Mateja Jamnik

Katherine M. Collins, Albert Q. Jiang, Simon Frieder et al. · cambridge

There is much excitement about the opportunity to harness the power of large language models (LLMs) when building problem-solving assistants. However, the standard methodology of evaluating LLMs relies on static pairs of inputs and outputs, and is insufficient for making an informed decision about which LLMs and under which assistive settings can they be sensibly used. Static assessment fails to account for the essential interactive element in LLM deployment, and therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models (InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we garner a more granular understanding of GPT-4 mathematical problem-solving through a series of case studies, contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models that communicate uncertainty respond well to user corrections, and are more interpretable and concise may constitute better assistants. Interactive evaluation is a promising way to navigate the capability of these models; humans should be aware of language models' algebraic fallibility and discern where they are appropriate to use.

Andrei Margeloiu, Nikola Simidjievski, Pietro Lio et al. · cambridge

Neural networks often struggle with high-dimensional but small sample-size tabular datasets. One reason is that current weight initialisation methods assume independence between weights, which can be problematic when there are insufficient samples to estimate the model's parameters accurately. In such small data scenarios, leveraging additional structures can improve the model's performance and training stability. To address this, we propose GCondNet, a general approach to enhance neural networks by leveraging implicit structures present in tabular data. We create a graph between samples for each data dimension, and utilise Graph Neural Networks (GNNs) to extract this implicit structure, and for conditioning the parameters of the first layer of an underlying predictor network. By creating many small graphs, GCondNet exploits the data's high-dimensionality, and thus improves the performance of an underlying predictor network. We demonstrate GCondNet's effectiveness on 12 real-world datasets, where it outperforms 14 standard and state-of-the-art methods. The results show that GCondNet is a versatile framework for injecting graph-regularisation into various types of neural networks, including MLPs and tabular Transformers. Code is available at https://github.com/andreimargeloiu/GCondNet.

Albert Q. Jiang, Sean Welleck, Jin Peng Zhou et al. · cambridge, uw

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Yuhuai Wu, Albert Q. Jiang, Wenda Li et al. · cambridge

Autoformalization is the process of automatically translating from natural language mathematics to formal specifications and proofs. A successful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence. While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal. We make the surprising observation that LLMs can correctly translate a significant portion ($25.3\%$) of mathematical competition problems perfectly to formal specifications in Isabelle/HOL. We demonstrate the usefulness of this process by improving a previously introduced neural theorem prover via training on these autoformalized theorems. Our methodology results in a new state-of-the-art result on the MiniF2F theorem proving benchmark, improving the proof rate from $29.6\%$ to $35.2\%$.

Mateo Espinosa Zarlenga, Pietro Barbiero, Gabriele Ciravegna et al. · cambridge

Deploying AI-powered systems requires trustworthy models supporting effective human interactions, going beyond raw prediction accuracy. Concept bottleneck models promote trustworthiness by conditioning classification tasks on an intermediate level of human-like concepts. This enables human interventions which can correct mispredicted concepts to improve the model's performance. However, existing concept bottleneck models are unable to find optimal compromises between high task accuracy, robust concept-based explanations, and effective interventions on concepts -- particularly in real-world conditions where complete and accurate concept supervisions are scarce. To address this, we propose Concept Embedding Models, a novel family of concept bottleneck models which goes beyond the current accuracy-vs-interpretability trade-off by learning interpretable high-dimensional concept representations. Our experiments demonstrate that Concept Embedding Models (1) attain better or competitive task accuracy w.r.t. standard neural models without concepts, (2) provide concept representations capturing meaningful semantics including and beyond their ground truth labels, (3) support test-time concept interventions whose effect in test accuracy surpasses that in standard concept bottleneck models, and (4) scale to real-world conditions where complete concept supervisions are scarce.

Albert Q. Jiang, Wenda Li, Szymon Tworkowski et al. · cambridge

In theorem proving, the task of selecting useful premises from a large library to unlock the proof of a given conjecture is crucially important. This presents a challenge for all theorem provers, especially the ones based on language models, due to their relative inability to reason over huge volumes of premises in text form. This paper introduces Thor, a framework integrating language models and automated theorem provers to overcome this difficulty. In Thor, a class of methods called hammers that leverage the power of automated theorem provers are used for premise selection, while all other tasks are designated to language models. Thor increases a language model's success rate on the PISA dataset from $39\%$ to $57\%$, while solving $8.2\%$ of problems neither language models nor automated theorem provers are able to solve on their own. Furthermore, with a significantly smaller computational budget, Thor can achieve a success rate on the MiniF2F dataset that is on par with the best existing methods. Thor can be instantiated for the majority of popular interactive theorem provers via a straightforward protocol we provide.

Katherine M. Collins, Matthew Barker, Mateo Espinosa Zarlenga et al. · cambridge

Placing a human in the loop may abate the risks of deploying AI systems in safety-critical settings (e.g., a clinician working with a medical AI system). However, mitigating risks arising from human error and uncertainty within such human-AI interactions is an important and understudied issue. In this work, we study human uncertainty in the context of concept-based models, a family of AI systems that enable human feedback via concept interventions where an expert intervenes on human-interpretable concepts relevant to the task. Prior work in this space often assumes that humans are oracles who are always certain and correct. Yet, real-world decision-making by humans is prone to occasional mistakes and uncertainty. We study how existing concept-based models deal with uncertain interventions from humans using two novel datasets: UMNIST, a visual dataset with controlled simulated uncertainty based on the MNIST dataset, and CUB-S, a relabeling of the popular CUB concept dataset with rich, densely-annotated soft labels from humans. We show that training with uncertain concept labels may help mitigate weaknesses of concept-based systems when handling uncertain interventions. These results allow us to identify several open challenges, which we argue can be tackled through future multidisciplinary research on building interactive uncertainty-aware systems. To facilitate further research, we release a new elicitation platform, UElic, to collect uncertain feedback from humans in collaborative prediction tasks.

