arXiv subjects

Søren Riis

Publications and source records attributed to Søren Riis.

At least 19 recordsLinked to original sources

arXiv2026

One Unit Separates Polynomial Time from Undecidability in Term Coding

Term coding provides a common algebraic framework for network coding, index coding and problems in extremal combinatorics. We exhibit a decision problem in which lowering an output threshold by just one changes the complexity from polynomial time to undecidability: no algorithm then halts with the correct answer on every input. The problem concerns dispersion, the maximum number of distinct output tuples obtainable by interpreting the function symbols in a tuple of terms on a finite alphabet. We restrict inputs by inequalities between terms and ask whether a given instance meets a prescribed output threshold for some alphabet size at least two. Both thresholds are considered on the same class of instances. A machine-checked Lean development and an interactive presentation of the paper accompany this work (GitHub: https://github.com/SR123/term-coding-disequality-lean; DOI: 10.5281/zenodo.22727895); Section 8 specifies its external input and verification limits.

math.CO↗

arXiv2026

Term Coding and Dispersion: Exact and Asymptotic Decision Problems

Let t be a tuple of r terms that, under an interpretation on an n-element alphabet A, defines a map from k-tuples over A to r-tuples over A. We study the decision theory of its maximum image size, separating exact perfect dispersion from asymptotic rate. Building on the term-cut theorem of Riis and Gadouleau, we prove that every eventual threshold strictly between consecutive integer powers is decidable in polynomial time. More precisely, if a threshold is eventually greater than n to the power d and grows strictly more slowly than n to the power d plus one, then the maximum image size eventually meets that threshold exactly when the term-cut exponent is at least d plus one. For the exact problem, we introduce the perfect-alphabet spectrum and prove that it is multiplicatively closed, that a nonempty spectrum forces full rate, and that the converse fails. We completely characterize the one-output case. On square instances, perfect dispersion is precisely finite square term bijectivity. We give explicit linear-size padding reductions from three-dimensional square bijectivity to perfect dispersion for every fixed output dimension of at least three. We also characterize scalar-linear witnesses by a determinant polynomial, obtaining decidability over fixed finite fields, over extensions of a fixed characteristic, and over arbitrary finite fields. General square bijectivity remains open. The principal mathematical results have been machine-checked in Lean.

cs.IT↗

arXiv2026

Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

Classical existence problems in extremal combinatorics ask whether finite operations can satisfy prescribed identities universally. Term Coding replaces this yes-or-no question by a graded one: for a finite system $Γ$, the maximum code size $S_n(Γ)$ is the largest number of satisfying assignments attainable on an $n$-element alphabet. We prove that normalisation and diversification associate $Γ$ with a labelled guessing game of guessing number $α$ and give finite-alphabet sandwich bounds. Consequently, $\log_n S_n(Γ)=α+o(1)$. Entropy and polymatroid inequalities provide systematic upper bounds. Examples include a five-cycle with exponent $5/2$, self-orthogonal Latin squares, and presentation-dependent exponents for universally equivalent identity systems. All theorems, lemmas and propositions in this paper have been machine-checked in the Lean 4 proof assistant; the development is available at https://github.com/SR123/term-coding-lean.

cs.IT↗

arXiv2026

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

cs.LG↗

arXiv2026

Drift and selection in LLM text ecosystems

The public text record -- the material from which both people and AI systems now learn -- is increasingly shaped by its own outputs. Generated text enters the public record, later agents learn from it, and the cycle repeats. Here we develop an exactly solvable mathematical framework for this recursive process, based on variable-order $n$-gram agents, and separate two forces acting on the public corpus. The first is drift: unfiltered reuse progressively removes rare forms, and in the infinite-corpus limit we characterise the stable distributions exactly. The second is selection: publication, ranking and verification filter what enters the record, and the outcome depends on what is selected. When publication merely reflects the statistical status quo, the corpus converges to a shallow state in which further lookahead brings no benefit. When publication is normative -- rewarding quality, correctness or novelty -- deeper structure persists, and we establish an optimal upper bound on the resulting divergence from shallow equilibria. The framework therefore identifies when recursive publication compresses public text and when selective filtering sustains richer structure, with implications for the design of AI training corpora.

