Related papers: Automated Tactics for Polynomial Reasoning in Lean…
Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift…
Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from…
Large Language Models (LLMs) have demonstrated significant potential in generating mathematical proofs. However, a persistent challenge is that LLMs occasionally make mistakes, while even a minor mistake can invalidate an entire proof.…
Chain-of-Thought (CoT) prompting has become the de facto method to elicit reasoning capabilities from large language models (LLMs). However, to mitigate hallucinations in CoT that are notoriously difficult to detect, current methods such as…
We present version 4.0 of the symbolic manipulation system FORM. The most important new features are manipulation of rational polynomials and the factorization of expressions. Many other new functions and commands are also added; some of…
Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general,…
Large Language Models (LLMs) hold the potential to revolutionize autoformalization. The introduction of Lean4, a mathematical programming language, presents an unprecedented opportunity to rigorously assess the autoformalization…
Recent advances in large language models (LLMs) have shown promise in formal theorem proving, yet evaluating semantic correctness remains challenging. Existing evaluations rely on indirect proxies such as lexical overlap with…
Automated theorem proving with large language models in Lean 4 is commonly approached through either step-level tactic prediction with tree search or whole-proof generation. These two paradigms represent opposite granularities for…
Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases,…
Can large language models be used to complete mathematical tasks that are traditionally performed either manually or with the aid of theorem provers? To answer this question, a state-of-the-art system, GPT-4, was provided with a concise…
We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a…
This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for…
This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for…
Autoformalization - automatically translating natural language mathematical texts into formal proof language such as Lean4 - can help accelerate AI-assisted mathematical research, be it via proof verification or proof search. I fine-tune…
We present a formally verified framework for patent analysis as a hybrid AI + Lean 4 pipeline. The DAG-coverage core (Algorithm 1b) is fully machine-verified once bounded match scores are fixed. Freedom-to-operate, claim-construction…
We identify a common scheme in several existing algorithms addressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as…
Lean 4 autoformalization has become increasingly popular in recent years, with frontier language models and open-weight autoformalizers now producing valid formalizations of mathematical theorems. However, these evaluations often rely on…
MerLean-Prover is an end-to-end Lean4 theorem prover that replaces sorry declarations with kernel-checkable proofs. It is built from three agent types (Planning, Check, and Lean) composed by a recursive outer loop whose unit of revision is…
In this paper we give an insight into the behaviour of signature-based Gr\"obner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the…