Adnan Mahmud

In 1963, Christopher Hooley proved that the number of positive integers up to x expressible as a sum of two positive cubes in more than one way grows like x^(2/3), up to logarithmic corrections. The proof is a delicate sieve argument that took real analytic effort. Sixty years on, verifying the theorem empirically across nine orders of magnitude takes twenty seconds of single-machine compute. This blog walks through what that shift actually means for the taxicab problem, produces the figures that make Hooley's asymptotic visible, revisits the classical modular obstructions that determine which integers can be sums of two cubes at all, and ends with an honest account of where the exercise stops paying off and what would be required to push past that boundary into the elliptic-curve territory beyond.

Foundation models have reshaped how we build general-purpose AI systems, but what would it mean to have one for Chain Event Graphs, a class of graphical statistical models developed at Warwick for representing asymmetric and context-specific conditional independence? The phrase does not map cleanly onto the standard concept, so this post works through four possible interpretations, narrows to the two most useful, and proposes a hybrid: a text-to-CEG model built by fine-tuning an existing LLM to output CEGs from natural-language descriptions of processes. The post then compares commercial and academic delivery routes for such a project side by side, grounds the compute and cost estimates in published fine-tuning benchmarks, and proposes a proof-of-concept phase to de-risk the scientific bet before scaling up, all within the University of Edinburgh's HPC ecosystem.

Pischedda, Gilli, and Gilli's recent (-ish) paper in the Journal of Conflict Resolution argues that the absolute lethality of insurgents' small arms, rather than their firepower relative to the government's, is a significant predictor of counterinsurgency outcomes. The paper is careful, robustly tested, and fills a genuine gap in the literature. It also carries identifiable statistical problems: an index built on untested equal-weighting assumptions, a secular trend that categorical controls cannot fully resolve, average effects that conceal theoretically important heterogeneity, and no out-of-sample validation. This post explains both what the paper gets right and where it falls short: a critique that ultimately sketches a predictive research programme designed to address each problem in sequence.

For a century, air dominance belonged to whichever nation fielded the most advanced crewed aircraft. That logic is being rewritten in real time, and not only on the battlefields of Ukraine and the Red Sea. This essay synthesises recent output from McKinsey, BCG, Deloitte, RAND, CNAS, and the Belfer Center to argue that the defence industry is undergoing a structural pivot from "exquisite platforms" toward "affordable mass" built on autonomy, networking, and software-defined coordination. The shift is driven less by enthusiasm for drones than by a cost-exchange arithmetic that Western procurement systems are not yet organised to absorb — with particular consequences for the United Kingdom, caught between GCAP, the Palantir partnership, and its own indigenous autonomy base.

Cellular automata look almost too simple to be computationally serious: a lattice of cells, a finite set of states, and a local update rule repeated over time. Yet, under the righttion, these evolving patterns can encode information, propagate signals, implement transformations, store state, and even simulate universal computation. This post gives a graduate-level account of how cellular automata compute, moving from formal definitions of configuration spaces and global maps to a mathematical example in which Rule 90 generates Pascal’s triangle modulo two. It then turns to gliders in Conway’s Game of Life and Rule 110’s universality, showing how local dynamics become computation through encoding, evolution, and decoding.

Deep learning produces models so opaque that an entire subfield exists to apologise for them; Inductive Logic Programming learns human-readable rules but chokes at scale. This review surveys how the two have been connected from 2018 to 2026, framed through the requirements of Genesis-AI: a Robot Scientist aiming to autonomously conduct yeast systems biology using ~10,000 computer-controlled micro-bioreactors at a thousand-fold cost-benefit over human scientists. The central finding is a critical integration gap that no existing system fills.

Large Language Models are increasingly embedded in the scientific method, from literature synthesis and experimental automation to hypothesis generation. Yet the field treats the primary challenges as engineering ones: reduce hallucination, improve reasoning, increase interpretability. This essay argues that the deeper challenges are constitutive rather than technical. Working through the Duhem-Quine thesis, Kuhnian paradigm theory, Lakatosian research programmes, reliabilism, the Gettier problem, and Wittgenstein's rule-following considerations, it audits a recent perspective paper co-authored with colleagues at KAUST, King's College London, and Tufts. The audit reveals four constitutive problems that engineering alone cannot resolve: the opacity of auxiliary assumptions bundled with every LLM output, the structural paradigm-conservatism of training-data-dependent systems, the semantic underdetermination of model outputs, and the social epistemology of AI-mediated scientific communities. What the field needs, and does not yet have, is a philosophy of AI-mediated science.

