This project looks at using diagrammatic reasoning to prove mathematical theorems. The work is motivated by a need for theorem provers whose reasoning is readily intelligible to human beings. It should also have practical applications in mathematics teaching. We focus on the continuous domain of analysis - a geometric subject, but one which is taught using a dry algebraic formalism which many students find hard. The geometric nature of the domain makes it suitable for a diagram-based approach. However it is a difficult domain, and there are several problems, including handling alternating quantifiers, sequences and generalisation. We developed representations and reasoning methods to solve these. Our diagram logic isn't complete, but does cover a reasonable range of theorems. It utilises computers to extend diagrammatic reasoning in new directions – including using animation. This work is tested for soundness, and evaluated empirically for ease of use. We demonstrate that computerised diagrammatic theorem proving is not only possible in the domain of real analysis, but that students perform better using it than with an equivalent algebraic computer system.

Science, Technology, Engineering, and Mathematics (STEM) experts employ many representations that novices find hard to use because they require a critical STEM skill, interpreting two-dimensional (2D) diagrams that represent three-dimensional (3D) information. The current research focuses on learning to interpret topographic maps. Understanding topographic maps requires knowledge of how to interpret the conventions of contour lines, and skill in visualizing that information in 3D (e.g. shape of the terrain). Novices find both tasks difficult. The present study compared two interventions designed to facilitate understanding for topographic maps to minimal text-only instruction. The 3D Visualization group received instruction using 3D gestures and models to help visualize three topographic forms. The Pattern Identification group received instruction using pointing and tracing gestures to help identify the contour patterns associated with the three topographic forms. The Text-based Instruction group received only written instruction explaining topographic maps. All participants then completed a measure of topographic map use. The Pattern Identification group performed better on the map use measure than participants in the Text-based Instruction group, but no significant difference was found between the 3D Visualization group and the other two groups. These results suggest that learning to identify meaningful contour patterns is an effective strategy for learning how to comprehend topographic maps. Future research should address if learning strategies for how to interpret the information represented on a diagram (e.g. identify patterns in the contour lines), before trying to visualize the information in 3D (e.g. visualize the 3D structure of the terrain), also facilitates students' comprehension of other similar types of diagrams. / Psychology

Two applications of Category Theory are considered. The link between them is applications to Physics and more specifically to Entropy. The first research chapter is broader in scope and not explicitly about Physics, although connections to Statistical Mechanics are made towards the end of the chapter. Matroids are abstract structures that describe dependence, and strong maps are certain structure-preserving functions between them with desirable properties. We examine properties of various categories of matroids and strong maps: we compute limits and colimits; we find free and cofree constructions of various subcategories; we examine factorisation structures, including a translation principle from geometric lattices; we find functors with convenient properties to/from vector spaces, multisets of vectors, geometric lattices, and graphs; we determine which widely used operations on matroids are functorial (these include deletion, contraction, series and parallel connection, and a simplification monad); lastly, we find a categorical characterisation of the greedy algorithm. In conclusion, this project determines which aspects of Matroid Theory are most and least conducive to categorical treatment. The purpose of the second research chapter is to provide a categorical framework for generalising and unifying notions of Entropy in various settings, exploiting the fact that Entropy is a monotone subadditive function. A categorical characterisation of Entropy through a category of thermodynamical systems and adiabatic processes is found. A modelling perspective (adiabatic categories) that directly generalises an existing model is compared to an axiomatisation through topological and linear structures (topological weak semimodules), where the latter is based on a categorification of semimodules. Properties of each class of categories are examined; most notably a cancellation property of adiabatic categories generalising an existing result, and an adjunction between the categories of weak semimodules and symmetric monoidal categories. An adjunction between categories of adiabatic categories and topological weak semimodules is found. We examine in which cases each of these classes of categories constitutes a traced monoidal category. Lastly, examples of physical applications are provided. In conclusion, this project uncovers a way of, and makes progress towards, retrieving the statistical formulation of Entropy from simple axioms.

