Research in universal algebra: constraint satisfaction and residual properties

普适代数研究：约束满足和剩余性质

• 批准号: RGPIN-2019-03931
• 负责人: Willard, Ross
• 金额: $ 1.24万
• 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=764423
• 关键词: Research; universal; algebra; constraint; satisfaction

Ordinary algebra studies the laws of addition, subtraction, multiplication and division of ordinary numbers. Modern algebra studies "nonstandard" systems of algebra which arise in various contexts. A simple example is Boolean algebra, which is the system of laws modeled by the logical operators AND, OR, and NOT as they operate on the two truth values 0 ("false") and 1 ("true"). Much more complicated systems of algebra, many of them bizarre, some of them useful in physics, chemistry, theoretical computer science and engineering, can be invented, studied, and modeled. Universal algebra is the general study of patterns in, and the limits of, nonstandard laws of algebra and their models. The research to be funded by this proposal has three major parts. First, my team and I will study and reformulate the solutions, announced independently in 2017 by Andrei Bulatov at Simon Fraser University and Dmitriy Zhuk at Lomonosov Moscow State University, of a 20-year-old conjecture in theoretical computer science. This conjecture, called the Constraint Satisfaction Problem Dichotomy Conjecture, asserts that, for a certain class of computational problems, each problem in the class is either computationally easy, or is impossibly hard (in a precise sense); in other words, no problem in the class has intermediate difficulty. The tools of universal algebra were especially useful in proving this conjecture. The goal of this part of my research will be to develop new theories within universal algebra to better explain the discoveries of Bulatov and Zhuk. In the second part, my team and I will work on two 40-year-old unsolved conjectures concerning patterns involving nonstandard laws of algebra and their finite models. The first conjecture describes circumstances which (it is believed) should imply that the laws of a nonstandard system of algebra will all be deducible from some fixed, finite set of basic laws. This conjecture is known to be true in a very large number of cases; our research will aim to extend the domain in which the conjecture is known to be true. The second conjecture concerns the distribution of certain "irreducible" models of a nonstandard system of algebra. Under rather weak assumptions, when an algebraic system has no irreducible models of infinite size, the irreducible models of the system of finite size seem to obey certain patterns of regularity that we currently cannot explain. My team and I hope to shed light on these mysteries by finding reasons to justify the observed patterns. The final part of this project involves exploratory investigations of two additional open problems in universal algebra. This research will serve the world-wide community of pure mathematicians and theoretical computer scientists who seek to understand abstract mathematical phenomena modeled by algebra. It will serve Canada by training students in cutting-edge research, and in bringing prestige to Canada through the solution to high-profile problems.

普通代数研究普通数的加法、减法、乘法和除法的规律。现代代数研究在各种背景下出现的“非标准”代数系统。一个简单的例子是布尔代数，它是由逻辑运算符 AND、OR 和 NOT 建模的法则系统，因为它们对两个真值 0（“假”）和 1（“真”）进行运算。更复杂的代数系统，其中许多是奇怪的，其中一些在物理、化学、理论计算机科学和工程中有用，可以被发明、研究和建模。泛代数是对代数及其模型的非标准定律的模式和限制的一般研究。该提案资助的研究分为三个主要部分。首先，我和我的团队将研究并重新制定由西蒙弗雷泽大学的 Andrei Bulatov 和莫斯科国立大学的 Dmitriy Zhuk 于 2017 年独立宣布的解决方案，该解决方案是理论计算机科学领域已有 20 年历史的猜想。这个猜想被称为约束满足问题二分猜想，它断言，对于某一类计算问题，该类中的每个问题要么计算上容易，要么很难（精确意义上）；换句话说，课堂上没有任何问题是中等难度的。泛代数工具对于证明这个猜想特别有用。我这部分研究的目标是在泛代数中发展新理论，以更好地解释布拉托夫和朱克的发现。在第二部分中，我和我的团队将研究两个 40 年来未解决的猜想，涉及涉及非标准代数定律及其有限模型的模式。第一个猜想描述的情况（据信）应该意味着非标准代数系统的定律都可以从一些固定的、有限的基本定律组中推导出来。众所周知，这个猜想在很多情况下都是正确的。我们的研究旨在扩展已知猜想正确的领域。第二个猜想涉及非标准代数系统的某些“不可约”模型的分布。在相当弱的假设下，当代数系统没有无限尺寸的不可约模型时，有限尺寸系统的不可约模型似乎遵循我们目前无法解释的某些规律性模式。我和我的团队希望通过寻找理由来证明观察到的模式的合理性，从而揭开这些谜团。该项目的最后部分涉及对通用代数中另外两个开放问题的探索性调查。这项研究将为全世界的纯数学家和理论计算机科学家提供服务，他们寻求理解代数建模的抽象数学现象。它将通过培训学生进行尖端研究来为加拿大服务，并通过解决引人注目的问题为加拿大带来声誉。