Research in universal algebra and constraint satisfaction

普适代数与约束满足研究

• 批准号: RGPIN-2014-04009
• 负责人: Willard, Ross
• 金额: $ 1.31万
• 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=668588
• 关键词: Research; universal; algebra; constraint; satisfaction

Ordinary algebra studies the laws of addition, subtraction, multiplication and division of ordinary numbers. Branches of modern algebra study certain "nonstandard" systems of algebra which arise in various contexts. A simple example is Boolean algebra, which is the system of laws modeled by the operators AND, OR, XOR ("exclusive OR"), and NOT as they operate on the two boolean truth values 0 ("false") and 1 ("true"). Much more complicated systems of algebra, most 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 seeks to solve several long-standing conjectures in universal algebra and theoretical computer science. 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; my research will aim to extend the domain in which the conjecture is known to be true.**The second conjecture is a famous problem from theoretical computer science called the "Constraint Satisfaction Problem Dichotomy Conjecture." This 15-year-old conjecture asserts that, for a certain class of computational problems, each problem in the class is either computationally relatively easy, or is impossibly hard (in a precise sense); in other words, there is no problem in the class with intermediate difficulty. It turns out that the tools of universal algebra are especially useful in tackling this problem. My research aims to significantly enlarge the cases for which the conjecture is confirmed. I will also work to extend the domain for which a related conjecture, regarding those computational problems in the class that can be solved very easily, is confirmed.**A final cluster of conjectures on which I will work 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. I hope to shed light on these mysteries by finding reasons to justify the observed patterns.**This is pure, curiosity-driven research. It 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。现代代数的分支研究在各种背景下出现的某些“非标准”代数系统。一个简单的例子是布尔代数，它是由运算符 AND、OR、XOR（“异或”）和 NOT 建模的法则系统，因为它们对两个布尔真值 0（“假”）和 1（“真的”）。更复杂的代数系统，其中大多数都很奇怪，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。**该提案资助的研究旨在解决泛代数和理论计算机科学中的几个长期存在的猜想。 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。我的研究旨在扩展已知猜想正确的领域。 **第二个猜想是理论计算机科学中的一个著名问题，称为“约束满足问题二分猜想”。 这个 15 年前的猜想断言，对于某一类计算问题，该类中的每个问题要么在计算上相对容易，要么极其困难（在精确意义上）；也就是说，中等难度的课是没有问题的。 事实证明，通用代数工具对于解决这个问题特别有用。 我的研究旨在大幅扩大证实该猜想的案例。 我还将努力扩展一个相关猜想的领域，该猜想涉及类中可以很容易解决的那些计算问题。**我将研究的最后一组猜想涉及某些“不可约”的分布“非标准代数系统的模型。 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。**这是纯粹的、好奇心驱动的研究。它将服务于全世界纯数学家和理论计算机科学家社区，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。