Related papers: Tutte's theorem as an educational formalization pr…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
One of the main prerequisites for understanding sheaves on elementary toposes is the proof that a (Lawvere-Tierney) topology on a topos induces a closure operator on it, and vice-versa. That standard theorem is usually presented in a…
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
We introduce ProofNet, a benchmark for autoformalization and formal proving of undergraduate-level mathematics. The ProofNet benchmarks consists of 371 examples, each consisting of a formal theorem statement in Lean 3, a natural language…
The block-cut tree decomposes a connected graph along its cutvertices, displaying its 2-connected components. The Tutte-decomposition extends this idea to 2-separators in 2-connected graphs, yielding a canonical tree-decomposition that…
Large language models (LLMs) often struggle with complex logical reasoning due to logical inconsistencies and the inherent difficulty of such reasoning. We use Lean, a theorem proving framework, to address these challenges. By formalizing…
We present ZFLean, a Lean 4 library for doing core mathematics inside a model of ZFC with the ergonomics expected of typed Mathlib developments. Building on Mathlib's ZFC model, we contribute a relational calculus for sets with rewriting…
We present an overview of how certain computational tools currently interact with mathematical practice, and reflect on the implications for research mathematics in the short to medium term, as the field navigates the emerging age of AI and…
We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method), their implementation and integration in educational tools. The project includes formal…
Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for…
LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's…
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…
The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed…
The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization…
This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
One of the ways to improve the performance of a target task is to learn the transfer of abundant knowledge of a pre-trained network. However, learning of the pre-trained network requires high computation capability and large-scale labeled…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents…
Recent advances in large language models show strong promise for formal reasoning. However, most LLM-based theorem provers have long been constrained by the need for expert-written formal statements as inputs, limiting their applicability…