No feature ranking can be simultaneously faithful, stable, and complete when features are collinear. For collinear pairs, ranking reduces to a coin flip. We prove this impossibility, quantify it for four model classes, resolve it via ensemble averaging (DASH), and machine-verify it with 305 Lean 4 theorems. We characterize the complete attribution design space: exactly two families of methods exist -- faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods like DASH (stable, reporting ties for symmetric features) -- and no method lies outside this dichotomy. The impossibility is quantitative: the attribution ratio diverges as 1/(1-rho^2) for gradient boosting, is infinite for Lasso, and converges for random forests. DASH (Diversified Aggregation of SHAP) is provably Pareto-optimal among unbiased aggregations, achieving the Cramer-Rao variance bound with a tight ensemble size formula. In a survey of 77 public datasets, 68% exhibit attribution instability. Switching to conditional SHAP does not escape the impossibility when features have equal causal effects. The framework includes practical diagnostics -- a Z-test workflow and single-model screening tool -- and has direct consequences for fairness auditing: SHAP-based proxy discrimination audits are provably unreliable under collinearity. The design space theorem, diagnostics, and impossibility are mechanically verified in Lean 4 (305 theorems from 16 axioms, 0 sorry) -- to our knowledge, the first formally verified impossibility in explainable AI.

Given the intractably large size of the space of proofs, any model that is capable of general deductive reasoning must generalize to proofs of greater complexity. Recent studies have shown that large language models (LLMs) possess some abstract deductive reasoning ability given chain-of-thought prompts. However, they have primarily been tested on proofs using modus ponens or of a specific size, and from the same distribution as the in-context examples. To measure the general deductive reasoning ability of LLMs, we test on a broad set of deduction rules and measure their ability to generalize to more complex proofs from simpler demonstrations from multiple angles: depth-, width-, and compositional generalization. To facilitate systematic exploration, we construct a new synthetic and programmable reasoning dataset that enables control over deduction rules and proof complexity. Our experiments on four LLMs of various sizes and training objectives show that they are able to generalize to compositional proofs. However, they have difficulty generalizing to longer proofs, and they require explicit demonstrations to produce hypothetical subproofs, specifically in proof by cases and proof by contradiction.

Liquid democracy with ranked delegations is a novel voting scheme that unites the practicability of representative democracy with the idealistic appeal of direct democracy: Every voter decides between casting their vote on a question at hand or delegating their voting weight to some other, trusted agent. Delegations are transitive, and since voters may end up in a delegation cycle, they are encouraged to indicate not only a single delegate, but a set of potential delegates and a ranking among them. Based on the delegation preferences of all voters, a delegation rule selects one representative per voter. Previous work has revealed a trade-off between two properties of delegation rules called anonymity and copy-robustness. To overcome this issue we study two fractional delegation rules: MIXEDBORDA BRANCHING, which generalizes a rule satisfying copy-robustness, and the RANDOMWALKRULE, which satisfies anonymity. Using the Markov chain tree theorem, we show that the two rules are in fact equivalent, and simultaneously satisfy generalized versions of the two properties. Combining the same theorem with Fulkerson's algorithm, we develop a polynomial-time algorithm for computing the outcome of the studied delegation rule. This algorithm is of independent interest, having applications in semi-supervised learning and graph theory.

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Some seemingly simple sequences of multiplication and addition grow so quickly that they question the very foundations of mathematics. Did you hear the one about the man who invented chess and got himself executed? Legend has it that a man called Sessa, who lived in India long ago, developed the rules for the game and presented them to a king. The king was delighted and offered the man his pick of reward. Sessa asked for a supposedly humble quantity of rice.

Gödel's seminal work directly contradicted one of the great minds of mathematics and limited the field forever Kurt Gödel, the man who ruined mathematics, was one of the most important thinkers of the 20th century. He was born in 1906, smack-bang in the middle of the greatest crisis that maths has ever known. Just a few decades later, he would help resolve this turmoil, but in doing so doom mathematicians to a smaller world than the one that came before. Mathematics, as an intellectual framework, is incredibly powerful. The entire point is taking one set of logical ideas and using them to build another, making maths the closest thing we have to a cognitive perpetual-motion machine - there is always a new mathematical idea lurking across the horizon, and we just need to assemble the steps to get there.

In Natural Language Processing (NLP), the Elo rating system, originally designed for ranking players in dynamic games such as chess, is increasingly being used to evaluate Large Language Models (LLMs) through A vs B paired comparisons.However, while popular, the system's suitability for assessing entities with constant skill levels, such as LLMs, remains relatively unexplored. We study two fundamental axioms that evaluation methods should adhere to: reliability and transitivity. We conduct an extensive evaluation of Elo behavior across simulated and real-world scenarios, demonstrating that individual Elo computations can exhibit significant volatility.We show that both axioms are not always satisfied, raising questions about the reliability of current comparative evaluations of LLMs.If the current use of Elo scores is intended to substitute the costly head-to-head comparison of LLMs, it is crucial to ensure the ranking is as robust as possible.Guided by the axioms, our findings offer concrete guidelines for enhancing the reliability of LLM evaluation methods, suggesting a need for reassessment of existing comparative approaches.

In the context of reinforcement learning from human feedback (RLHF), the reward function is generally derived from maximum likelihood estimation of a random utility model based on pairwise comparisons made by humans. The problem of learning a reward function is one of preference aggregation that, we argue, largely falls within the scope of social choice theory. From this perspective, we can evaluate different aggregation methods via established axioms, examining whether these methods meet or fail well-known standards. We demonstrate that both the Bradley-Terry-Luce Model and its broad generalizations fail to meet basic axioms. In response, we develop novel rules for learning reward functions with strong axiomatic guarantees. A key innovation from the standpoint of social choice is that our problem has a structure, which greatly restricts the space of feasible rules and leads to a new paradigm that we call .