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: $ 3.5万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712746
• 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的相互作用。第一个领域涉及简单计算机（“自动机”）及其功能的数学模型。第二个领域涉及纯数学的基础：数字理论，组合学和代数。尽管这两个领域在表面上似乎很分开，但实际上它们是密切相关的。我们如何利用自动机理论的见解来促进纯数学，反之亦然？我们可以使用自动化的定理提供“纯粹是机械上”证明我们的数学见解吗？ 举个例子，考虑添加数理论：如何将数字表示为给定集的成员之和的研究。这是一个纯粹的数学领域，其中包括沃林（Waring）定理的整数力量（由希尔伯特（Hilbert）证明），以及关于素数总和的戈德巴赫（Goldbach）猜想（仍然未证实）。最近，Cilleruelo，Luca和Baxter证明，对于基地B 5，每个自然数字最多是三个数字的总和，其基本-B表示为palindrome（这个数字读取相同的前向和后退，例如英语单词雷达）。但是他们无法证明这一点是B = 2、3、4。 我和我的合作者使用自动机理论和决策程序完成了其余案例的allindromes的加性理论。例如，为了处理基本b = 2的情况，我们改写了“每个自然数字最多是四个二元palindromes的总和”作为有关特定自动机的计算行为的主张。然后，我们使用了已知的决定该类别自动机的通用性问题的过程证明我们的自动机A具有指定的行为。这只是这种方法可以接受的许多类似问题之一。 我建议将这些想法应用于数字理论，组合学和代数的许多其他问题。我将确定合适的问题，搜索可以解决它们的适当计算模型，并应用决策程序来证明定理。我还将指导免费软件的准备，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，并已被其他研究人员使用。 我的工作积极涉及对高度合格的人员进行培训，从本科生到博士后研究人员。这些对我的工作至关重要，无论是解决问题还是写作软件。