower the user in such an environment.

In producing diagrams for a variety of contexts, people use a small set of schematic figures to convey certain context specific concepts, where the forms themselves suggest meanings. These same schematic figures are interpreted appropriately in context. Three examples will support these conclusions: lines, crosses, and blobs in sketch maps; bars and lines in graphs; and arrows in diagrams of complex systems.

With increased use of multimedia and computers in education, the use of animation to illustrate dynamics is becoming more commonplace. Previous research suggests that diagrams may reduce cognitive processing as all information is perceptually available, making it more explicit and therefore requiring less inferencing (e.g. Simon and Larkin 1987). Animation, therefore, may be expected to enhance learning, especially when illustrating dynamic processes, as motion is depicted more visually explicitly, thus reducing cognitive processing. However, although animation may increase explicit perceptually available information, it may not automatically improve understanding. Visual explicitness itself does not necessarily guarantee accurate perception of specific information, nor does perception of information guarantee comprehension. Initial studies suggest that certain characteristics of diagrammatic animation have significant effects on cognitive interaction with material and therefore on comprehension. Current computer technology not only enables improved graphical animated illustration, but also provides the facility to physically interact with information on the screen. This in itself may influence the kind of learning that takes place. This paper presents research investigating how different ways of both representing and interacting with animated diagrams influence the kinds of cognitive interactions that may take place.

The use of graphical media in synchronous communication has received relatively little attention. This paper reports the results of an experimental study of graphical communication that systematically compares interaction in a task-oriented dialogue with and without a shared virtual whiteboard. Observations of the interaction show that a wide variety of communicative functions can potentially be served by graphical interaction. Analyses of both overall performance and communicative process demonstrate that graphical communication can provide a clear transactional advantage in communication. The results also show that participants develop their use of graphics, producing progressively more abstract graphical representations as their experience increases.

In architectural design, diagramming has an equally important role in functional studies and in aesthetic studies. Diagrams are used to create and explore alternative schemes at the very early stages. They are also used to explain concepts once a project is completed. Learning to diagram is an important part of architectural education. A particular diagramming vocabulary can help to guide students into an appreciation and consciousness of aesthetics. As an introduction to theories of modernism, students have been instructed in the use of a set of diagrams that express abstract qualities of architectural aesthetics. The exercises are designed to wean students from a naïve aesthetic that merely mimics popular taste and introduce them to the field of aesthetics as an intellectual discipline. The diagramming vocabulary has been developed from the “seven invariables,” described by Bruno Zevi in The Modern Language of Architecture. Students apply the diagrams to analyze examples of famous buildings. They then design a house, applying the aesthetic principles expressed by the diagrams. The resulting designs are compared to previous designs produced by the students to reveal the change that is due in part to learning the diagramming vocabulary.

Deduction by a computer studied so far has been centered around symbolic reasoning with formulas. Recently, attention has been directed to reasoning with diagrams as well, in order to augment the deficiency of reasoning with symbols only. In this paper, we propose a visual reasoning system called JVenn which attains a unique amalgamation of the diagrammatic reasoning and the symbolic reasoning, having perspicuity of diagrams and strictness of symbols complementarily. JVenn is unique particularly in the points that it has the strategy for proving a chain of syllogisms, allows for an interplay between diagrams and symbols, and guides reasoning with the beauty measure for diagrams.

This paper presents one approach to the formalisation of diagrammatic proofs as an alternative to algebraic logic. An idea of ‘generic diagrams’ is developed whereby one diagram (or rather, one sequence of diagrams) can be used to prove many instances of a theorem. This allows the extension of Jamnik’s ideas in the Diamond system to continuous domains. The domain is restricted to non-recursive proofs in real analysis whose statement and proof have a strong geometric component. The aim is to develop a system of diagrams and redraw rules to allow a mechanised construction of sequences of diagrams constituting a proof. This approach involves creating a diagrammatic theory. The method is justified formally by (a) a diagrammatic axiomatisation, and (b) an appeal to analysis, viewing the diagram as an object inℝ2. The idea is to then establish an isomorphism between diagrams acted on by redraw rules and instances of a theorem acted on by rewrite rules. We aim to implement these ideas in an interactive prover entitled Rap (the Real Analysis Prover).

This paper extends work that developed a programmed model of reasoning about geometric propositions. The system reasons by manipulating representations of diagrams and noticing newly emerged facts that are construed as inferences. The system has been explored as a means of verifying diagrammatic demonstrations of classical geometric propositions and for constructing diagrammatic demonstrations of conclusions supplied for the system. The process of discovering propositions to be demonstrated is a more difficult task. This paper argues that central to the discovery process is systematic manipulation of diagrams - playing - and observing consistent relations among features of the diagram as manipulations are made and observed. The play results in the creation of an “episode” of diagram behaviors which is examined for regularities from which a general proposition might be proposed. The paper illustrates this process and discusses the advantages and limitations of this system and of other computational models of diagrammatic reasoning.

Many diagrams can be thought of as graphical representations used to support the solution of problems. This paper discusses how computation based on pixel-level rewrites can produce a rich form of diagrammatic computation making use of intermediate graphical constructions not explicit in the input or output of the problems it is solving.