Mateo Espinosa Zarlenga, Katherine M. Collins, Krishnamurthy Dvijotham et al. · cambridge

Concept Bottleneck Models (CBMs) tackle the opacity of neural architectures by constructing and explaining their predictions using a set of high-level concepts. A special property of these models is that they permit concept interventions, wherein users can correct mispredicted concepts and thus improve the model's performance. Recent work, however, has shown that intervention efficacy can be highly dependent on the order in which concepts are intervened on and on the model's architecture and training hyperparameters. We argue that this is rooted in a CBM's lack of train-time incentives for the model to be appropriately receptive to concept interventions. To address this, we propose Intervention-aware Concept Embedding models (IntCEMs), a novel CBM-based architecture and training paradigm that improves a model's receptiveness to test-time interventions. Our model learns a concept intervention policy in an end-to-end fashion from where it can sample meaningful intervention trajectories at train-time. This conditions IntCEMs to effectively select and receive concept interventions when deployed at test-time. Our experiments show that IntCEMs significantly outperform state-of-the-art concept-interpretable models when provided with test-time concept interventions, demonstrating the effectiveness of our approach.

Mateo Espinosa Zarlenga, Pietro Barbiero, Zohreh Shams et al. · cambridge

Recent work on interpretability has focused on concept-based explanations, where deep learning models are explained in terms of high-level units of information, referred to as concepts. Concept learning models, however, have been shown to be prone to encoding impurities in their representations, failing to fully capture meaningful features of their inputs. While concept learning lacks metrics to measure such phenomena, the field of disentanglement learning has explored the related notion of underlying factors of variation in the data, with plenty of metrics to measure the purity of such factors. In this paper, we show that such metrics are not appropriate for concept learning and propose novel metrics for evaluating the purity of concept representations in both approaches. We show the advantage of these metrics over existing ones and demonstrate their utility in evaluating the robustness of concept representations and interventions performed on them. In addition, we show their utility for benchmarking state-of-the-art methods from both families and find that, contrary to common assumptions, supervision alone may not be sufficient for pure concept representations.

Xiangjian Jiang, Andrei Margeloiu, Nikola Simidjievski et al.

Tabular biomedical data poses challenges in machine learning because it is often high-dimensional and typically low-sample-size (HDLSS). Previous research has attempted to address these challenges via local feature selection, but existing approaches often fail to achieve optimal performance due to their limitation in identifying globally important features and their susceptibility to the co-adaptation problem. In this paper, we propose ProtoGate, a prototype-based neural model for feature selection on HDLSS data. ProtoGate first selects instance-wise features via adaptively balancing global and local feature selection. Furthermore, ProtoGate employs a non-parametric prototype-based prediction mechanism to tackle the co-adaptation problem, ensuring the feature selection results and predictions are consistent with underlying data clusters. We conduct comprehensive experiments to evaluate the performance and interpretability of ProtoGate on synthetic and real-world datasets. The results show that ProtoGate generally outperforms state-of-the-art methods in prediction accuracy by a clear margin while providing high-fidelity feature selection and explainable predictions. Code is available at https://github.com/SilenceX12138/ProtoGate.

Andrei Margeloiu, Xiangjian Jiang, Nikola Simidjievski et al.

Data collection is often difficult in critical fields such as medicine, physics, and chemistry. As a result, classification methods usually perform poorly with these small datasets, leading to weak predictive performance. Increasing the training set with additional synthetic data, similar to data augmentation in images, is commonly believed to improve downstream classification performance. However, current tabular generative methods that learn either the joint distribution $ p(\mathbf{x}, y) $ or the class-conditional distribution $ p(\mathbf{x} \mid y) $ often overfit on small datasets, resulting in poor-quality synthetic data, usually worsening classification performance compared to using real data alone. To solve these challenges, we introduce TabEBM, a novel class-conditional generative method using Energy-Based Models (EBMs). Unlike existing methods that use a shared model to approximate all class-conditional densities, our key innovation is to create distinct EBM generative models for each class, each modelling its class-specific data distribution individually. This approach creates robust energy landscapes, even in ambiguous class distributions. Our experiments show that TabEBM generates synthetic data with higher quality and better statistical fidelity than existing methods. When used for data augmentation, our synthetic data consistently improves the classification performance across diverse datasets of various sizes, especially small ones. Code is available at https://github.com/andreimargeloiu/TabEBM.

Albert Q. Jiang, Wenda Li, Mateja Jamnik · cambridge

Autoformalization is the task of translating natural language materials into machine-verifiable formalisations. Progress in autoformalization research is hindered by the lack of a sizeable dataset consisting of informal-formal pairs expressing the same essence. Existing methods tend to circumvent this challenge by manually curating small corpora or using few-shot learning with large language models. But these methods suffer from data scarcity and formal language acquisition difficulty. In this work, we create $\texttt{MMA}$, a large, flexible, multilingual, and multi-domain dataset of informal-formal pairs, by using a language model to translate in the reverse direction, that is, from formal mathematical statements into corresponding informal ones. Experiments show that language models fine-tuned on $\texttt{MMA}$ produce $16-18\%$ of statements acceptable with minimal corrections on the $\texttt{miniF2F}$ and $\texttt{ProofNet}$ benchmarks, up from $0\%$ with the base model. We demonstrate that fine-tuning on multilingual formal data results in more capable autoformalization models even when deployed on monolingual tasks.

