On the Completeness of Spider Diagrams Augmented with Constants

Diagrammatic reasoning can be described formally by a number of diagrammatic logics; spider diagrams are one of these, and are used for expressing logical statements about set membership and containment. Here, existing work on spider diagrams is extended to include constant spiders that represent specific individuals. We give a formal syntax and semantics for the extended diagram language before introducing a collection of reasoning rules encapsulating logical equivalence and logical consequence. We prove that the resulting logic is sound, complete and decidable

A cognitive exploration of the “non-visual” nature of geometric proofs

Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and differences between diagrams and text (visual thinking versus language thinking). Traditional text-based proofs are considered (by many to be) more rigorous than diagrams alone. In this paper we focus on human perceptual-cognitive characteristics that may encourage textual modes for proofs because of the ergonomic affordances of text relative to diagrams. We suggest that visual-spatial perception of physical objects, where an object is perceived with greater acuity through foveal vision rather than peripheral vision, is similar to attention navigating a conceptual visual-spatial structure. We suggest that attention has foveal-like and peripheral-like characteristics and that textual modes appeal to what we refer to here as foveal-focal attention, an extension of prior work in focused attention

Equivalences in Euler-based diagram systems through normal forms

AbstractThe form of information presented can influence its utility for the conveying of knowledge by affecting an interpreter’s ability to reason with the information. There are distinct types of representational systems (for example, symbolic versus diagrammatic logics), various sub-systems (for example, propositional versus predicate logics), and even within a single representational system there may be different means of expressing the same piece of information content. Thus, to display information, choices must be made between its different representations, depending upon many factors such as: the context, the reasoning tasks to be considered, user preferences or desires (for example, for short symbolic sentences or minimal clutter within diagrammatic systems). The identification of all equivalent representations with the same information content is a sensible precursor to attempts to minimise a metric over this class. We posit that defining notions of semantic redundancy and identifying the syntactic properties that encapsulate redundancy can help in achieving the goal of completely identifying equivalences within a single notational system or across multiple systems, but that care must be taken when extending systems, since refinements of redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler-diagram-based notations. Such notations can be used to represent logical information and have applications including visualisation of database queries, social network visualisation, statistical data visualisation, and as the basis of more expressive diagrammatic logics such as constraint languages used in software specification and reasoning. The development of the new associated machinery and concepts required is important in its own right since it increases the growing body of knowledge on diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow for a much greater degree of flexibility of representation. We give syntactic properties that encapsulate semantic equivalence in both systems, whilst observing that the same semantic concept of redundancy is significantly more difficult to realise as syntactic properties in the extended system with projections.</jats:p

Evaluation of the usability of constraint diagrams as a visual modelling language: theoretical and empirical investigations

This research evaluates the constraint diagrams (CD) notation, which is a formal representation for program specification that has some promise to be used by people who are not expert in software design. Multiple methods were adopted in order to provide triangulated evidence of the potential benefits of constraint diagrams compared with other notational systems. Three main approaches were adopted for this research. The first approach was a semantic and task analysis of the CD notation. This was conducted by the application of the Cognitive Dimensions framework, which was used to examine the relative strengths and weaknesses of constraint diagrams and conventional notations in terms of the perceptive facilitation or impediments of these different representations. From this systematic analysis, we found that CD cognitively reduced the cost of exploratory design, modification, incrementation, searching, and transcription activities with regard to the cognitive dimensions: consistency, visibility, abstraction, closeness of mapping, secondary notation, premature commitment, role-expressiveness, progressive evaluation, diffuseness, provisionality, hidden dependency, viscosity, hard mental operations, and error-proneness. The second approach was an empirical evaluation of the comprehension of CD compared to natural language (NL) with computer science students. This experiment took the form of a web-based competition in which 33 participants were given instructions and training on either CD or the equivalent NL specification expressions, and then after each example, they responded to three multiple-choice questions requiring the interpretation of expressions in their particular notation. Although the CD group spent more time on the training and had less confidence, they obtained comparable interpretation scores to the NL group and took less time to answer the questions, although they had no prior experience of CD notation. The third approach was an experiment on the construction of CD. 20 participants were given instructions and training on either CD or the equivalent NL specification expressions, and then after each example, they responded to three questions requiring the construction of expressions in their particular notation. We built an editor to allow the construction of the two notations, which automatically logged their interactions. In general, for constructing program specification, the CD group had more accurate answers, they had spent less time in training, and their returns to the training examples were fewer than those of the NL group. Overall it was found that CD is understandable, usable, intuitive, and expressive with unambiguous semantic notation

Cognitive Foundations for Visual Analytics

In this report, we provide an overview of scientific/technical literature on information visualization and VA. Topics discussed include an update and overview of the extensive literature search conducted for this study, the nature and purpose of the field, major research thrusts, and scientific foundations. We review methodologies for evaluating and measuring the impact of VA technologies as well as taxonomies that have been proposed for various purposes to support the VA community. A cognitive science perspective underlies each of these discussions