Abstract Concept Modelling in Conceptual Spaces: A Study on Chess Strategies

We present a conceptual space framework for modelling abstract concepts that unfold over time, demonstrated through a chess-based proof-of-concept. Strategy concepts, such as attack or sacrifice, are represented as geometric regions across interpretable quality dimensions, with chess games instantiated and analysed as trajectories whose directional movement toward regions enables recognition of intended strategies. This approach also supports dual-perspective modelling, capturing how players interpret identical situations differently. Our implementation demonstrates the feasibility of trajectory-based concept recognition, with movement patterns aligning with expert commentary. This work explores extending the conceptual spaces theory to temporally realised, goal-directed concepts. The approach establishes a foundation for broader applications involving sequential decision-making and supports integration with knowledge evolution mechanisms for learning and refining abstract concepts over time.

Trapped in the past? Disentangling fluid and crystallized intelligence of large language models using chess

Large Language Models (LLMs) exhibit remarkable capabilities, yet it remains unclear to what extent these reflect sophisticated recall (crystallized intelligence) or reasoning ability (fluid intelligence). We introduce chess as a controlled testbed for disentangling these faculties. Leveraging the game's structure and scalable engine evaluations, we construct a taxonomy of positions varying in training corpus proximity--ranging from common states solvable by memorization to novel ones requiring first-principles reasoning. We systematically evaluate multiple GPT generations under varying reasoning intensities. Our analysis reveals a clear gradient: performance consistently degrades as fluid intelligence demands increase. Notably, in out-of-distribution tasks, performance collapses to random levels. While newer models improve, progress slows significantly for tasks outside the training distribution. Furthermore, while reasoning-augmented inference improves performance, its marginal benefit per token decreases with distributional proximity. These results suggest current architectures remain limited in systematic generalization, highlighting the need for mechanisms beyond scale to achieve robust fluid intelligence.

Quantum King-Ring Domination in Chess: A QAOA Approach

The Quantum Approximate Optimization Algorithm (QAOA) is extensively benchmarked on synthetic random instances such as MaxCut, TSP, and SAT problems, but these lack semantic structure and human interpretability, offering limited insight into performance on real-world problems with meaningful constraints. We introduce Quantum King-Ring Domination (QKRD), a NISQ-scale benchmark derived from chess tactical positions that provides 5,000 structured instances with one-hot constraints, spatial locality, and 10--40 qubit scale. The benchmark pairs human-interpretable coverage metrics with intrinsic validation against classical heuristics, enabling algorithmic conclusions without external oracles. Using QKRD, we systematically evaluate QAOA design choices and find that constraint-preserving mixers (XY, domain-wall) converge approximately 13 steps faster than standard mixers (p<10^{-7}, d\approx0.5) while eliminating penalty tuning, warm-start strategies reduce convergence by 45 steps (p<10^{-127}, d=3.35) with energy improvements exceeding d=8, and Conditional Value-at-Risk (CVaR) optimization yields an informative negative result with worse energy (p<10^{-40}, d=1.21) and no coverage benefit. Intrinsic validation shows QAOA outperforms greedy heuristics by 12.6\% and random selection by 80.1\%. Our results demonstrate that structured benchmarks reveal advantages of problem-informed QAOA techniques obscured in random instances. We release all code, data, and experimental artifacts for reproducible NISQ algorithm research.

Automatic Adaptation to Concept Complexity and Subjective Natural Concepts: A Cognitive Model based on Chunking

A key issue in cognitive science concerns the fundamental psychological processes that underlie the formation and retrieval of multiple types of concepts in short-term and long-term memory (STM and LTM, respectively). We propose that chunking mechanisms play an essential role and show how the CogAct computational model grounds concept learning in fundamental cognitive processes and structures (such as chunking, attention, STM and LTM). First are the in-principle demonstrations, with CogAct automatically adapting to learn a range of categories from simple logical functions, to artificial categories, to natural raw (as opposed to natural pre-processed) concepts in the dissimilar domains of literature, chess and music. This kind of adaptive learning is difficult for most other psychological models, e.g., with cognitive models stopping at modelling artificial categories and (non-GPT) models based on deep learning requiring task-specific changes to the architecture. Secondly, we offer novel ways of designing human benchmarks for concept learning experiments and simulations accounting for subjectivity, ways to control for individual human experiences, all while keeping to real-life complex categories. We ground CogAct in simulations of subjective conceptual spaces of individual human participants, capturing humans subjective judgements in music, with the models learning from raw music score data without bootstrapping to pre-built knowledge structures. The CogAct simulations are compared to those obtained by a deep-learning model. These findings integrate concept learning and adaptation to complexity into the broader theories of cognitive psychology. Our approach may also be used in psychological applications that move away from modelling the average participant and towards capturing subjective concept space.

Emergent World Beliefs: Exploring Transformers in Stochastic Games

Transformer-based large language models (LLMs) have demonstrated strong reasoning abilities across diverse fields, from solving programming challenges to competing in strategy-intensive games such as chess. Prior work has shown that LLMs can develop emergent world models in games of perfect information, where internal representations correspond to latent states of the environment. In this paper, we extend this line of investigation to domains of incomplete information, focusing on poker as a canonical partially observable Markov decision process (POMDP). We pretrain a GPT-style model on Poker Hand History (PHH) data and probe its internal activations. Our results demonstrate that the model learns both deterministic structure, such as hand ranks, and stochastic features, such as equity, without explicit instruction. Furthermore, by using primarily nonlinear probes, we demonstrated that these representations are decodeable and correlate with theoretical belief states, suggesting that LLMs are learning their own representation of the stochastic environment of Texas Hold'em Poker.