Andrei Margeloiu, Nikola Simidjievski, Pietro Lio et al. · cambridge

Tabular biomedical data is often high-dimensional but with a very small number of samples. Although recent work showed that well-regularised simple neural networks could outperform more sophisticated architectures on tabular data, they are still prone to overfitting on tiny datasets with many potentially irrelevant features. To combat these issues, we propose Weight Predictor Network with Feature Selection (WPFS) for learning neural networks from high-dimensional and small sample data by reducing the number of learnable parameters and simultaneously performing feature selection. In addition to the classification network, WPFS uses two small auxiliary networks that together output the weights of the first layer of the classification model. We evaluate on nine real-world biomedical datasets and demonstrate that WPFS outperforms other standard as well as more recent methods typically applied to tabular data. Furthermore, we investigate the proposed feature selection mechanism and show that it improves performance while providing useful insights into the learning task.

Lucie Charlotte Magister, Pietro Barbiero, Dmitry Kazhdan et al.

The opaque reasoning of Graph Neural Networks induces a lack of human trust. Existing graph network explainers attempt to address this issue by providing post-hoc explanations, however, they fail to make the model itself more interpretable. To fill this gap, we introduce the Concept Encoder Module, the first differentiable concept-discovery approach for graph networks. The proposed approach makes graph networks explainable by design by first discovering graph concepts and then using these to solve the task. Our results demonstrate that this approach allows graph networks to: (i) attain model accuracy comparable with their equivalent vanilla versions, (ii) discover meaningful concepts that achieve high concept completeness and purity scores, (iii) provide high-quality concept-based logic explanations for their prediction, and (iv) support effective interventions at test time: these can increase human trust as well as significantly improve model performance.

Pietro Barbiero, Gabriele Ciravegna, Francesco Giannini et al.

Deep learning methods are highly accurate, yet their opaque decision process prevents them from earning full human trust. Concept-based models aim to address this issue by learning tasks based on a set of human-understandable concepts. However, state-of-the-art concept-based models rely on high-dimensional concept embedding representations which lack a clear semantic meaning, thus questioning the interpretability of their decision process. To overcome this limitation, we propose the Deep Concept Reasoner (DCR), the first interpretable concept-based model that builds upon concept embeddings. In DCR, neural networks do not make task predictions directly, but they build syntactic rule structures using concept embeddings. DCR then executes these rules on meaningful concept truth degrees to provide a final interpretable and semantically-consistent prediction in a differentiable manner. Our experiments show that DCR: (i) improves up to +25% w.r.t. state-of-the-art interpretable concept-based models on challenging benchmarks (ii) discovers meaningful logic rules matching known ground truths even in the absence of concept supervision during training, and (iii), facilitates the generation of counterfactual examples providing the learnt rules as guidance.

Konstantin Hemker, Nikola Simidjievski, Mateja Jamnik

Technological advances in medical data collection, such as high-throughput genomic sequencing and digital high-resolution histopathology, have contributed to the rising requirement for multimodal biomedical modelling, specifically for image, tabular and graph data. Most multimodal deep learning approaches use modality-specific architectures that are often 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 multimodal 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 multimodal survival analysis on Whole Slide Images and Multi-omic data on four cancer datasets from The Cancer Genome Atlas (TCGA). HEALNet achieves state-of-the-art performance compared to other end-to-end trained fusion models, substantially improving over unimodal and multimodal baselines whilst being robust in scenarios with missing modalities.

Navindu Leelarathna, Andrei Margeloiu, Mateja Jamnik et al. · cambridge

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.

Yael Konforti, Mateo Espinosa Zarlenga, Elaf Almahmoud et al.

Despite strong average-case performance, deep learning models often exhibit systematic errors on specific population groups, known as error slices. Identifying these groups and the root causes of their failures is critical for model debugging and bias mitigation. However, existing error Slice Discovery Methods (SDMs) typically generate explanations disconnected from the model's inference process, thus only approximating the underlying error source and may be inaccurate. We address this limitation by leveraging Concept Bottleneck Models (CBMs), whose predictions are directly dependent on human-understandable semantic concepts. Since downstream task failures in CBMs commonly arise from concept mispredictions, concept representations provide a strong candidate for error slice identification, offering fine-grained explanations directly linked to the error source. Building on this insight, we introduce CB-SLICE, a concept-based SDM that groups samples with shared concept prediction failures and identifies the keyword concepts most responsible for each slice's failure mode. Across multiple benchmarks, we show that CB-SLICE outperforms state-of-the-art methods in uncovering well-known biases while providing richer and more faithful explanations of model errors.

Yana Lishkova, Paul Scherer, Steffen Ridderbusch et al.

By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function $\mathcal{L}_d$ which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise.

Mateo Espinosa Zarlenga, Swami Sankaranarayanan, Jerone T. A. Andrews et al.

Deep neural networks trained via empirical risk minimisation often exhibit significant performance disparities across groups, particularly when group and task labels are spuriously correlated (e.g., "grassy background" and "cows"). Existing bias mitigation methods that aim to address this issue often either rely on group labels for training or validation, or require an extensive hyperparameter search. Such data and computational requirements hinder the practical deployment of these methods, especially when datasets are too large to be group-annotated, computational resources are limited, and models are trained through already complex pipelines. In this paper, we propose Targeted Augmentations for Bias Mitigation (TAB), a simple hyperparameter-free framework that leverages the entire training history of a helper model to identify spurious samples, and generate a group-balanced training set from which a robust model can be trained. We show that TAB improves worst-group performance without any group information or model selection, outperforming existing methods while maintaining overall accuracy.

Konstantin Hemker, Zohreh Shams, Mateja Jamnik

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).

Dmitry Kazhdan, Botty Dimanov, Lucie Charlotte Magister et al.

Explainable AI (XAI) underwent a recent surge in research on concept extraction, focusing on extracting human-interpretable concepts from Deep Neural Networks. An important challenge facing concept extraction approaches is the difficulty of interpreting and evaluating discovered concepts, especially for complex tasks such as molecular property prediction. We address this challenge by presenting GCI: a (G)raph (C)oncept (I)nterpretation framework, used for quantitatively measuring alignment between concepts discovered from Graph Neural Networks (GNNs) and their corresponding human interpretations. GCI encodes concept interpretations as functions, which can be used to quantitatively measure the alignment between a given interpretation and concept definition. We demonstrate four applications of GCI: (i) quantitatively evaluating concept extractors, (ii) measuring alignment between concept extractors and human interpretations, (iii) measuring the completeness of interpretations with respect to an end task and (iv) a practical application of GCI to molecular property prediction, in which we demonstrate how to use chemical functional groups to explain GNNs trained on molecular property prediction tasks, and implement interpretations with a 0.76 AUCROC completeness score.