This thesis is about diagrammatic reasoning and error-finding in conceptual modeling diagrams. Specifically, the differences of the cognitive strategies and behaviors of notation-familiar participants versus domain-familiar participants working on conceptual modeling diagrams are inspected. The domain-familiar participants are experienced in the topic being represented, but they do not have any formal training in software development representations. On the other hand, the notation-familiar participants are educated in software representations, but unfamiliar with the topic represented. The main experiment and the follow-up experiment also aim to study how some properties of diagrams affect the error-finding behaviors. The participant groups&rsquo / performances in the main experiment are investigated and compared by the analysis of verbal protocol data and eye movement data. The combination of the two different methods enhances detailed analyses. In the follow-up experiment, only eye movement data is involved to evaluate how some properties of diagrams affect problem-solving. By means of both experiments, it is concluded that diagrammatic complexity has a negative effect on reasoning whereas the degree of causal chaining improves diagrammatic reasoning. In the main experiment, some differences in the diagrammatic reasoning processes between the groups are observed, too. The notation-familiar participants are observed to be more successful in error-finding although they are unfamiliar with the topic. This study underlines the interaction of cognitive science and software engineering by integrating eye movement data, verbal protocol analysis and performance data into the cognitive inspection of software engineering notations.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation addresses the features of diagrammatic reasoning from the perspective of syllogistics. With the emergence of modern logic, diagrams where not considered as legitimate elements of decision methods, operating only as illustrative and heuristic tools. More recently there were questioning for what characteristics of diagrammatic methods are distinguished from purely sentential methods and how these distinctive features determine the possible misleading and inaccurate character of diagrams. Exploring this debate within syllogistics, I show that diagrammatic methods are more complex and dynamic systems than they appear for two reasons. On the one hand, diagrammatic systems are distinguished from sentential systems by their semiotic constitution. A diagram uses a spatial relationship to represent some aspect of the logical domain, while sentential systems uses a symbol for represent the same aspect. On the other hand, in order to generate an isomorphic representation with this spatial relation, diagrammatic reasoning involves substantial cognitive and perceptual capabilities that provide advantages for some utilities. / A presente dissertação versa sobre as características do raciocínio diagramático a partir da lógica silogística. No surgimento da lógica moderna diagramas foram descartados enquanto legítimos elementos de métodos de decisão, operando apenas como ferramentas ilustrativas e heurísticas. Mais recentemente houve questionamento por qual razão métodos diagramáticos seriam distintos de métodos puramente sentenciais e como essas características distintivas determinariam o caráter possivelmente enganoso e pouco preciso dos diagramas. Explorando esse debate a partir da silogística, mostramos que métodos diagramáticos são sistemas mais complexos e dinâmicos do que aparentam em dois sentidos. Por um lado, sistemas diagramáticos distinguem-se de sistemas sentencias pela sua constituição semiótica, na medida em que utilizam uma relação espacial para representar algum aspecto do domínio lógico, enquanto que sistemas sentenciais utilizam um símbolo para representar o mesmo aspecto. Por outra via, ao utilizar essa propriedade para gerar uma representação isomórfica, o raciocínio diagramático envolve substancialmente capacidades cognitivas e perceptuais que proporcionam vantagens para determinadas utilidades.

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The study addresses the automated translation of architectural drawings from 2D Computer Aided Drafting (CAD) data into a Building Information Model (BIM), with emphasis on the nature, possible role, and limitations of a drafting language Knowledge Representation (KR) on the problem and process. The central idea is that CAD to BIM translation is a complex diagrammatic interpretation problem requiring a domain (drafting language) KR to render it tractable and that such a KR can take the form of an information model. Formal notions of drawing-as-language have been advanced and studied quite extensively for close to 25 years. The analogy implicitly encourages comparison between problem structures in both domains, revealing important similarities and offering guidance from the more mature field of Natural Language Understanding (NLU). The primary insight we derive from NLU involves the central role that a formal language description plays in guiding the process of interpretation (inferential reasoning), and the notable absence of a comparable specification for architectural drafting. We adopt a modified version of Engelhard's approach which expresses drawing structure in terms of a symbol set, a set of relationships, and a set of compositional frameworks in which they are composed. We further define an approach for establishing the features of this KR, drawing upon related work on conceptual frameworks for diagrammatic reasoning systems. We augment this with observation of human subjects performing a number of drafting interpretation exercises and derive some understanding of its inferential nature therefrom. We consider this indicative of the potential range of inferential processes a computational drafting model should ideally support. The KR is implemented as an information model using the EXPRESS language because it is in the public domain and is the implementation language of the target Industry Foundation Classes (IFC) model. We draw extensively from the IFC library to demonstrate that it can be applied in this manner, and apply the MVD methodology in defining the scope and interface of the DOM and IFC. This simplifies the IFC translation process significantly and minimizes the need for mapping. We conclude on the basis of selective implementations that a model reflecting the principles and features we define can indeed provide needed and otherwise unavailable support in drafting interpretation and other problems involving reasoning with this class of diagrammatic representations.