cs.CL↗

arXiv2026

Mastering NIM and Impartial Games with Weak Neural Networks: An AlphaZero-inspired Multi-Frame Approach

We study impartial games under fixed-latency, fixed-scale quantised inference (FSQI). In this fixed-scale, bounded-range regime, we prove that inference is simulable by constant-depth polynomial-size Boolean circuits (AC0). This yields a worst-case representational barrier: single-frame agents in the FSQI/AC0 regime cannot strongly master NIM, because optimal play depends on the global nim-sum (parity). Under our stylised deterministic rollout interface, a single rollout policy head from the structured family analysed here reveals only one fixed linear functional of the invariant, so increasing rollout budget alone does not recover the missing bits. We derive two structural bypasses: (1) a multi-policy-head rollout architecture that recovers the full invariant via distinct rollout channels, and (2) a multi-frame architecture that tracks local nimber differences and supports restoration. Experiments across multiple settings are consistent with these predictions: single-head baselines stay near chance, while two-frame models reach near-perfect restoration accuracy and multi-head FSM-controlled shootouts achieve perfect win/loss position classification. Overall, the empirical results support the view that explicit structural priors (history/differences or multiple rollout channels) are important in the FSQI/AC0 regime.

cs.AI↗

arXiv2026

Coin flipping and waiting times paradoxes: Why fair coins are exceptional

Penney's Ante exhibits non-transitivity when two target strings race to appear in a shared stream of coin tosses. We study instead independent string races, where each player observes their own independent and identically distributed (i.i.d.) coin/die stream (possibly biased), and the winner is the player whose target appears first (under an explicit tie convention). We derive compact generating-function formulas for waiting times and a Hadamard-generating-function calculus for head-to-head odds. Our main theorem shows that for a fair -sided die, stochastic dominance induces a total pre-order on all strings, ordered by expected waiting time. For binary coins, we also prove a converse: total comparability under stochastic dominance characterises the fair coin (), and any bias yields patterns whose waiting times are incomparable under stochastic dominance. In contrast, bias allows both (i) reversals between mean waiting time and win probability and (ii) non-transitive cycles; we give explicit examples and certified computational classifications for short patterns.

math.PR↗

arXiv2026

Impartial Games: A Challenge for Reinforcement Learning

AlphaZero-style reinforcement learning (RL) algorithms have achieved superhuman performance in many complex board games such as Chess, Shogi, and Go. However, we showcase that these algorithms encounter significant and fundamental challenges when applied to impartial games, a class where players share game pieces and optimal strategy often relies on abstract mathematical principles. Specifically, we utilise the game of Nim as a concrete and illustrative case study to reveal critical limitations of AlphaZero-style and similar self-play RL algorithms. We introduce a novel conceptual framework distinguishing between champion and expert mastery to evaluate RL agent performance. Our findings reveal that while AlphaZero-style agents can achieve champion-level play on very small Nim boards, their learning progression severely degrades as the board size increases. This difficulty stems not merely from complex data distributions or noisy labels, but from a deeper representational bottleneck: the inherent struggle of generic neural networks to implicitly learn abstract, non-associative functions like parity, which are crucial for optimal play in impartial games. This limitation causes a critical breakdown in the positive feedback loop essential for self-play RL, preventing effective learning beyond rote memorisation of frequently observed states. These results align with broader concerns regarding AlphaZero-style algorithms' vulnerability to adversarial attacks, highlighting their inability to truly master all legal game states. Our work underscores that simple hyperparameter adjustments are insufficient to overcome these challenges, establishing a crucial foundation for the development of fundamentally novel algorithmic approaches, potentially involving neuro-symbolic or meta-learning paradigms, to bridge the gap towards true expert-level AI in combinatorial games.

