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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635176
• 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、乘法和除法定律，现代代数的分支研究在各种情况下出现的某些“非标准”代数系统，一个简单的例子是布尔代数，它是由 布尔代数 建模的法则系统。 AND、OR、XOR（“异或”）和 NOT，因为它们对两个布尔真值 0（“假”）和 1（“真”）进行运算。代数中的大部分都是奇怪的，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。通用代数是对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普遍代数和理论计算机科学中的几个长期存在的猜想。第一个猜想描述了（据信）应该暗示非标准系统定律的情况。代数都可以从一些固定的、有限的基本定律中推导出来，在很多情况下，这个猜想都是正确的；我的研究旨在扩展已知的猜想的范围。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”。这个已有 15 年历史的猜想断言，对于某一类计算问题，该类中的每个问题都是要么计算上相对容易，要么很难（精确意义上）；换句话说，在中等难度的课程中没有问题。事实证明，通用代数的工具在解决这个问题时特别有用。旨在显着扩大猜想得到证实的情况，我还将努力扩展相关猜想的领域，涉及可以很容易解决的类中的那些计算问题。最后一组猜想。我将关注的问题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大小的不可约模型时，有限大小系统的不可约模型似乎遵循我们所认为的某些规律模式。目前无法解释。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，它将服务于全世界的纯数学家和理论计算机科学家。理解代数建模的抽象数学现象。它将通过培训学生进行前沿研究，并通过解决引人注目的问题为加拿大带来声誉。