Daniel Raggi, Gem Stapleton, Mateja Jamnik et al.

The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to abstractly encode a wide variety of representations from three core perspectives: syntax, entailment, and their properties. By introducing the concept of a construction space, we are able to encode each of these core components under a single, unifying paradigm. Using our Representational Systems Theory, it becomes possible to structurally transform representations in one system into representations in another. An intrinsic facet of our structural transformation technique is representation selection based on properties that representations possess, such as their relative cognitive effectiveness or structural complexity. A major theoretical barrier to providing general structural transformation techniques is a lack of terminating algorithms. Representational Systems Theory permits the derivation of partial transformations when no terminating algorithm can produce a full transformation. Since Representational Systems Theory provides a universal approach to encoding representational systems, a further key barrier is eliminated: the need to devise system-specific structural transformation algorithms, that are necessary when different systems adopt different formalisation approaches. Consequently, Representational Systems Theory is the first general framework that provides a unified approach to encoding representations, supports representation selection via structural transformations, and has the potential for widespread practical application.

Paul Scherer, Pietro Liò, Mateja Jamnik

In this paper we study the practicality and usefulness of incorporating distributed representations of graphs into models within the context of drug pair scoring. We argue that the real world growth and update cycles of drug pair scoring datasets subvert the limitations of transductive learning associated with distributed representations. Furthermore, we argue that the vocabulary of discrete substructure patterns induced over drug sets is not dramatically large due to the limited set of atom types and constraints on bonding patterns enforced by chemistry. Under this pretext, we explore the effectiveness of distributed representations of the molecular graphs of drugs in drug pair scoring tasks such as drug synergy, polypharmacy, and drug-drug interaction prediction. To achieve this, we present a methodology for learning and incorporating distributed representations of graphs within a unified framework for drug pair scoring. Subsequently, we augment a number of recent and state-of-the-art models to utilise our embeddings. We empirically show that the incorporation of these embeddings improves downstream performance of almost every model across different drug pair scoring tasks, even those the original model was not designed for. We publicly release all of our drug embeddings for the DrugCombDB, DrugComb, DrugbankDDI, and TwoSides datasets.

Johan Commelin, Mateja Jamnik, Rodrigo Ochigame et al.

Artificial intelligence is transforming mathematics at a speed and scale that demand active engagement from the mathematical community. We examine five areas where this transformation is particularly pressing: values, practice, teaching, technology, and ethics. We offer recommendations on safeguarding our intellectual autonomy, rethinking our practice, broadening curricula, building academically oriented infrastructure, and developing shared ethical principles - with the aim of ensuring that the future of mathematics is shaped by the community itself.

Oscar Hill, Mateo Espinosa Zarlenga, Mateja Jamnik

Although concept-based models promise interpretability by explaining predictions with human-understandable concepts, they typically rely on exhaustive annotations and treat concepts as flat and independent. To circumvent this, recent work has introduced Hierarchical Concept Embedding Models (HiCEMs) to explicitly model concept relationships, and Concept Splitting to discover sub-concepts using only coarse annotations. However, both HiCEMs and Concept Splitting are restricted to shallow hierarchies. We overcome this limitation with Multi-Level Concept Splitting (MLCS), which discovers multi-level concept hierarchies from only top-level supervision, and Deep-HiCEMs, an architecture that represents these discovered hierarchies and enables interventions at multiple levels of abstraction. Experiments across multiple datasets show that MLCS discovers human-interpretable concepts absent during training and that Deep-HiCEMs maintain high accuracy while supporting test-time concept interventions that can improve task performance.

Tiansi Dong, Henry He, Pietro Liò et al.

This paper compares three methodological categories of neural reasoning: LLM reasoning, supervised learning-based reasoning, and explicit model-based reasoning. LLMs remain unreliable and struggle with simple decision-making that animals can master without extensive corpora training. Through disjunctive syllogistic reasoning testing, we show that reasoning via supervised learning is less appealing than reasoning via explicit model construction. Concretely, we show that an Euler Net trained to achieve 100.00% in classic syllogistic reasoning can be trained to reach 100.00% accuracy in disjunctive syllogistic reasoning. However, the retrained Euler Net suffers severely from catastrophic forgetting (its performance drops to 6.25% on already-learned classic syllogistic reasoning), and its reasoning competence is limited to the pattern level. We propose a new version of Sphere Neural Networks that embeds concepts as circles on the surface of an n-dimensional sphere. These Sphere Neural Networks enable the representation of the negation operator via complement circles and achieve reliable decision-making by filtering out illogical statements that form unsatisfiable circular configurations. We demonstrate that the Sphere Neural Network can master 16 syllogistic reasoning tasks, including rigorous disjunctive syllogistic reasoning, while preserving the rigour of classical syllogistic reasoning. We conclude that neural reasoning with explicit model construction is the most reliable among the three methodological categories of neural reasoning.

Shubhra Mishra, Yuka Machino, Gabriel Poesia et al.

