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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664841
• 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和互动。使用自动机理论的见解有助于纯数学，反之亦然？作为纯数学的夏天，脚趾数字包括Waring的著名成果，例如Hilbert的整数总和（Hilbert证明），以及有关PRAPERS的Goldbach猜想（仍然没有证明） ，Baxter证明，对于B≥5的基础，每个自然数字是三个数字的总和。 ***我的合作者和剩余的al palindromes，例如使用自动机和复仇者来处理基本b = 2的情况，我们改写了“每个自然数字。关于此类的特定自动机的索赔，我只是与该方法相似的问题之一。 ，搜索可以解决它们的适当计算模型，并应用了Hematicians，计算机科学家可以在自己的工作中使用此方法。 D人员，从地下学生到博士后研究人员，这对于我的工作至关重要，无论是解决问题还是写作软件。