Diagrams in mechanised reasoning systems are typically encoded into symbolic representations that can be easily processed with rule-based expert systems. This relies on human experts to define diagramto-symbol mapping and the set of rules to reason with the symbols. We present a new method of using Deep artificial Neural Networks (DNN) to learn continuous, vector-form representations of diagrams without any human input, and entirely from datasets of diagrammatic reasoning problems. Based on this DNN, we developed a novel reasoning system, Euler-Net, to solve syllogisms with Euler diagrams. Euler-Net takes two diagrams representing the premises in a syllogism as input, and outputs either a categorical (subset, intersection or disjoint) or diagrammatic conclusion (generating an Euler diagram representing the conclusion) to the syllogism. Euler-Net can achieve 99.5% accuracy for generating syllogism conclusions, and learns meaningful representations. We propose that our framework can be applied to other types of diagrams, especially the ones we are less sure how to formalise symbolically. Recently, DNNs have achieved human comparable performance in several tasks such as image recognition [6], natural language translation [7]. DNNs' success in

International initiatives such as the Molecular Taxonomy of Breast Cancer International Consortium are collecting multiple data sets at different genome-scales with the aim to identify novel cancer bio-markers and predict patient survival. To analyze such data, several machine learning, bioinformatics, and statistical methods have been applied, among them neural networks such as autoencoders. Although these models provide a good statistical learning framework to analyze multi-omic and/or clinical data, there is a distinct lack of work on how to integrate diverse patient data and identify the optimal design best suited to the available data.In this paper, we investigate several autoencoder architectures that integrate a variety of cancer patient data types (e.g., multi-omics and clinical data). We perform extensive analyses of these approaches and provide a clear methodological and computational framework for designing systems that enable clinicians to investigate cancer traits and translate the results into clinical applications. We demonstrate how these networks can be designed, built, and, in particular, applied to tasks of integrative analyses of heterogeneous breast cancer data. The results show that these approaches yield relevant data representations that, in turn, lead to accurate and stable diagnosis.

In order to effectively communicate information, the choice of representation is important. Ideally, a chosen representation will aid readers in making desired inferences. In this paper, we develop the theory of observation: what it means for one statement to be observable from another. Using observability, we give a formal characterization of the observational advantages of one representation of information over another. By considering observational advantages, people will be able to make better informed choices of representations of information. To demonstrate the benefit of observation and observational advantages, we apply these concepts to set theory and Euler diagrams. In particular, we can show that Euler diagrams have significant observational advantages over set theory. This formally justifies Larkin and Simon's claim that "a diagram is (sometimes) worth ten thousand words".

International initiatives such as the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) are collecting multiple data sets at different genome-scales with the aim to identify novel cancer bio-markers and predict patient survival. To analyse such data, several machine learning, bioinformatics and statistical methods have been applied, among them neural networks such as autoencoders. Although these models provide a good statistical learning framework to analyse multi-omic and/or clinical data, there is a distinct lack of work on how to integrate diverse patient data and identify the optimal design best suited to the available data.In this paper, we investigate several autoencoder architectures that integrate a variety of cancer patient data types (e.g., multi-omics and clinical data). We perform extensive analyses of these approaches and provide a clear methodological and computational framework for designing systems that enable clinicians to investigate cancer traits and translate the results into clinical applications. We demonstrate how these networks can be designed, built and, in particular, applied to tasks of integrative analyses of heterogeneous breast cancer data. The results show that these approaches yield relevant data representations that, in turn, lead to accurate and stable diagnosis.

This thesis is on the automation of diagrammatic proofs, a novel approach to mechanised mathematical reasoning. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there are some conjectures which humans can prove by the use of geometric operations on diagrams that somehow represent these conjectures, so called diagrammatic proofs. Insight is often more clearly perceived in these diagrammatic proofs than in the algebraic proofs. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete rather than general diagrams are used to prove ground instances of a universally quanti ed theorem. The diagrammatic proof is constructed by applying geometric operations to the diagram. These operations are the inference steps of the proof. A general schematic proof is extracted from the ground instances of a proof. It is represented as a recursive program that consists of a general number of applications of geometric operations. When given a particular diagram, a schematic proof generates a proof for that diagram. To verify that the schematic proof produces a correct proof of the conjecture for each ground instance we check its correctness in a theory of diagrams. We use the constructive !-rule and schematic proofs to make a transition from v vi Declaration I hereby declare that I composed this thesis entirely myself and that it describes my own research.