cs.LG↗

arXiv2024

A heuristic search algorithm for discovering large Condorcet domains

The study of large Condorcet domains (CD) has been a significant area of interest in voting theory. In this paper, our goal is to search for large CDs that are hitherto unknown. With a straightforward combinatorial definition, searching for large CDs is naturally suited for algorithmic optimisations. For each value of n>2, one can ask for the size of the largest CD, thus finding the largest CDs provides an important benchmark for heuristic-based combinatorial optimisation algorithms. Despite extensive research over the past three decades, the CD sizes identified in 1996 remain the best known for many values of n. When n>8, conducting an exhaustive search becomes computationally unfeasible, thereby prompting the use of heuristic methods. To address this, we developed a novel heuristic search algorithm in which a specially designed heuristic function, backed by a lookup database, directs the search towards promising branches in the search tree. Our algorithm found new large CDs of size 1082 (surpassing the previous record of 1069) for n=10, and 2349 (improving the previous 2324) for n=11. Notably, these newly discovered CDs exhibit characteristics distinct from those of known CDs.

cs.DM↗

arXiv2024

Bipartite peak-pit domains

In this paper, we introduce the class of bipartite peak-pit domains. This is a class of Condorcet domains which include both the classical single-peaked and single-dipped domains. Our class of domains can be used to model situations where some alternatives are ranked based on a most preferred location on a societal axis, and some are ranked based on a least preferred location. This makes it possible to model situations where agents have different rationales for their ranking depending on which of two subclasses of the alternatives one is considering belong to. The class of bipartite peak-pit domains includes most peak-pit domains for $n\leq 7$ alternatives, and the largest Condorcet domains for each $n\leq 8$. In order to study the maximum possible size of a bipartite peak-pit domain we introduce set-alternating schemes. This is a method for constructing well-structured peak-pit domains which are copious and connected. We show that domains based on these schemes always have size at least $2^{n-1}$ and some of them have sizes larger than the domains of Fishburn's alternating scheme. We show that the maximum domain size for sufficiently high $n$ exceeds $2.1973^n$. This improves the previous lower bound for peak-pit domains $2.1890^n$ from \cite{karpov2023constructing}, which was also the highest asymptotic lower bound for the size of the largest Condorcet domains.

cs.DM↗

arXiv2024

CDL: A fast and flexible library for the study of permutation sets with structural restrictions

In this paper, we introduce CDL, a software library designed for the analysis of permutations and linear orders subject to various structural restrictions. Prominent examples of these restrictions include pattern avoidance, a topic of interest in both computer science and combinatorics, and "never conditions" utilized in social choice and voting theory. CDL offers a range of fundamental functionalities, including identifying the permutations that meet specific restrictions and determining the isomorphism of such sets. To facilitate exploration of large permutation sets or domains, CDL incorporates multiple search strategies and heuristics.

cs.MS↗

arXiv2024

Arrow's single peaked domains, richness, and domains for plurality and the Borda count

In this paper we extend the study of Arrow's generalisation of Black's single-peaked domain and connect this to domains where voting rules satisfy different versions of independence of irrelevant alternatives. First we report on a computational generation of all non-isomorphic Arrow's single-peaked domains on $n\leq 9$ alternatives. Next, we introduce a quantitative measure of richness for domains, as the largest number $r$ such that every alternative is given every rank between 1 and $r$ by the orders in the domain. We investigate the richness of Arrow's single-peaked domains and prove that Black's single-peaked domain has the highest possible richness, but it is not the only domain which attains the maximum. After this we connect Arrow's single-peaked domains to the discussion by Dasgupta, Maskin and others of domains on which plurality and the Borda count satisfy different versions of Independence of Irrelevant alternatives (IIA). For Nash's version of IIA and plurality, it turns out the domains are exactly the duals of Arrow's single-peaked domains. As a consequence there can be at most two alternatives which are ranked first in any such domain. For the Borda count both Arrow's and Nash's versions of IIA lead to a maximum domain size which is exponentially smaller than $2^{n-1}$, the size of Black's single-peaked domain.

econ.TH↗

arXiv2024

Local Diversity of Condorcet Domains

Several of the classical results in social choice theory demonstrate that in order for many voting systems to be well-behaved the set domain of individual preferences must satisfy some kind of restriction, such as being single-peaked on a political axis. As a consequence it becomes interesting to measure how diverse the preferences in a well-behaved domain can be. In this paper we introduce an egalitarian approach to measuring preference diversity, focusing on the abundance of distinct suborders one subsets of the alternative. We provide a common generalisation of the frequently used concepts of ampleness and copiousness. We give a detailed investigation of the abundance for Condorcet domains. Our theorems imply a ceiling for the local diversity in domains on large sets of alternatives, which show that in this measure Black's single-peaked domain is in fact optimal. We also demonstrate that for some numbers of alternatives, there are Condorcet domains which have largest local diversity without having maximum order.