Beyond Accuracy: A Geometric Stability Analysis of Large Language Models in Chess Evaluation

The evaluation of Large Language Models (LLMs) in complex reasoning domains typically relies on performance alignment with ground-truth oracles. In the domain of chess, this standard manifests as accuracy benchmarks against strong engines like Stockfish. However, high scalar accuracy does not necessarily imply robust conceptual understanding. This paper argues that standard accuracy metrics fail to distinguish between genuine geometric reasoning and the superficial memorization of canonical board states. To address this gap, we propose a Geometric Stability Framework, a novel evaluation methodology that rigorously tests model consistency under invariant transformations-including board rotation, mirror symmetry, color inversion, and format conversion. We applied this framework to a comparative analysis of six state-of-the-art LLMs including GPT-5.1, Claude Sonnet 4.5, and Kimi K2 Turbo, utilizing a dataset of approximately 3,000 positions. Our results reveal a significant Accuracy-Stability Paradox. While models such as GPT-5.1 achieve near-optimal accuracy on standard positions, they exhibit catastrophic degradation under geometric perturbation, specifically in rotation tasks where error rates surge by over 600%. This disparity suggests a reliance on pattern matching over abstract spatial logic. Conversely, Claude Sonnet 4.5 and Kimi K2 Turbo demonstrate superior dual robustness, maintaining high consistency across all transformation axes. Furthermore, we analyze the trade-off between helpfulness and safety, identifying Gemini 2.5 Flash as the leader in illegal state rejection (96.0%). We conclude that geometric stability provides an orthogonal and essential metric for AI evaluation, offering a necessary proxy for disentangling reasoning capabilities from data contamination and overfitting in large-scale models.

Liquid Reasoning Transformers: A Sudoku-Based Prototype for Chess-Scale Algorithmic Tasks

The Liquid Reasoning Transformer (LRT) is a transformer architecture designed for inference with adaptive depths using iterative changes, discard-based correction, and a learned stopping mechanism. Instead of relying on a single feedforward pass, the model updates a recurrent reasoning token across multiple internal steps, allowing it to correct early errors and allocate computation based on input difficulty. We evaluate the LRT on Sudoku as a controlled testbed for structured reasoning and show that it achieves strong performance, reaching 98.68% digit accuracy and 36.30% full-puzzle accuracy without using symbolic rules or search. Analyzing internal patterns shows that the discard and stop gates play different, important roles in stabilizing inferences and adjusting computational depth. We discuss how these mechanisms extend naturally to chess-scale reasoning tasks and outline extensions for multi-token reasoning and larger domains.

Machine Learning: Progress and Prospects

This Inaugural Lecture was given at Royal Holloway University of London in 1996. It covers an introduction to machine learning and describes various theoretical advances and practical projects in the field. The Lecture here is presented in its original format, but a few remarks have been added in 2025 to reflect recent developments, and the list of references has been updated to enhance the convenience and accuracy for readers. When did machine learning start? Maybe a good starting point is 1949, when Claude Shannon proposed a learning algorithm for chess-playing programs. Or maybe we should go back to the 1930s when Ronald Fisher developed discriminant analysis - a type of learning where the problem is to construct a decision rule that separates two types of vectors. Or could it be the 18th century when David Hume discussed the idea of induction? Or the 14th century, when William of Ockham formulated the principle of "simplicity" known as "Ockham's razor" (Ockham, by the way, is a small village not far from Royal Holloway). Or it may be that, like almost everything else in Western civilisation and culture, the origin of these ideas lies in the Mediterranean. After all, it was Aristotle who said that "we learn some things only by doing things". The field of machine learning has been greatly influenced by other disciplines and the subject is in itself not a very homogeneous discipline, but includes separate, overlapping subfields. There are many parallel lines of research in ML: inductive learning, neural networks, clustering, and theories of learning. They are all part of the more general field of machine learning.

CVChess: A Deep Learning Framework for Converting Chessboard Images to Forsyth-Edwards Notation

Chess has experienced a large increase in viewership since the pandemic, driven largely by the accessibility of online learning platforms. However, no equivalent assistance exists for physical chess games, creating a divide between analog and digital chess experiences. This paper presents CVChess, a deep learning framework for converting chessboard images to Forsyth-Edwards Notation (FEN), which is later input into online chess engines to provide you with the best next move. Our approach employs a convolutional neural network (CNN) with residual layers to perform piece recognition from smartphone camera images. The system processes RGB images of a physical chess board through a multistep process: image preprocessing using the Hough Line Transform for edge detection, projective transform to achieve a top-down board alignment, segmentation into 64 individual squares, and piece classification into 13 classes (6 unique white pieces, 6 unique black pieces and an empty square) using the residual CNN. Residual connections help retain low-level visual features while enabling deeper feature extraction, improving accuracy and stability during training. We train and evaluate our model using the Chess Recognition Dataset (ChessReD), containing 10,800 annotated smartphone images captured under diverse lighting conditions and angles. The resulting classifications are encoded as an FEN string, which can be fed into a chess engine to generate the most optimal move

Latent Planning via Embedding Arithmetic: A Contrastive Approach to Strategic Reasoning

Planning in high-dimensional decision spaces is increasingly being studied through the lens of learned representations. Rather than training policies or value heads, we investigate whether planning can be carried out directly in an evaluation-aligned embedding space. We introduce SOLIS, which learns such a space using supervised contrastive learning. In this representation, outcome similarity is captured by proximity, and a single global advantage vector orients the space from losing to winning regions. Candidate actions are then ranked according to their alignment with this direction, reducing planning to vector operations in latent space. We demonstrate this approach in chess, where SOLIS uses only a shallow search guided by the learned embedding to reach competitive strength under constrained conditions. More broadly, our results suggest that evaluation-aligned latent planning offers a lightweight alternative to traditional dynamics models or policy learning.