Document image analysis is the study of converting documents from paper form to an electronic form that captures the information content of the document. Necessary processing includes recognition of document layout (to determine reading order, and to distinguish text from diagrams), recognition of text (called Optical Character Recognition, OCR), and processing of diagrams and photographs. The processing of diagrams has been an active research area for several decades. A selection of existing diagram recognition techniques are presented in this paper. Challenging problems in diagram recognition include (1) the great diversity of diagram types, (2) the difficulty of adequately describing the syntax and semantics of diagram notations, and (3) the need to handle imaging noise. Recognition techniques that are discussed include blackboard systems, stochastic grammars, Hidden Markov Models, and graph grammars.

The goal of the paper is to explicate some common formal logic underlying various notational systems used in visual modeling. The idea is to treat the notational diversity as the diversity of visualizations of the same basic specificational format. It is argued that the task can be well approached in the arrow-diagram logic framework where specifications are directed graphs carrying a structure of diagram predicates and operations.

Shorter product cycles, lower prices of products, and the production of goods that are tailored to the customers needs made knowledge based product configuration systems a great success of AI technology. However, configuration knowledge bases tend to become large and complex. Therefore, knowledge acquisition and maintenance are crucial phases in the life-cycle of a configuration system. We will show how to meet this challenge by extending a standard design language from the area of Software Engineering with classical description concepts for expressing configuration knowledge. We automatically translate this graphical depiction into logical sentences which can be exploited by a general inference engine to solve the configuration task. In order to cope with usability restrictions of diagrammatic notations for large applications, we introduce the usage of contextual diagrams. This mechanism makes the conceptual model more readable and understandable and supports intuitively the acquisition of functional configuration knowledge.

This paper presents an approach to evaluating the intelligibility of diagrammatic languages used in the specification of software. Research suggests that specification languages can be assessed in terms of properties that influence the intelligibility of representations produced using the languages. The paper describes the properties identified and highlights three in particular that have been shown to influence the intelligibility of representations: motivation of symbols in the language; the extent to which the language allows exploitation of human visual perception; and the amount of structure inherent in the language. The paper argues that the first two of these properties are not present to any great extent in diagrammatic languages used in software specification. In order to enhance the intelligibility of software specifications, we suggest that more attention should be paid to ways in which these languages can exploit the amount of structure inherent in the language.

We present a general framework for using diagram sequences as plan specifications. We also present an implemented system based on that framework, which generates imperative programs from diagram sequences similar to those used in teaching programming. The specific notations we use in the system are based closely on the diagrams typically used for teaching introductory programming, but the framework is general enough to account for and express many uses of diagram sequences. The system and the underlying theory highlight some areas where planning, reasoning about action, the refinement calculus and diagrammatic reasoning are synergistic. For example, by framing the definition of algorithms as a type of plan specification, it becomes clear that decomposition of a planning problem into sub-plans is analogous to refinement in the software engineering sense. More importantly, the system gives insight into the underlying structure of the largely informal use of diagrams that is routinely found in the explanation of algorithms. Obvious applications include teaching (since the inspiration for the system is a common method for teaching) and software engineering, where diagrams are often used to specify type systems rigorously (e.g. class diagrams), but specify program dynamics informally.

A software tool named MetaBuilder is described. MetaBuilder’s purpose is to enable the rapid creation of computerised diagram editing tools for structured diagrammatic notations. At its heart is an object-oriented, graphical metamodelling technique - a diagrammatic notation for describing other diagrammatic notations.The notation is based upon the concept of a mathematical graph consisting of nodes and edges. Construction of a “target tool” proceeds by drawing a metamodel of the target notation. Items in the target notation are modelled as “classes” and the syntax of the target notation such as connectivity between elements are expressed as “relationships” between the classes. Once the metamodel is complete, a new tool can be generated automatically. Thus the time to develop such notation specific drawing tools can be dramatically reduced. As the design of a piece of software can be expressed diagrammatically, the MetaBuilder software can be used to build itself!

Labeled graphs are a subclass of the class of all diagrams which are widely used in various disciplines. In consequence, graph grammars have been developed as powerful and intuitive means for the generation and manipulation of such kind of diagrams. As the operations of graph grammars are local and non-deterministic in principle, additional concepts of control are required. While textual control structures for the application of graph grammars are well-known already, more intuitive diagrammatic control mechanisms are still missing. We use a combination of the UML Activity Diagrams and the earlier Dijkstra Schemas for this purpose. We present a control flow editor which the user can use to build up diagrammatic control structures for the execution of graph grammatical diagram constructions. The control flow diagrams are interpreted and animated such that the the user can observe an automatic diagram construction running along the specified control flow diagram.

People depend on various external representations in various design situations. These external representations are necessary at the time of creation in early stages of a design task, as they help the designer visualize what they are thinking and continue with their task in the process of reflection-in-action. Designers in domains such as architecture have drawn diagrams, or sketches, as the external representations. We take writing and programming as two example domains, and argue that two-dimensional positioning serve the same purpose for these domains as diagrams do for architectural design. We describe two tools, ART for writing and RemBoard for component-based programming, which help writers or programmers visualize what they are thinking through positioning parts of writing or software components on a two-dimensional space. We examine the issues that are necessary for this, and explore how they were handled in the two tools.