Right now you are reading a sentence. Earlier, you might have been looking at a realistic picture, such as a photograph, or an outline drawing in a set of instructions. If you are a programmer, you work with sentence-like structures, such as code, or a system diagram. These are all graphic representations. To varying degrees, the effectiveness of every graphic representation relies on its ability to convey the designer’s intended meaning and elicit the intended reaction from its audience. However, the design of graphic representations, even in technical domains such as visual programming language design or interactive information visualization, currently relies heavily on general principles based solely on practice, intuition, and informal measures of effectiveness from the applied art and craft of design (as opposed to scientific analysis or theory). There is an increasing demand for a scientific understanding of design and its evaluation from stakeholders (who seek evidence for effectiveness) and designers (who seek to advance their field). Because both the creation of graphic displays and their perception are literally embodied experiences, a model was developed with an embodiment orientation, specifically based on how graphics are perceptually and cognitively processed. In my research, I found that graphic representations are constituted of two properties, pictorial and symbolic information, that emerge through two interrelated aspects of perception. In sighted individuals, for example, every graphic representation makes use of biological capabilities to process visual sensation (i.e., light hitting the retina), which are processed in relation to culturally-learned capabilities (i.e., writing). I observed how graphic representations – such as pictures, diagrams, and sentences – are “naturally selected” (i.e., during different phases of design or problem solving). From these observations, I developed a model that distinguishes and predicts the effectiveness of pictures, diagrams, and sentences, in terms of how object relations and attributes are pictorially or symbolically represented, relative to the functional roles of those representations, contexts, and in some cases, individual perceptual-cognitive differences among perceivers. This model is a step toward a science of graphics that could lead to evaluation techniques for information systems, theories for inclusive design, and ergonomically designed software programming tools.

Right now you are reading a sentence. Earlier, you might have been looking at a realistic picture, such as a photograph, or an outline drawing in a set of instructions. If you are a programmer, you work with sentence-like structures, such as code, or a system diagram. These are all graphic representations. To varying degrees, the effectiveness of every graphic representation relies on its ability to convey the designer’s intended meaning and elicit the intended reaction from its audience. However, the design of graphic representations, even in technical domains such as visual programming language design or interactive information visualization, currently relies heavily on general principles based solely on practice, intuition, and informal measures of effectiveness from the applied art and craft of design (as opposed to scientific analysis or theory). There is an increasing demand for a scientific understanding of design and its evaluation from stakeholders (who seek evidence for effectiveness) and designers (who seek to advance their field). Because both the creation of graphic displays and their perception are literally embodied experiences, a model was developed with an embodiment orientation, specifically based on how graphics are perceptually and cognitively processed. In my research, I found that graphic representations are constituted of two properties, pictorial and symbolic information, that emerge through two interrelated aspects of perception. In sighted individuals, for example, every graphic representation makes use of biological capabilities to process visual sensation (i.e., light hitting the retina), which are processed in relation to culturally-learned capabilities (i.e., writing). I observed how graphic representations – such as pictures, diagrams, and sentences – are “naturally selected” (i.e., during different phases of design or problem solving). From these observations, I developed a model that distinguishes and predicts the effectiveness of pictures, diagrams, and sentences, in terms of how object relations and attributes are pictorially or symbolically represented, relative to the functional roles of those representations, contexts, and in some cases, individual perceptual-cognitive differences among perceivers. This model is a step toward a science of graphics that could lead to evaluation techniques for information systems, theories for inclusive design, and ergonomically designed software programming tools.