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.

从广义上讲，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”）的数学模型及其功能。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很分离，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？我们可以使用自动定理证明来“纯粹机械地”证明我们的数学见解吗？举个例子，考虑加性数论：研究如何将数字表示为给定集合的成员之和。这是纯数学中一个经过深入研究的领域，其中包括诸如关于整数幂和的华林定理（由希尔伯特证明）和关于素数和的哥德巴赫猜想（尚未证明）等著名成果。最近，Cilleruelo、Luca 和 Baxter 证明，对于以 b 为 5 的基数，每个自然数最多是三个以 b 为基数的表示形式为回文的数字之和（即向前和向后读相同的数字，如英语单词雷达）。但他们无法证明基数 b = 2, 3, 4 的这一点。我和我的合作者使用自动机理论和决策程序完成了其余情况的回文加法理论。例如，为了处理基数 b = 2 的情况，我们将断言“每个自然数最多是四个二进制回文数之和”重新表述为关于特定自动机 A 的计算行为的断言。然后我们使用了一个已知的决策此类自动机的普遍性问题的过程，以证明我们的自动机 A 具有指定的行为。这只是适合这种方法的许多类似问题之一。我建议将这些想法应用到数论、组合学和代数中的许多其他问题。我将识别合适的问题，寻找可以解决这些问题的合适的计算模型，并应用决策程序来证明定理。我还将指导自由软件的编写，以便其他数学家和计算机科学家可以在自己的工作中使用这种方法。我的学生 Hamoon Mousavi 已经创建了一些名为 Walnut 的开源软件，并正在被其他研究人员使用。我的工作积极涉及高素质人员的培训，从本科生到博士后研究人员。这些对于我的工作至关重要，无论是解决问题还是编写软件。