The Reorderable Matrix is a visualization method for tabular data. This paper deals with the algorithmic problems related to ordering the rows and columns in a Reorderable Matrix. We establish links between ordering the matrix and the well-known and much studied problem of drawing graphs. First, we show that, as in graph drawing, our problem allows different aesthetic criterions which reduce to known NP-complete problems. Second, we apply and compare two simple heuristics to the problem of reordering the Reorderable Matrix: a two-dimensional sort and a graph drawing algorithm.

There are currently several interesting works on interactive concept map construction. This simple representation of knowledge - the concept maps - is widely accepted as a promising device for helping in complex tasks such as planning and learning. Moreover, several psychologists (mainly from the constructivist stream) argue that the use of concept maps in teaching can bring relevant improvements in students. Nevertheless, as far as we know, these tools for interactive construction of concept map diagrams have a passive role in the sense that their main concerns rely upon interface and generality. If a Machine Learning based module was added to such frameworks, the computer could have an active role in participating in the concept map construction.This paper presents Clouds, a module that uses Inductive Learning methods to help a user build her own concept maps. It uses each new entry on the map as an input for the learning algorithms, which can be used later for suggesting new concepts and relations.

In this paper, a two-dimensional, diagrammatic representation of the space of intervals, called an MR-diagram, is presented, together with another diagrammatic notations based on it, like the so called W-diagram and some other auxiliary notations. Examples of the use of the notation in the algebra of interval relations, in interval arithmetic, and for solving a simple common-sense problem involving time intervals, are given.

Static graphics are often used to present key aspects of dynamic instructional content (e.g. chemical reaction diagrams in Chemistry and weather maps in Meteorology). However, because static graphics represent the dynamics of a situation by implication only, students sometimes find them difficult to interpret properly. In contrast, animated graphics can represent dynamics quite explicitly. Studying relevant animations may help students become more adept at learning from related static representations (see Lowe, 1995a). In the case of beginning students of meteorology, being able to predict a subsequent pattern of meteorological markings from a given weather map is a very important skill (but one they learn slowly and imperfectly by studying static weather maps).

There are many different ways of looking at diagrams. Here, however, we consider them in relation to human communication. In particular, we draw upon some concepts taken from semiotics and linguistics, with a view to showing how a semiotically based approach can help to make more explicit the way in which diagrams contribute to the interpersonal communication process which is known as discourse.

Diagrams have a long history as visual aids which assist in structuring and simplifying potentially complex reasoning tasks. Recent years have witnessed a rapid, ongoing popularisation of diagrammatic notations in the specification, modelling and programming of computing systems, leading to diagrammatic languages requiring increasingly complex semantic interpretations. A general theory of diagrammatic languages, following those of more typical text-based languages, requires as a minimum an account of diagram syntax, of semantics, and of interpretations: the relationships between syntax and semantics. A satisfactory theory of diagrams must also account for more cognitive aspects, notably the ways in which pragmatic features of diagrams contribute to their effectiveness for human users, and how individual cognitive differences affect human interpretation of diagrams.

Our approach aims at a general description that is common to all types of grids used in diagrammatic representations despite their individual differences. Based on our analysis, we specify different types of spatial knowledge and single out in which way a particular type of grid represents a particular type of spatial knowledge. This specification identifies the various contributions of grids to diagrammatic representations. It turns out that grids in maps and especially in schematic maps have two complementary functions. First, they enable inferences that are not possible using only the spatial map features. Second, they provide additional design freedom, as important information that is not represented in the schematic map itself, can be encoded in the grid structure.

This paper gives a brief overview of FG, a formal system for doing Euclidean geometry whose basic syntactic elements are geometric diagrams, and which has been implimentented as the computer system CDEG. The computational complexity of determining whether or not a given diagram is satisfiable is also briefly discussed.

Bar charts are common data representations in scientific and technical papers. In order to recognize the printed bar charst, we present a new Hough based bar chart recognition algorithm which combines syntactic analysis into segmentation. We first detect the most salient feature in any bar chart, bar patterns, using syntactic analysis in the Hough domain. Then we group text primitives according to their centroids distribution in the Hough space. Finally, we interweave the two extracting processes to refine the recognition results. Our recognition algorithm is not dependent heavily on a priori knowledge and can recognize bar charts lying in arbitrary directions, such as oblique or skewed bar charts, or even hand-drawn bar charts.

Many automatic graph layout algorithms have been implemented to display relational data in a graphical (usually node-arc) manner. The success of these algorithms is typically measured by their computational efficiency and the extent to which they conform to aesthetic criteria (for example, minimising the number of crossings, maximising symmetry). Little research has been performed on the usability aspects of such algorithms: do they produce graph drawings that make the embodied information easy to use and understand? Is the computational effort expended on conforming to the assumed aesthetic criteria justifiable with espect to better usability? This paper reports on usability studies to investigate automatic graph layout algorithms with respect to human use.