Related papers: Finding Proofs in Tarskian Geometry

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular,…

This paper describes the formal verification of two Turing machines using the program verifier Dafny. Both machines are deciders, so we prove total correctness. They are typical first examples of Turing machines used in any course of…

Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We consider also a special categorical invariant $K_\Theta (H)$ of this geometry. The…

The proofs of K. Oka's Coherence Theorems are based on Weierstrass' Preparation (division) Theorem. Here we formulate and prove a Weak Coherence Theorem without using Weierstrass' Preparation Theorem, but only with power series expansions:…

In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for…

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.…

We propose using mechanistic interpretability -- techniques for reverse engineering model weights into human-interpretable algorithms -- to derive and compactly prove formal guarantees on model performance. We prototype this approach by…

Motivated by certain applications from physics, biochemistry, economics, and computer science, in which the objects under investigation are not accessible because of various limitations, we propose a trial-and-error model to examine…

Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to…

Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree…

Solving large scale Optimal Transport (OT) in machine learning typically relies on sampling measures to obtain a tractable discrete problem. While the discrete solver's accuracy is controllable, the rate of convergence of the discretization…

We introduce PHYSICS, a comprehensive benchmark for university-level physics problem solving. It contains 1297 expert-annotated problems covering six core areas: classical mechanics, quantum mechanics, thermodynamics and statistical…

This thesis deals with applications of experimental mathematics to a number of problems. The first problem is related to random graph statistics. We consider a certain class of Galton-Watson random trees and look at the total height…

We describe an experiment in LLM-assisted autoformalization that produced over 85,000 lines of Isabelle/HOL code covering all 39 sections of Munkres' Topology (general topology, Chapters 2--8), from topological spaces through dimension…

We investigate the problem $\#\mathsf{IndSub}(\Phi)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $\Phi$. This continues the work of Jerrum and Meeks who proved the problem to be…

This paper describes a large set of related theorem proving problems obtained by translating theorems from the HOL4 standard library into multiple logical formalisms. The formalisms are in higher-order logic (with and without type…

Proof engineering efforts using interactive theorem proving have yielded several impressive projects in software systems and mathematics. A key obstacle to such efforts is the requirement that the domain expert is also an expert in the…

We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is…

Correctness proofs for floating point programs are difficult to verify. To simplify the task, a similar, but less complex system, known as logarithmic arithmetic can be used. The Boyer-Moore Theorem Prover, NQTHM, mechanically verified the…

In the last three decades, the $k$-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a…