The 2025 UK Strategic Defence Review commits £4 billion to autonomy this Parliament and mandates 10% of the defence budget for artificial intelligence and autonomous systems from April 2025. Yet existing certification frameworks cannot adequately address machine learning systems. This analysis examines how this structural mismatch between surging investment and immature assurance frameworks creates significant commercial opportunities for specialists in probabilistic reasoning, epistemic and aleatoric uncertainty representation, and interpretable artificial intelligence. Drawing on policy documents from the UK Ministry of Defence, NATO, and the European Union, this blog maps specific funding programmes, capability gaps, and market entry points across the defence innovation ecosystem.

Autonomous systems present tort law with a structural epistemological barrier: the opacity of machine learning decision-making renders traditional fault and causation analysis intractable, leaving victims facing insurmountable information asymmetries whilst manufacturers struggle to explain algorithmic behaviour even with full system access. This paper proposes that accountability must be engineered into autonomous systems ex ante rather than reconstructed ex post, outlining a Bayesian accountability infrastructure comprising three integrated components: continuous evidential logging via sufficient statistics, causal graphical models enabling computable counterfactual queries, and uncertainty-aware behavioural envelopes that distinguish statistically anticipated failures from systematic miscalibration. The framework transforms liability adjudication from adversarial narrative construction to principled probabilistic evaluation, preserving tech-impartiality whilst addressing the evidentiary gaps identified in recent European legislative proposals.

Autonomous systems operating in contested environments require real-time uncertainty quantification under adversarial interference and sensor degradation, yet existing approaches struggle to balance computational tractability with probabilistic rigour. This review aims to sketch whether modular Bayesian frameworks can reconcile this tension by decomposing complex digital twins into linked components that enable efficient computation whilst maintaining principled uncertainty propagation. Through thematic synthesis of foundational works and recent contributions spanning probabilistic graphical models, multi-fidelity modelling, and Bayesian verification methods, the review identifies patterns across four dimensions: modular decomposition strategies, information-passing mechanisms, computational efficiency approaches, and uncertainty quantification methods.

JPMorgan's unprecedented analysis of the U.S. defence-industrial complex exposes a startling paradox: despite $148 billion in annual R&D investment, only 16% of SBIR-funded companies successfully transition to production, with less than 1% achieving Programme of Record status. This comprehensive examination reveals how resource abundance creates its own inefficiencies—the "capital intensity paradox"—while exploring alternative models from technologically advanced allies, partnership-dependent strategies, and emerging nuclear diplomacy frameworks. The findings challenge fundamental assumptions about defence innovation, suggesting that efficiency matters more than scale and that resource-constrained countries may actually achieve superior innovation outcomes.

The future of robotics demands a fundamental rethinking of autonomous systems operating in contested environments where multiple agents interact, compete, and coordinate. Traditional single-agent approaches prove inadequate for scenarios involving GPS-jammed drones, underwater vehicles avoiding obstacles, or ground robots in disaster zones. This post explores how next-generation digital twins must evolve beyond simple digital replicas to comprehensive multi-agent simulation environments that integrate neurosymbolic AI, Bayesian inference, and multi-agent reinforcement learning.

An exploration of one of mathematics' most influential failures: David Sprecher's 1996 attempt to make Kolmogorov's superposition theorem computationally practical. This venture consequently sparked a decade of rigorous investigation, revealing fatal issues with monotonicity and continuity, before ultimately leading to Köppen's recursive solution and the modern breakthroughs in Kolmogorov-Arnold Networks that power (at least has strong potential) today's machine learning applications.

This exploration examines the potential parallels between verification principles central to Logical Positivism and contemporary approaches to ensuring reliability in LLM outputs for scientific discovery. We identify four key verification traps that may constrain LLMs' scientific potential and propose an alternative approach inspired by Epicurus' principle of multiple explanations. The discussion culminates in a research proposal for developing a balanced framework that ensures reliability while preserving LLMs' capacity for creative scientific thinking.

In this post, we present a comprehensive taxonomy of logical reasoning, systematically charting the landscape from fundamental deductive and non-deductive frameworks to specialized logical systems and meta-logical properties. Building on this structured taxonomy, we then explore the implications of this mapping for understanding and evaluating reasoning processes in LLMs. The discussion is anchored in the goal of establishing clearer conceptual boundaries for assessing LLM reasoning performance.

Evaluating hallucinations in LLM outputs is anything but straightforward. Over the past few years, researchers have developed a wide array of metrics—from ROUGE and BLEU to embedding-based and graph-based techniques—each with its own strengths and blind spots. This post walks through a structured taxonomy of these metrics, classifying them by lexical, semantic, factual, logical, and pragmatic dimensions.

But we do not stop there. The latest addition to this taxonomy is a unified Meta-Metric Framework that integrates diverse metrics into a single, adaptive evaluation pipeline. Inspired by ensemble learning, this framework uses dynamic weighting, dimensional aggregation, and task-aware scoring to deliver a more robust, interpretable, and generalizable assessment of hallucination across domains. Whether you're working on summarization, open-domain QA, or factual dialogue systems, this meta-metric approach offers a practical path toward better model accountability and fidelity.