The evolution of mathematics has been guided in part by interestingness. From researchers choosing which problems to tackle next, to students deciding which ones to engage with, people's choices are often guided by judgments about how interesting or challenging problems are likely to be. As AI systems, such as LLMs, increasingly participate in mathematics with people -- whether for advanced research or education -- it becomes important to understand how well their judgments align with human ones. Our work examines this alignment through two empirical studies of human and LLM assessment of mathematical interestingness and difficulty, spanning a range of mathematical experience. We study two groups: participants from a crowdsourcing platform and International Math Olympiad competitors. We show that while many LLMs appear to broadly agree with human notions of interestingness, they mostly do not capture the distribution observed in human judgments. Moreover, most LLMs only somewhat align with why humans find certain math problems interesting, showing weak correlation with human-selected interestingness rationales. Together, our findings highlight both the promises and limitations of current LLMs in capturing human interestingness judgments for mathematical AI thought partnerships.

Shea Cardozo, Gabriel Islas Montero, Dmitry Kazhdan et al.

Recent work has suggested post-hoc explainers might be ineffective for detecting spurious correlations in Deep Neural Networks (DNNs). However, we show there are serious weaknesses with the existing evaluation frameworks for this setting. Previously proposed metrics are extremely difficult to interpret and are not directly comparable between explainer methods. To alleviate these constraints, we propose a new evaluation methodology, Explainer Divergence Scores (EDS), grounded in an information theory approach to evaluate explainers. EDS is easy to interpret and naturally comparable across explainers. We use our methodology to compare the detection performance of three different explainers - feature attribution methods, influential examples and concept extraction, on two different image datasets. We discover post-hoc explainers often contain substantial information about a DNN's dependence on spurious artifacts, but in ways often imperceptible to human users. This suggests the need for new techniques that can use this information to better detect a DNN's reliance on spurious correlations.

Yuval Shalev, Zifeng Ding, Mateja Jamnik

Language agents can adapt from experience in interactive environments, but current reflection-based methods can only self-correct within a single task instance. Whether such experience can be distilled into reusable lessons that improve performance on future unseen tasks remains unclear. We address this problem by introducing the In-context Training (ICT) task, a framework for evaluating cross-task self-improvement in language agents. In ICT, a reflector model observes trajectories collected by an actor model and generates system prompts intended to improve the actor's performance on future unseen tasks. We then propose an RL-based training pipeline for learning such reflections directly from experience, without human-provided examples. Across ALFWorld and MiniHack, our trained reflectors outperform an untrained baseline on most held-out task families, showing that the ability to learn from experience can itself be learned. In some cases, we observe generalisation beyond the benchmark on which the reflector was trained, to substantially different environments. Finally, we introduce MetaGym, a generic Python library for constructing meta-environments, enabling future research on self-improving language agents.

Andy Lo, Albert Q. Jiang, Wenda Li et al. · cambridge

Ontologies are useful for automatic machine processing of domain knowledge as they represent it in a structured format. Yet, constructing ontologies requires substantial manual effort. To automate part of this process, large language models (LLMs) have been applied to solve various subtasks of ontology learning. However, this partial ontology learning does not capture the interactions between subtasks. We address this gap by introducing OLLM, a general and scalable method for building the taxonomic backbone of an ontology from scratch. Rather than focusing on subtasks, like individual relations between entities, we model entire subcomponents of the target ontology by finetuning an LLM with a custom regulariser that reduces overfitting on high-frequency concepts. We introduce a novel suite of metrics for evaluating the quality of the generated ontology by measuring its semantic and structural similarity to the ground truth. In contrast to standard metrics, our metrics use deep learning techniques to define more robust distance measures between graphs. Both our quantitative and qualitative results on Wikipedia show that OLLM outperforms subtask composition methods, producing more semantically accurate ontologies while maintaining structural integrity. We further demonstrate that our model can be effectively adapted to new domains, like arXiv, needing only a small number of training examples. Our source code and datasets are available at https://github.com/andylolu2/ollm.

Laura Wenderoth, Konstantin Hemker, Nikola Simidjievski et al.

Integrating AI in healthcare can greatly improve patient care and system efficiency. However, the lack of explainability in AI systems (XAI) hinders their clinical adoption, especially in multimodal settings that use increasingly complex model architectures. Most existing XAI methods focus on unimodal models, which fail to capture cross-modal interactions crucial for understanding the combined impact of multiple data sources. Existing methods for quantifying cross-modal interactions are limited to two modalities, rely on labelled data, and depend on model performance. This is problematic in healthcare, where XAI must handle multiple data sources and provide individualised explanations. This paper introduces InterSHAP, a cross-modal interaction score that addresses the limitations of existing approaches. InterSHAP uses the Shapley interaction index to precisely separate and quantify the contributions of the individual modalities and their interactions without approximations. By integrating an open-source implementation with the SHAP package, we enhance reproducibility and ease of use. We show that InterSHAP accurately measures the presence of cross-modal interactions, can handle multiple modalities, and provides detailed explanations at a local level for individual samples. Furthermore, we apply InterSHAP to multimodal medical datasets and demonstrate its applicability for individualised explanations.

Xiangjian Jiang, Nikola Simidjievski, Mateja Jamnik

Heterogeneous tabular data poses unique challenges in generative modelling due to its fundamentally different underlying data structure compared to homogeneous modalities, such as images and text. Although previous research has sought to adapt the successes of generative modelling in homogeneous modalities to the tabular domain, defining an effective generator for tabular data remains an open problem. One major reason is that the evaluation criteria inherited from other modalities often fail to adequately assess whether tabular generative models effectively capture or utilise the unique structural information encoded in tabular data. In this paper, we carefully examine the limitations of the prevailing evaluation framework and introduce $\textbf{TabStruct}$, a novel evaluation benchmark that positions structural fidelity as a core evaluation dimension. Specifically, TabStruct evaluates the alignment of causal structures in real and synthetic data, providing a direct measure of how effectively tabular generative models learn the structure of tabular data. Through extensive experiments using generators from eight categories on seven datasets with expert-validated causal graphical structures, we show that structural fidelity offers a task-independent, domain-agnostic evaluation dimension. Our findings highlight the importance of tabular data structure and offer practical guidance for developing more effective and robust tabular generative models. Code is available at https://github.com/SilenceX12138/TabStruct.

