相关论文: Schemes in Lean

This note discusses our formalisation in Lean of the classification of the groups of order $p q$ for (not necessarily distinct) prime numbers $p$ and $q$, together with various intermediate results such as the characterisation of internal…

Phasers pose an interesting synchronization mechanism that generalizes many collective synchronization patterns seen in parallel programming languages, including barriers, clocks, and point-to-point synchronization using latches or…

We propose a formalism to model and reason about reconfigurable multi-agent systems. In our formalism, agents interact and communicate in different modes so that they can pursue joint tasks; agents may dynamically synchronize, exchange…

A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists,…

The aim of this work is to develop general optimization methods for finite difference schemes used to approximate linear differential equations. The specific case of the transport equation is exposed. In particular, the minimization of the…

This paper introduces a novel Large Language Models (LLMs)-assisted agent that automatically converts natural-language descriptions of power system optimization scenarios into compact, solver-ready formulations and generates corresponding…

We discuss the notion of linearization through examples, which include the Price map, PageRank, representation theory, the Euler characteristic and quantum invariants. We also review categorification, which adds an additional layer of…

Using AI to write formal proofs for mathematical problems is a challenging task that has seen some advancements in recent years. Automated systems such as Lean can verify the correctness of proofs written in formal language, yet writing the…

We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…

In this paper we will obtain some further properties for specializations in a scheme. Using these results, we will take a picture for a scheme and a picture for a morphism of schemes. In particular, we will prove that every morphism of…

We examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through…

Extracting structured information from unstructured text is crucial for modeling real-world processes, but traditional schema mining relies on semi-structured data, limiting scalability. This paper introduces schema-miner, a novel tool that…

A physical theory consists of the mathematical formalism and an interpretation, which contains the definition of symbols, measurement assignments, concepts and principles, and an ontology. We present a scheme to classify these different…

This paper presents the results achieved while pursuing the verification and validation of a train system behavior at the first steps of development in an industrial context. A method is proposed, supported by preliminary results through…

This work characterizes large language models' chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become…

In the last decades, several objects such as grammars, economical agents, laws of physics... have been defined as algorithms. In particular, after Brouwer, Heyting, and Kolomogorov, mathematical proofs have been defined as algorithms. In…

In this paper, we study numbers $n$ that can be factored in three different ways as $n = A_1 B_1 = A_2 B_2 = A_3 B_3$ where $A_1$, $A_2$, $A_3$ are close to each other and $B_1$, $B_2$, $B_3$ are close to each other.

Human language and logic abilities are computationally quantified within the well-studied grammar-automata hierarchy. We identify three hierarchical tiers and two corresponding transitions and show their correspondence to specific abilities…

Large Language Models (LLMs) have been successful in mathematical reasoning tasks such as formal theorem proving when integrated with interactive proof assistants like Lean. Existing approaches involve training or fine-tuning an LLM on a…

Machine learning algorithms for prediction are increasingly being used in critical decisions affecting human lives. Various fairness formalizations, with no firm consensus yet, are employed to prevent such algorithms from systematically…