A Common Type of Rigorous Proof that Resists Hilbert’s Programme

Following Hilbert, there seems to be a simple and clear definition of mathematical proof: it is a sequence of formulae each of which is either an axiom or follows from earlier formulae by a rule of inference. Automated theorem provers are based on this Hilbertian concept of proof, in which the formulae and rules of inference are represented in a formal logic. These logic- based proofs are typically an order of magnitude longer than the rigorous proofs produced by human mathematicians. There is a consensus, however, that rigorous proofs could, in principle, be unpacked into logical proofs, but this programme is rarely carried out because it would be tedious and uninformative. We argue that, for at least one class of rigorous proofs, which we will call schematic proofs, such a simple unpacking is not available. We will illustrate schematic proofs by analysing Cauchy’s faulty proof of Euler’s Theorem V-E+F = 2, as reported in [Lakatos, 1976] and giving further examples from [Nelsen, 1993]. We will then give a logic-based account of schematic proofs, distinguishing them from Hilbertian proofs, and showing why they are error prone

Automating Diagrammatic Proofs of Arithmetic Arguments

Centre for Intelligent Systems and their ApplicationsThis 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 quantified theorem. The diagrammatic proof in constructed by applying geometric operations to the diagram. These operations are in 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 gien 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 omega-rule and schematic proofs to make a translation from concrete instances to a general argument about the diagrammatic proof.The realisation of our ideas is a diagrammatic reasoning system DIAMOND. DIAMOND allows a user to interactively construct instances of a diagrammatic proof. It then automatically abstracts these into a general schematic proof and checks the correctness of this proof using an inductive theorem prover

CGXplain: Rule-Based Deep Neural Network Explanations Using Dual Linear Programs

Rule-based surrogate models are an effective and interpretable way to approximate a Deep Neural Network's (DNN) decision boundaries, allowing humans to easily understand deep learning models. Current state-of-the-art decompositional methods, which are those that consider the DNN's latent space to extract more exact rule sets, manage to derive rule sets at high accuracy. However, they a) do not guarantee that the surrogate model has learned from the same variables as the DNN (alignment), b) only allow to optimise for a single objective, such as accuracy, which can result in excessively large rule sets (complexity), and c) use decision tree algorithms as intermediate models, which can result in different explanations for the same DNN (stability). This paper introduces the CGX (Column Generation eXplainer) to address these limitations - a decompositional method using dual linear programming to extract rules from the hidden representations of the DNN. This approach allows to optimise for any number of objectives and empowers users to tweak the explanation model to their needs. We evaluate our results on a wide variety of tasks and show that CGX meets all three criteria, by having exact reproducibility of the explanation model that guarantees stability and reduces the rule set size by >80% (complexity) at equivalent or improved accuracy and fidelity across tasks (alignment).Comment: Accepted at ICLR 2023 Workshop on Trustworthy Machine Learning for Healthcar

Abstract Diagrammatic Reasoning with Multiplex Graph Networks

Abstract reasoning, particularly in the visual domain, is a complex human ability, but it remains a challenging problem for artificial neural learning systems. In this work we propose MXGNet, a multilayer graph neural network for multi-panel diagrammatic reasoning tasks. MXGNet combines three powerful concepts, namely, object-level representation, graph neural networks and multiplex graphs, for solving visual reasoning tasks. MXGNet first extracts object-level representations for each element in all panels of the diagrams, and then forms a multi-layer multiplex graph capturing multiple relations between objects across different diagram panels. MXGNet summarises the multiple graphs extracted from the diagrams of the task, and uses this summarisation to pick the most probable answer from the given candidates. We have tested MXGNet on two types of diagrammatic reasoning tasks, namely Diagram Syllogisms and Raven Progressive Matrices (RPM). For an Euler Diagram Syllogism task MXGNet achieves state-of-the-art accuracy of 99.8%. For PGM and RAVEN, two comprehensive datasets for RPM reasoning, MXGNet outperforms the state-of-the-art models by a considerable margin

HEALNet -- Hybrid Multi-Modal Fusion for Heterogeneous Biomedical Data

Technological advances in medical data collection such as high-resolution histopathology and high-throughput genomic sequencing have contributed to the rising requirement for multi-modal biomedical modelling, specifically for image, tabular, and graph data. Most multi-modal deep learning approaches use modality-specific architectures that are trained separately and cannot capture the crucial cross-modal information that motivates the integration of different data sources. This paper presents the Hybrid Early-fusion Attention Learning Network (HEALNet): a flexible multi-modal fusion architecture, which a) preserves modality-specific structural information, b) captures the cross-modal interactions and structural information in a shared latent space, c) can effectively handle missing modalities during training and inference, and d) enables intuitive model inspection by learning on the raw data input instead of opaque embeddings. We conduct multi-modal survival analysis on Whole Slide Images and Multi-omic data on four cancer cohorts of The Cancer Genome Atlas (TCGA). HEALNet achieves state-of-the-art performance, substantially improving over both uni-modal and recent multi-modal baselines, whilst being robust in scenarios with missing modalities.Comment: 7 pages body, 5 pages appendi

Extrapolatable Relational Reasoning With Comparators in Low-Dimensional Manifolds

While modern deep neural architectures generalise well when test data is sampled from the same distribution as training data, they fail badly for cases when the test data distribution differs from the training distribution even along a few dimensions. This lack of out-of-distribution generalisation is increasingly manifested when the tasks become more abstract and complex, such as in relational reasoning. In this paper we propose a neuroscience-inspired inductive-biased module that can be readily amalgamated with current neural network architectures to improve out-of-distribution (o.o.d) generalisation performance on relational reasoning tasks. This module learns to project high-dimensional object representations to low-dimensional manifolds for more efficient and generalisable relational comparisons. We show that neural nets with this inductive bias achieve considerably better o.o.d generalisation performance for a range of relational reasoning tasks. We finally analyse the proposed inductive bias module to understand the importance of lower dimension projection, and propose an augmentation to the algorithmic alignment theory to better measure algorithmic alignment with generalisation

Enhancing Representation Learning on High-Dimensional, Small-Size Tabular Data: A Divide and Conquer Method with Ensembled VAEs

Variational Autoencoders and their many variants have displayed impressive ability to perform dimensionality reduction, often achieving state-of-the-art performance. Many current methods however, struggle to learn good representations in High Dimensional, Low Sample Size (HDLSS) tasks, which is an inherently challenging setting. We address this challenge by using an ensemble of lightweight VAEs to learn posteriors over subsets of the feature-space, which get aggregated into a joint posterior in a novel divide-and-conquer approach. Specifically, we present an alternative factorisation of the joint posterior that induces a form of implicit data augmentation that yields greater sample efficiency. Through a series of experiments on eight real-world datasets, we show that our method learns better latent representations in HDLSS settings, which leads to higher accuracy in a downstream classification task. Furthermore, we verify that our approach has a positive effect on disentanglement and achieves a lower estimated Total Correlation on learnt representations. Finally, we show that our approach is robust to partial features at inference, exhibiting little performance degradation even with most features missing