Drift and selection in text ecosystems
An exactly solvable n-gram framework for studying neutral drift and normative selection in LLM text ecosystems. The model separates mathematical mechanisms from claims about empirical text systems.
Research
Mathematical structure is the thread running through my work: the resources a proof needs, the information a network carries, the orderings a voting rule permits, and the structure a learner discovers.
01 · Logic & proof complexity
Counting principles and the resources needed for proof
Some mathematical assertions express a simple counting obstruction. The difficulty lies in what a formal system must do to establish it. My work studies relationships between such principles and the size of proofs in restricted proof systems.
In Count(q) Does Not Imply Count(p), I classify implications between modular counting principles by the prime factors of the moduli. This makes the relationship between the principles an exact mathematical question.
The counting result more precisely
For the family of counting principles studied in the paper, Count(q) implies Count(p) exactly when every prime factor of p divides q. The journal article appeared in Annals of Pure and Applied Logic 90 (1997), 1–56; the earlier BRICS report appeared in 1994.
A second strand concerns proof complexity, including A complexity gap for tree resolution and work with Stefan Dantchev on resolution proofs and relativisation.
02 · Information flow
Local guesses, global constraints
Imagine a graph whose vertices carry hidden symbols. Each vertex must guess its own symbol using only the information available from its neighbours. Which graphs allow many assignments on which every guess is correct?
My work on guessing numbers connects this question to communication networks. In the 2011 paper with Maximilien Gadouleau, guessing graphs relate compatible configurations to independent sets, and graph constructions provide examples with large guessing numbers.
A strategy value and a graph invariant
For an alphabet of size s, a fixed local strategy f has a set Fix(f) of assignments on which all guesses are correct. Its achieved value is logs|Fix(f)|. The graph’s guessing number maximises that quantity over admissible strategies. Changing a strategy changes its achieved value; it does not change the underlying graph invariant.
Related work with collaborators connects Boolean-network fixed points, coding theory, and non-Shannon information inequalities.
03 · Collective choice
Orderings that keep majority choice consistent
Majority preferences can cycle: a group can prefer A to B, B to C, and C to A. A Condorcet domain restricts the allowed individual preference orders so that these majority cycles are avoided.
With Charles Leedham-Green and Klas Markström, I proved that a Condorcet domain on eight alternatives has at most 224 orders, and that the maximum is attained by a unique domain up to isomorphism. The article appeared online in 2023 and in volume 62 of Social Choice and Welfare in 2024.
What the number means
The 224 counts admissible linear preference orders on the same eight alternatives. It is an extremal result about the domain of permitted rankings, rather than a bound on the number of voters. Isomorphism identifies domains differing only by the appropriate relabelling of alternatives.
Related collaborative work studies domains on at most seven alternatives, larger constructions, heuristic search, and software for working with restricted permutation sets.
04 · Games & learning
Competitive performance and structural mastery
Impartial games offer a useful setting for examining learning because exact mathematical strategies can be compared with the behaviour of a trained agent. The rules are simple enough to explore, while the structure can challenge learning algorithms.
In Impartial Games: A Challenge for Reinforcement Learning, Bei Zhou and I distinguish “champion” performance from “expert” mastery. Our experiments investigate settings in which strong competitive play does not amount to learning a strategy that generalises across the game’s state space.
Scope of the comparison
The paper studies specified impartial-game environments and AlphaZero-style learning. The experiments are evidence about those systems and tasks. Counterplay illustrates the question with transparent scripted opponents and exact Nim strategy. Its results are teaching examples, not measurements from the paper’s trained agents.
My subsequent Mastering NIM preprint explores a multi-frame approach. I also contributed to the consortium behind Humanity’s Last Exam, whose benchmark paper appeared in Nature in 2026.
Current research · Preprints
New questions and connections
Coin races and waiting times
With Mike Paterson, I study waiting-time comparisons for coin patterns and the special role of fair coins. The accompanying tool explores independent biased-coin races.
Term coding
An entropic framework for extremal combinatorics and a guessing-number sandwich theorem.
Read the preprint →Larger Condorcet domains
Work with Alexander Karpov, Klas Markström, and Bei Zhou on improved lower bounds for the maximum size of Condorcet domains.
Read the preprint →