FRG: Collaborative Research: Computability-Theoretic Aspects of Combinatorics

可计算性理论起源于图灵和其他人对算法概念的精确数学定义的发展。这一领域最富有成果的发展之一是对数学算法内容的研究。与此同时，对数学基础的关注导致了对数学推理的不同形式系统的分析，以及对这些系统的相对力量的研究。可计算理论方法在这个项目中发挥了关键作用。组合学是数学的一个领域，其方法和结果是大量数学推理的基础，因此对其算法内容和形式化其方法所需的系统的研究尤为重要。可计算组合学有着悠久的历史，但最近人们对这一事实的兴趣激增，这一事实强调了可计算性——理论上自然的概念往往是组合自然的，反之亦然。本项目旨在加强可计算性理论和组合学之间的联系，通过增加我们对组合原理的算法内容的理解，通过协调一致的小组努力，利用我们近年来学到的知识进一步系统化该领域，努力解决一些突出的问题，并探索其面向外部的方面。该项目将解决主要的开放问题，如确定欣德曼定理是否在算术上成立，以及它是否等同于其对长度最多为2的和的限制；确定了对的Ramsey定理的一阶部分，并阐明了该定理与其稳定版本之间的可计算性理论关系；并确定拉弗定理的理论证明强度。更重要的是，它将专注于研究方向的发展，特别是与可计算理论和组合学之间的联系日益增长的兴趣相关的研究，包括对组合学家感兴趣的方法论问题的研究，例如涉及Hindman定理和组合学中超过滤器的使用的问题；随着蒙塔尔班对拉弗定理的新证明，受有关这些定理的计算内容的问题的启发，对组合定理的新证明的发展；对不同的一般方法进行了分析和比较，如使用超滤、拓扑动力学和概率方法；定理之间的比较概念的发展，虽然在精神上仍然是可计算理论的，但更接近于组合差异的直接反映；将组合原理分解为计算上和组合上更简单的部分；组合定义对象作为计算预言器的有用性研究；以及研究组合定义二阶对象存在的一阶结果。因此，该项目旨在利用其pi，高级人员和主要合作者的可计算理论，证明理论和组合专业知识，沿着大量相关方向发展可计算理论与组合学之间联系的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Computability theory arose out of the development by Turing and others of a mathematically precise definition of the notion of algorithm. One of the most fruitful developments in this field has been the study of the algorithmic content of mathematics. At the same time, concerns about the foundations of mathematics have led to the analysis of different formal systems for mathematical reasoning, and the study of the relative power of such systems. The computability-theoretic approach has turned out to play a key role in this project. Combinatorics is an area of mathematics whose methods and results underlie a great deal of mathematical reasoning, so the study of its algorithmic content and of the systems required to formalize its methods is particularly important. Computable combinatorics has a long history, but has seen a recent surge of interest highlighting the fact that computability-theoretically natural notions tend to be combinatorially natural, and vice-versa. This project aims to strengthen the connections between computability theory and combinatorics by increasing our understanding of the algorithmic content of combinatorial principles, through a concerted group effort to use what we have learned in recent years to systematize the area further, work toward solving a number of its outstanding questions, and explore its outward-facing aspects.This project will address major open problems such as determining whether Hindman's Theorem holds arithmetically, and whether it is equivalent to its restriction to sums of length at most two; determining the first-order part of Ramsey's Theorem for pairs, and clarifying the computability-theoretic relationship between this principle and its stable version; and determining the exact proof-theoretic strength of Laver's Theorem. More importantly, it will focus on the development of lines of research particularly relevant to the growing interest in connections between computability theory and combinatorics, including the study of questions of methodological interest to combinatorialists, for instance ones involving Hindman's Theorem and the use of ultrafilters in combinatorics; the development of new proofs of combinatorial theorems inspired by questions regarding the computational content of these theorems, along the lines of Montalban's new proof of Laver's Theorem; the analysis and comparison of different general methods, such as the use of ultrafilters, topological dynamics, and probabilistic methods; the development of notions of comparison between theorems that, while still computability-theoretic in spirit, are closer to being direct reflections of combinatorial differences; the understanding of splittings of combinatorial principles into computationally and combinatorially simpler parts; the study of the usefulness of combinatorially-defined objects as computational oracles; and the study of the first-order consequences of the existence of combinatorially-defined second-order objects. This project is thus aimed at developing the study of the connections between computability theory and combinatorics along a large number of related directions, making use of the computability-theoretic, proof-theoretic, and combinatorial expertise of its PIs, senior personnel, and key collaborators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.