econ.TH↗

arXiv2023

Condorcet Domains of Degree at most Seven

In this paper we give the first explicit enumeration of all maximal Condorcet domains on $n\leq 7$ alternatives. This has been accomplished by developing a new algorithm for constructing Condorcet domains, and an implementation of that algorithm which has been run on a supercomputer. We follow this up by the first survey of the properties of all maximal Condorcet domains up to degree 7, with respect to many properties studied in the social sciences and mathematical literature. We resolve several open questions posed by other authors, both by examples from our data and theorems. We give a new set of results on the symmetry properties of Condorcet domains which unify earlier works. Finally we discuss connections to other domain types such as non-dictatorial domains and generalisations of single-peaked domains. All our data is made freely available for other researches via a new website.

cs.DM↗

arXiv2023

The Largest Condorcet Domains on 8 Alternatives

In this note, we report on a record-breaking Condorcet domain (CD) for n=8 alternatives. We show that there exists a CD of size 224, which is optimal and essentially unique (up to isomorphism). If we consider the underlying permutations and focus on Condorcet domains containing the identity permutation, 56 isomorphic such Condorcet domains exist. Our work sheds light on the structure of CDs and UCDs and has potential applications in voting theory and social choice.

math.CO↗

arXiv2015

Network Communication with operators in Dedekind Finite and Stably Finite Rings

Messages in communication networks often are considered as "discrete" taking values in some finite alphabet (e.g. a finite field). However, if we want to consider for example communication based on analogue signals, we will have to consider messages that might be functions selected from an infinite function space. In this paper, we extend linear network coding over finite/discrete alphabets/message space to the infinite/continuous case. The key to our approach is to view the space of operators that acts linearly on a space of signals as a module over a ring. It turns out that modules over many rings $R$ leads to unrealistic network models where communication channels have unlimited capacity. We show that a natural condition to avoid this is equivalent to the ring $R$ being Dedekind finite (or Neumann finite) i.e. each element in $R$ has a left inverse if and only if it has a right inverse. We then consider a strengthened capacity condition and show that this requirement precisely corresponds to the class of (faithful) modules over stably finite rings (or weakly finite). The introduced framework makes it possible to compare the performance of digital and analogue techniques. It turns out that within our model, digital and analogue communication outperforms each other in different situations. More specifically we construct: 1) A communications network where digital communication outperforms analogue communication. 2) A communication network where analogue communication outperforms digital communication. The performance of a communication network is in the finite case usually measured in terms band width (or capacity). We show this notion also remains valid for finite dimensional matrix rings which make it possible (in principle) to establish gain of digital versus analogue (analogue versus digital) communications.

math.RA↗

arXiv2014

Graph Guessing Games and non-Shannon Information Inequalities

Guessing games for directed graphs were introduced by Riis for studying multiple unicast network coding problems. In a guessing game, the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph. They later guess simultaneously the outcome of their own die. Their objective is to find a strategy which maximises the probability that they all guess correctly. The performance of the optimal strategy for a graph is measured by the guessing number of the digraph. Christofides and Markström studied guessing numbers of undirected graphs and defined a strategy which they conjectured to be optimal. One of the main results of this paper is a disproof of this conjecture. The main tool so far for computing guessing numbers of graphs is information theoretic inequalities. In the paper we show that Shannon's information inequalities, which work particularly well for a wide range of graph classes, are not sufficient for computing the guessing number. Finally we pose a few more interesting questions some of which we can answer and some which we leave as open problems.

cs.IT↗

arXiv2014

Fixed points of Boolean networks, guessing graphs, and coding theory

In this paper, we are interested in the number of fixed points of functions $f:A^n\to A^n$ over a finite alphabet $A$ defined on a given signed digraph $D$. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on $D$. We then discover relationships between the number of fixed points of $f$ and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.

cs.DM↗