Computability and Decision Procedures for Number Theory and Combinatorics

数论和组合学的可计算性和决策程序

• 批准号: RGPIN-2018-04118
• 负责人: Shallit, Jeffrey
• 金额: $ 6.99万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750229
• 关键词: Computability; Decision; Procedures; Number; Theory

Broadly speaking, my research involves two different areas and how they interact. The first area concerns mathematical models of simple computers ("automata") and their capabilities. The second area concerns the basics of pure mathematics: number theory, combinatorics, and algebra. Although these two areas superficially seem quite separated, in reality they are closely connected. How can we use insights from automata theory to contribute to pure mathematics, and vice versa? And can we use automated theorem-proving to prove our mathematical insights "purely mechanically"?To take an example, consider additive number theory: the study of how to represent numbers as the sum of members of a given set. This is a well-studied area of pure mathematics that includes such celebrated results as Waring's theorem on sums of powers of integers (proved by Hilbert), and the Goldbach conjecture about sums of primes (still unproved). Recently, Cilleruelo, Luca, and Baxter proved that, for bases b 5, every natural number is the sum of at most three numbers whose base-b representation is a palindrome (a number that reads the same forwards and backwards, like the English word radar). But they were unable to prove this for bases b = 2, 3, 4.My collaborators and I completed the additive theory of the palindromes for the remaining cases, using automata theory and a decision procedure. For example, to handle the case of base b = 2, we rephrased the assertion "every natural number is the sum of at most four binary palindromes" as a claim about the computational behavior of a particular automaton A. We then used a known decision procedure for the universality problem for this class of automata to prove that our automaton A has the specified behavior. This is just one of many similar problems that are amenable to this approach.I propose to apply these ideas to many other problems in number theory, combinatorics, and algebra. I will identify suitable problems, search for appropriate computational models that can resolve them, and apply decision procedures to prove the theorems. I will also direct the preparation of free software, so that other mathematicians and computer scientists can use this approach in their own work. Already some open-source software, called Walnut, has been created by my student Hamoon Mousavi, and is being used by other researchers.My work actively involves the training of highly-qualified personnel, ranging from undergraduate students to postdoctoral researchers. These are essential to my work, both for solving problems and for writing software.

总的来说，我的研究涉及两个不同的领域以及它们如何相互作用。第一个领域涉及简单计算机（“自动机”）的数学模型及其能力。第二个领域涉及纯数学的基础知识：数论、组合学和代数。虽然这两个领域表面上看起来是分开的，但实际上它们是紧密相连的。我们如何利用自动机理论的见解为纯数学做出贡献，反之亦然？我们能否使用自动定理证明来“纯粹机械地”证明我们的数学见解？举个例子，考虑加性数论：研究如何将数字表示为给定集合中成员的总和。这是纯数学中一个被充分研究的领域，包括一些著名的结果，如关于整数幂和的沃林定理（由希尔伯特证明），以及关于素数和的哥德巴赫猜想（尚未被证明）。最近，Cilleruelo、Luca和Baxter证明，对于基数b5，每个自然数都是最多三个以b为基数的数字的和，这些数字的基数b表示为回文（一种向前和向后读相同的数字，就像英语单词radar）。但是他们不能证明b = 2,3,4。我和我的合作者利用自动机理论和决策程序，完成了剩余情况下回文的加性理论。例如，为了处理基数b = 2的情况，我们将断言“每个自然数是最多四个二进制回文的和”重新表述为关于特定自动机a的计算行为的声明。然后，我们使用已知的决策过程来证明这类自动机a具有指定的行为。这只是适用于这种方法的许多类似问题之一。我建议将这些思想应用于数论、组合学和代数中的许多其他问题。我将确定合适的问题，寻找合适的计算模型来解决它们，并应用决策过程来证明定理。我还将指导自由软件的准备工作，以便其他数学家和计算机科学家可以在他们自己的工作中使用这种方法。我的学生Hamoon Mousavi已经开发了一些开源软件，叫做Walnut，并且正在被其他研究人员使用。我的工作积极涉及高素质人才的培养，从本科生到博士后研究人员。这些对我的工作都是必不可少的，无论是解决问题还是编写软件。