Xiangjian Jiang, Mingxuan Liu, Nikola Simidjievski et al.

Generative modelling is a demanding test of foundation models, because it requires robust, holistic representation learning for a given data modality, rather than optimisation for a supervised prediction target alone. While recent work on tabular foundation models has achieved remarkable progress in predictive modelling, generative tabular foundation models remain underexplored. Existing tabular foundation generators, in particular, have not yet consistently matched strong dataset-specific generators in synthetic data quality. A key reason is their misalignment with the distinctive causal structural prior of heterogeneous tabular data. In this paper, we address this gap by introducing a novel tabular foundation model, \textbf{TabFORGE}, built on pretrained \textbf{Tab}ular \textbf{FO}undational \textbf{R}epresentations for \textbf{GE}neration. TabFORGE is designed to utilise the implicitly learned causal information underlying diverse tabular datasets in a unified latent space induced by a pretrained causality-aware feature encoder. It further decouples latent modelling from decoding through a two-stage design: we first pretrain a score-based diffusion transformer, and then pretrain a denoising-aligned decoder using the denoised latent embeddings. This design elegantly mitigates the distribution shifts in latent embeddings that typically arise between training and inference. We evaluate TabFORGE comprehensively against 22 benchmark methods on 45 real-world datasets. Our results show that TabFORGE effectively learns and leverages generalisable tabular representations, enabling efficient generation of high-quality synthetic tabular data, particularly with strong structural fidelity.

Xiangjian Jiang, Nikola Simidjievski, Mateja Jamnik

Evaluating tabular generators remains a challenging problem, as the unique causal structural prior of heterogeneous tabular data does not lend itself to intuitive human inspection. Recent work has introduced structural fidelity as a tabular-specific evaluation dimension to assess whether synthetic data complies with the causal structures of real data. However, existing benchmarks often neglect the interplay between structural fidelity and conventional evaluation dimensions, thus failing to provide a holistic understanding of model performance. Moreover, they are typically limited to toy datasets, as quantifying existing structural fidelity metrics requires access to ground-truth causal structures, which are rarely available for real-world datasets. In this paper, we propose a novel evaluation framework that jointly considers structural fidelity and conventional evaluation dimensions. We introduce a new evaluation metric, $\textbf{global utility}$, which enables the assessment of structural fidelity even in the absence of ground-truth causal structures. In addition, we present $\textbf{TabStruct}$, a comprehensive evaluation benchmark offering large-scale quantitative analysis on 13 tabular generators from nine distinct categories, across 29 datasets. Our results demonstrate that global utility provides a task-independent, domain-agnostic lens for tabular generator performance. We release the TabStruct benchmark suite, including all datasets, evaluation pipelines, and raw results. Code is available at https://github.com/SilenceX12138/TabStruct.

Pietro Barbiero, Mateo Espinosa Zarlenga, Alberto Termine et al.

We argue that existing definitions of interpretability are not actionable in that they fail to inform users about general, sound, and robust interpretable model design. This makes current interpretability research fundamentally ill-posed. To address this issue, we propose a definition of interpretability that is general, simple, and subsumes existing informal notions within the interpretable AI community. We show that our definition is actionable, as it directly reveals the foundational properties, underlying assumptions, principles, data structures, and architectural features necessary for designing interpretable models. Building on this, we propose a general blueprint for designing interpretable models and introduce the first open-sourced library with native support for interpretable data structures and processes.

Andrei Margeloiu, Adrián Bazaga, Nikola Simidjievski et al.

Tabular data is prevalent in many critical domains, yet it is often challenging to acquire in large quantities. This scarcity usually results in poor performance of machine learning models on such data. Data augmentation, a common strategy for performance improvement in vision and language tasks, typically underperforms for tabular data due to the lack of explicit symmetries in the input space. To overcome this challenge, we introduce TabMDA, a novel method for manifold data augmentation on tabular data. This method utilises a pre-trained in-context model, such as TabPFN, to map the data into an embedding space. TabMDA performs label-invariant transformations by encoding the data multiple times with varied contexts. This process explores the learned embedding space of the underlying in-context models, thereby enlarging the training dataset. TabMDA is a training-free method, making it applicable to any classifier. We evaluate TabMDA on five standard classifiers and observe significant performance improvements across various tabular datasets. Our results demonstrate that TabMDA provides an effective way to leverage information from pre-trained in-context models to enhance the performance of downstream classifiers. Code is available at https://github.com/AdrianBZG/TabMDA.

Mateo Espinosa Zarlenga, Zohreh Shams, Mateja Jamnik

In recent years, there has been significant work on increasing both interpretability and debuggability of a Deep Neural Network (DNN) by extracting a rule-based model that approximates its decision boundary. Nevertheless, current DNN rule extraction methods that consider a DNN's latent space when extracting rules, known as decompositional algorithms, are either restricted to single-layer DNNs or intractable as the size of the DNN or data grows. In this paper, we address these limitations by introducing ECLAIRE, a novel polynomial-time rule extraction algorithm capable of scaling to both large DNN architectures and large training datasets. We evaluate ECLAIRE on a wide variety of tasks, ranging from breast cancer prognosis to particle detection, and show that it consistently extracts more accurate and comprehensible rule sets than the current state-of-the-art methods while using orders of magnitude less computational resources. We make all of our methods available, including a rule set visualisation interface, through the open-source REMIX library (https://github.com/mateoespinosa/remix).

