FRG: Collaborative Research: Computability-Theoretic Aspects of Combinatorics

可计算性理论产生于图灵和其他人对算法概念的数学精确定义的发展。在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。可计算性理论的方法已被证明在这个项目中发挥了关键作用。组合数学是一个数学领域，其方法和结果是大量数学推理的基础，因此研究其算法内容和形式化其方法所需的系统尤为重要。可计算组合学有着悠久的历史，但最近的兴趣激增，突出了一个事实，即可计算性理论上自然的概念往往是组合自然的，反之亦然。该项目旨在通过增加我们对组合原理的算法内容的理解来加强可计算性理论和组合学之间的联系，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，努力解决一些悬而未决的问题，并探索其外向的方面。这个项目将解决主要的开放问题，如确定是否欣德曼定理持有算术，以及它是否等价于它对长度至多为2的和的限制;确定Ramsey定理的一阶部分，并澄清该原理与其稳定版本之间的可计算性理论关系;以及确定Laver定理的精确证明理论强度。更重要的是，它将侧重于研究线的发展，特别是与可计算性理论和组合学之间日益增长的兴趣有关，包括对组合学家感兴趣的方法学问题的研究，例如涉及Hindman定理和组合学中超滤波器的使用;发展新的证明组合定理的启发问题有关的计算内容，这些定理，沿着线路蒙塔尔班的新证明拉弗定理;分析和比较不同的一般方法，如使用超滤波器，拓扑动力学和概率方法;定理之间的比较概念的发展，虽然仍然是计算理论的精神，更接近于直接反映组合差异;组合原则分裂成计算和组合更简单的部分的理解;研究组合定义的对象作为计算预言机的有用性;以及研究组合定义的二阶对象存在的一阶后果。因此，该项目旨在发展可计算性理论和组合学之间的联系的研究，沿着大量相关的方向，利用其PI，高级人员，该奖项反映了NSF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进行评估，被认为值得支持审查标准。

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.