Applied Mathematical Logic

应用数理逻辑

• 批准号: 0097881
• 负责人: Kenneth Kunen
• 金额: $ 13.6万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0097881&HistoricalAwards=false
• 关键词: Applied; Mathematical; Logic; Applied; Mathematical; Logic

The investigator's research integrates a number of diverse areas inmathematics: logic, set theory, algebra, topology, and analysis,as well as some automated reasoning techniques from computer science.In topology, the investigator focuses on properties ofStone spaces, compact homogeneous spaces, and Bohr topologies.Topology and analysis are integrated in this research; measure theory isused to construct topological spaces with interesting properties,while the topological properties of a space are used to prove theoremsabout the possible measures which can exist on the space.The investigator studies Bohr topologies, which involve giving a topologyto arbitrary abstract groups or other structures; this subject has itsroots in the harmonic analysis of the 1930s; modern questions in thisarea relate to general topology, Fourier series, and functional analysis.Logic and set theory are relevant because results in topology and measuretheory are frequently independent of the usual axioms of set theory; when aresult is proved independent, the methods used are those of formal logic. In algebra, the investigator works on algebraic systems such asquasigroups and loops. Automated reasoning tools are very usefulhere, primarily in the study of non-associative systems. Thesesystems are described by fairly simple axioms, and a computer searchcan often reveal interesting new consequences of these axioms. However, the investigator combines the computer use withclassical arguments involving combinatorics and group theory. There are two distinct, but related, threads to this research.The first thread involves the expansion of our knowledge of traditionalpure mathematics. There is no specific practical application in mindhere, although topology arises naturally in an attempt to generalizeproperties of the geometry of physical space, and measuretheory is a natural extension of the notion of probability.The second thread involves automated reasoning (AR) tools. AR allows thecomputer to derive logical conclusions from given knowledge. This subjecthas been in existence since the 1960s, but it is only in recent years thatthe hardware and software have become powerful enough to discover conclusionswhich could not have been discovered without computer assistance.This second thread is a continuation of the investigator's work inimproving the AR tools and using these tools to create new resultsin mathematics. This is of interest not only for themathematics itself, but because it demonstrates the power of the tools,which can then be applied to reasoning tasks in other areas of scienceand engineering, as well as to autonomous decision making by robotic agents.

The investigator's research integrates a number of diverse areas inmathematics: logic, set theory, algebra, topology, and analysis,as well as some automated reasoning techniques from computer science.In topology, the investigator focuses on properties ofStone spaces, compact homogeneous spaces, and Bohr topologies.Topology and analysis are integrated in this research;量度理论被用来构建具有有趣属性的拓扑空间，而空间的拓扑特性则用于证明定理在空间上可能存在的措施。该主题在1930年代的谐波分析中具有iSroots；本区中的现代问题与一般拓扑，傅立叶系列和功能分析有关。遗传学和集合理论是相关的，因为拓扑和测量结果的结果经常与集合理论的通常公理无关。当证明阿苏族独立时，所使用的方法是形式逻辑的方法。在代数中，研究人员从事代数系统（例如assquasigroups and Loops）。 自动推理工具非常有用，主要是在非缔合系统的研究中。 这些系统用相当简单的公理来描述，并且计算机搜索通常会揭示这些公理的有趣新后果。但是，研究者将计算机使用与涉及组合学和群体理论的阶级论证相结合。 这项研究有两个不同但相关的线程。第一个线程涉及我们对传统数学知识的扩展。 尽管拓扑自然而然地出现了拓扑结构，但没有特定的实际应用，这些拓扑是为了尝试对物理空间的几何形状进行概括，而测量值是概率概念的自然扩展。第二个线程涉及自动推理（AR）工具。 AR允许计算机从给定的知识中得出逻辑结论。 自1960年代以来，这一主题就存在，但是直到近年来，硬件和软件才变得足够强大，无法发现没有计算机援助的结论。这第二个线程是继续研究AR工具的工作的继续，并使用这些工具来创建新的结果。 这不仅对themathematics本身引起了人们的关注，而且因为它证明了工具的力量，然后可以将其应用于Scienceand工程其他领域的推理任务，以及机器人代理商的自主决策。