Haoyan Luo, Mateo Espinosa Zarlenga, Mateja Jamnik

Activation steering controls LLM behaviour towards target behaviour by intervening in internal representations, yet it often degrades reasoning and retrieval performance. We argue that a primary cause of this trade-off is attention rerouting: steering vectors alter query-key matching, shifting attention away from contextually important tokens toward less informative ones. To address this, we propose Steering via Key-Orthogonal Projections (SKOP), a steering method that constrains harmful attention rerouting without eliminating steering efficacy. SKOP achieves this by preserving attention patterns on a small set of focus tokens the model relies on for reasoning and retrieval, while allowing redistribution among less critical tail tokens. Across multiple steering benchmarks, we show that SKOP achieves the best joint steering-utility trade-off, reducing utility degradation by 5-7x while retaining over 95% of vanilla steering efficacy. Our results further suggest that, in long-context retrieval settings where vanilla steering approaches are ineffective, SKOP can maintain robust performance by avoiding attention rerouting.

Naveen Raman, Mateo Espinosa Zarlenga, Juyeon Heo et al.

Concept-based explainability methods use human-understandable intermediaries to produce explanations for machine learning models. These methods assume concept predictions can help understand a model's internal reasoning. In this work, we assess the degree to which such an assumption is true by analyzing whether concept predictors leverage "relevant" features to make predictions, a term we call locality. Concept-based models that fail to respect localities also fail to be explainable because concept predictions are based on spurious features, making the interpretation of the concept predictions vacuous. To assess whether concept-based models respect localities, we construct and use three metrics to characterize when models respect localities, complementing our analysis with theoretical results. Each of our metrics captures a different notion of perturbation and assess whether perturbing "irrelevant" features impacts the predictions made by a concept predictors. We find that many concept-based models used in practice fail to respect localities because concept predictors cannot always clearly distinguish distinct concepts. Based on these findings, we propose suggestions for alleviating this issue.

Boshko Koloski, Andrei Margeloiu, Xiangjian Jiang et al.

Tabular deep-learning methods require embedding numerical and categorical input features into high-dimensional spaces before processing them. Existing methods deal with this heterogeneous nature of tabular data by employing separate type-specific encoding approaches. This limits the cross-table transfer potential and the exploitation of pre-trained knowledge. We propose a novel approach that first transforms tabular data into text, and then leverages pre-trained representations from LLMs to encode this data, resulting in a plug-and-play solution to improv ing deep-learning tabular methods. We demonstrate that our approach improves accuracy over competitive models, such as MLP, ResNet and FT-Transformer, by validating on seven classification datasets.

Mateo Espinosa Zarlenga, Gabriele Dominici, Pietro Barbiero et al.

In this paper, we investigate how concept-based models (CMs) respond to out-of-distribution (OOD) inputs. CMs are interpretable neural architectures that first predict a set of high-level concepts (e.g., stripes, black) and then predict a task label from those concepts. In particular, we study the impact of concept interventions (i.e., operations where a human expert corrects a CM's mispredicted concepts at test time) on CMs' task predictions when inputs are OOD. Our analysis reveals a weakness in current state-of-the-art CMs, which we term leakage poisoning, that prevents them from properly improving their accuracy when intervened on for OOD inputs. To address this, we introduce MixCEM, a new CM that learns to dynamically exploit leaked information missing from its concepts only when this information is in-distribution. Our results across tasks with and without complete sets of concept annotations demonstrate that MixCEMs outperform strong baselines by significantly improving their accuracy for both in-distribution and OOD samples in the presence and absence of concept interventions.

Zak Buzzard, Konstantin Hemker, Nikola Simidjievski et al.

Computational analysis of whole slide images (WSIs) has seen significant research progress in recent years, with applications ranging across important diagnostic and prognostic tasks such as survival or cancer subtype prediction. Many state-of-the-art models process the entire slide - which may be as large as $150,000 \times 150,000$ pixels - as a bag of many patches, the size of which necessitates computationally cheap feature aggregation methods. However, a large proportion of these patches are uninformative, such as those containing only healthy or adipose tissue, adding significant noise and size to the bag. We propose Pathology Transformer with Hierarchical Selection (PATHS), a novel top-down method for hierarchical weakly supervised representation learning on slide-level tasks in computational pathology. PATHS is inspired by the cross-magnification manner in which a human pathologist examines a slide, recursively filtering patches at each magnification level to a small subset relevant to the diagnosis. Our method overcomes the complications of processing the entire slide, enabling quadratic self-attention and providing a simple interpretable measure of region importance. We apply PATHS to five datasets of The Cancer Genome Atlas (TCGA), and achieve superior performance on slide-level prediction tasks when compared to previous methods, despite processing only a small proportion of the slide.

Pietro Barbiero, Mateo Espinosa Zarlenga, Francesco Giannini et al.

This paper argues that interpretability research in Artificial Intelligence is fundamentally ill-posed as existing definitions of interpretability are not *actionable*: they fail to provide formal principles from which concrete modelling and inferential rules can be derived. We posit that for a definition of interpretability to be actionable, it must be given in terms of *symmetries*. We hypothesise that four symmetries suffice to (i) motivate core interpretability properties, (ii) characterize the class of interpretable models, and (iii) derive a unified formulation of interpretable inference (e.g., alignment, interventions, and counterfactuals) as a form of Bayesian inversion.

Konstantin Hemker, Andrew H. Song, Cristina Almagro-Pérez et al.

Spatial transcriptomics (ST) provides spatially resolved measurements of gene expression, enabling characterization of the molecular landscape of human tissue beyond histological assessment as well as localized readouts that can be aligned with morphology. Concurrently, the success of multimodal foundation models that integrate vision with complementary modalities suggests that morphomolecular coupling between local expression and morphology can be systematically used to improve histological representations themselves. We introduce Spatial Expression-Aligned Learning (SEAL), a vision-omics self-supervised learning framework that infuses localized molecular information into pathology vision encoders. Rather than training new encoders from scratch, SEAL is designed as a parameter-efficient vision-omics finetuning method that can be flexibly applied to widely used pathology foundation models. We instantiate SEAL by training on over 700,000 paired gene expression spot-tissue region examples spanning tumor and normal samples from 14 organs. Tested across 38 slide-level and 15 patch-level downstream tasks, SEAL provides a drop-in replacement for pathology foundation models that consistently improves performance over widely used vision-only and ST prediction baselines on slide-level molecular status, pathway activity, and treatment response prediction, as well as patch-level gene expression prediction tasks. Additionally, SEAL encoders exhibit robust domain generalization on out-of-distribution evaluations and enable new cross-modal capabilities such as gene-to-image retrieval. Our work proposes a general framework for ST-guided finetuning of pathology foundation models, showing that augmenting existing models with localized molecular supervision is an effective and practical step for improving visual representations and expanding their cross-modal utility.

Daniel Raggi, Gem Stapleton, Mateja Jamnik et al.

Humans use representations flexibly. We draw diagrams, change representations and exploit creative analogies across different domains. We want to harness this kind of power and endow machines with it to make them more compatible with human use. Previously we developed Representational Systems Theory (RST) to study the structure and transformations of representations. In this paper we present Oruga (caterpillar in Spanish; a symbol of transformation), an implementation of various aspects of RST. Oruga consists of a core of data structures corresponding to concepts in RST, a language for communicating with the core, and an engine for producing transformations using a method we call structure transfer. In this paper we present an overview of the core and language of Oruga, with a brief example of the kind of transformation that structure transfer can execute.

Daniel Raggi, Gem Stapleton, Mateja Jamnik et al.

Representation choice is of fundamental importance to our ability to communicate and reason effectively. A major unsolved problem, addressed in this paper, is how to devise representational-system (RS) agnostic techniques that drive representation transformation and choice. We present a novel calculus, called structure transfer, that enables representation transformation across diverse RSs. Specifically, given a source representation drawn from a source RS, the rules of structure transfer allow us to generate a target representation for a target RS. The generality of structure transfer comes in part from its ability to ensure that the source representation and the generated target representation satisfy any specified relation (such as semantic equivalence). This is done by exploiting schemas, which encode knowledge about RSs. Specifically, schemas can express preservation of information across relations between any pair of RSs, and this knowledge is used by structure transfer to derive a structure for the target representation which ensures that the desired relation holds. We formalise this using Representational Systems Theory, building on the key concept of a construction space. The abstract nature of construction spaces grants them the generality to model RSs of diverse kinds, including formal languages, geometric figures and diagrams, as well as informal notations. Consequently, structure transfer is a system-agnostic calculus that can be used to identify alternative representations in a wide range of practical settings.

Urška Matjašec, Nikola Simidjievski, Mateja Jamnik

Tree-based models are often robust to uninformative features and can accurately capture non-smooth, complex decision boundaries. Consequently, they often outperform neural network-based models on tabular datasets at a significantly lower computational cost. Nevertheless, the capability of traditional tree-based ensembles to express complex relationships efficiently is limited by using a single feature to make splits. To improve the efficiency and expressiveness of tree-based methods, we propose Random Oblique Fast Interpretable Greedy-Tree Sums (RO-FIGS). RO-FIGS builds on Fast Interpretable Greedy-Tree Sums, and extends it by learning trees with oblique or multivariate splits, where each split consists of a linear combination learnt from random subsets of features. This helps uncover interactions between features and improves performance. The proposed method is suitable for tabular datasets with both numerical and categorical features. We evaluate RO-FIGS on 22 real-world tabular datasets, demonstrating superior performance and much smaller models over other tree- and neural network-based methods. Additionally, we analyse their splits to reveal valuable insights into feature interactions, enriching the information learnt from SHAP summary plots, and thereby demonstrating the enhanced interpretability of RO-FIGS models. The proposed method is well-suited for applications, where balance between accuracy and interpretability is essential.

Alicja Ziarko, Albert Q. Jiang, Bartosz Piotrowski et al.

Text embeddings are essential for many tasks, such as document retrieval, clustering, and semantic similarity assessment. In this paper, we study how to contrastively train text embedding models in a compute-optimal fashion, given a suite of pre-trained decoder-only language models. Our innovation is an algorithm that produces optimal configurations of model sizes, data quantities, and fine-tuning methods for text-embedding models at different computational budget levels. The resulting recipe, which we obtain through extensive experiments, can be used by practitioners to make informed design choices for their embedding models. Specifically, our findings suggest that full fine-tuning and low-rank adaptation fine-tuning produce optimal models at lower and higher computational budgets respectively.

Tiansi Dong, Mateja Jamnik, Pietro Liò

The success of Large Language Models (LLMs), e.g., ChatGPT, is witnessed by their planetary popularity, their capability of human-like communication, and also by their steadily improved reasoning performance. However, it remains unclear whether LLMs reason. It is an open problem how traditional neural networks can be qualitatively extended to go beyond the statistic paradigm and achieve high-level cognition. Here, we present a novel qualitative extension by generalising computational building blocks from vectors to spheres. We propose Sphere Neural Networks (SphNNs) for human-like reasoning through model construction and inspection, and develop SphNN for syllogistic reasoning, a microcosm of human rationality. SphNN is a hierarchical neuro-symbolic Kolmogorov-Arnold geometric GNN, and uses a neuro-symbolic transition map of neighbourhood spatial relations to transform the current sphere configuration towards the target. SphNN is the first neural model that can determine the validity of long-chained syllogistic reasoning in one epoch without training data, with the worst computational complexity of O(N). SphNN can evolve into various types of reasoning, such as spatio-temporal reasoning, logical reasoning with negation and disjunction, event reasoning, neuro-symbolic unification, and humour understanding (the highest level of cognition). All these suggest a new kind of Herbert A. Simon's scissors with two neural blades. SphNNs will tremendously enhance interdisciplinary collaborations to develop the two neural blades and realise deterministic neural reasoning and human-bounded rationality and elevate LLMs to reliable psychological AI. This work suggests that the non-zero radii of spheres are the missing components that prevent traditional deep-learning systems from reaching the realm of rational reasoning and cause LLMs to be trapped in the swamp of hallucination.