Finite and Infinite Model Theory and Applications

有限和无限模型理论及应用

• 批准号: 0801256
• 负责人: Charles Steinhorn
• 金额: $ 12.31万
• 依托单位: Vassar College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801256&HistoricalAwards=false
• 关键词: Finite; Infinite; Model; Theory; Applications

The project focuses on questions dealing with classes of finite structures, and variants and extensions of o-minimality. Most of the effort directed toward the study of finite structures involves asymptotic classes of finite structures, their infinite analogues (measurable structures), and robust classes of finite structures. Broadly speaking, this research has as its aim the development of a model theory for classes of finite structures that is in analogy with mainstream model theory for infinite structures. Other ongoing work involving finite models will solve a conjecture on the relative expressive power of two logics, which in particular will demonstrate that one of these logics cannot capture the polynomial time computational complexity class. The projected research devoted to problems about extensions of o-minimality concentrates on the continued development of a model theory for ordered structures of rank greater than one. Important foundational results already have been obtained and the proposed investigations build on these. Some of this work appears to have intriguing applications to preference and utility theory in mathematical economics, and possibly to other aspects of economic theory. Another aspect of the research to be undertaken dealing with extensions of o-minimality concerns questions arising from previous work on expansions of o-minimal structures whose definable open sets form an o-minimal reduct of the original structure.The research outlined above has as its foundation model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures that can be expressed in a formal mathematical language such as predicate logic. This distinctive point of view can provide understanding and insights into such structures that otherwise could not be easily obtained. One of the two principal aspects of the project deals with classes of finite structures, that is, classes of mathematical structures whose domain consists of a finite set. Finite structures in general are central to computer science: any database can be interpreted as a finite structure in the sense in which they are studied in here, and a particular class of finite structures called finite fields are especially important in cryptology.The second major aspect of this project focuses on structures that include and behave in significant respects like the ordered field of real numbers, that is, the real numbers together with the polynomial and algebraic functions that are studied in calculus and describe a wide range of phenomena in the physical and life sciences, as well as in the more quantitative social sciences. Research arising from the model-theoretic point of view has deepened our understanding of familiar mathematical systems in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, relational database theory, and estimation theory in statistics. Some of the most intriguing applications of the proposed research relate to and unify within a single framework both the neoclassical theory of utility in economics and contingent valuation theory that has become prominent in environmental economics, for example. During the period of the award the principal investigator also intends to continue to direct the Vassar Science Scholars Program, an academic year science and mathematics outreach program for students from a local high school with inner city demographics which he initiated and has directed since its inception.

该项目着重于处理有限结构的类别以及O最小性的变体和扩展的问题。针对有限结构研究的大多数努力都涉及有限结构的渐近类，它们的无限类似物（可测量的结构）和有限的结构类别。从广义上讲，这项研究的目的是开发用于有限结构类别的模型理论，该理论与无限结构的主流模型理论类似。其他涉及有限模型的正在进行的工作将解决两个逻辑的相对表达能力的猜想，这特别表明，其中一种无法捕获多项式时间计算复杂度类别。预计的研究致力于有关O-最低态度扩展的问题，集中在持续开发模型理论的秩序结构大于一个的秩序结构上。已经获得了重要的基础结果，并在这些基础上进行了拟议的调查。这项工作中的某些工作似乎在数学经济学中以及可能在经济理论的其他方面中对偏好和效用理论具有有趣的应用。要处理的研究的另一个方面是处理O最低性的扩展，涉及以前关于O-Winimal结构扩展的工作引起的问题，其可确定的开放式集合形成了原始结构的O最低点降低。模型理论家研究熟悉的数学结构的属性，可以用正式的数学语言（例如谓词逻辑）表示。这种独特的观点可以为否则无法轻易获得的这种结构提供理解和见解。该项目的两个主要方面之一涉及有限结构类别，即，域由有限集组成的数学结构类别。通常，有限的结构是计算机科学的核心：在这里研究的任何数据库都可以解释为有限结构，并且在加密学中，特定类别的有限结构称为有限领域尤其重要。该项目的第二个主要方面着重于结构上，包括在重要方面与现实数字相同的重要领域，这些结构与真实的数字相同，这些都在现实数字上，这些结构是现实的，并且是现实的，并且是现实的，并且是现实的，并且是现实数字的，并且是现实数字的，这些范围既有又有效果。并描述物理和生命科学以及更具定量的社会科学中的广泛现象。从模型理论的角度进行的研究加深了我们对数学科学不同领域熟悉的数学系统的理解，例如对真实功能的分析和几何形状，神经网，关系数据库理论和统计学中的估计理论。例如，提出的研究的一些最有趣的应用与单一框架有关并统一经济学的新古典主义理论和偶然的估值理论，例如在环境经济学中变得突出。在奖项期间，首席调查员还打算继续指导Vassar Science Scholars计划，这是一项学年的科学和数学外展计划，适用于一所当地高中的学生，具有内城市人口统计，他发起并自从其成立以来就一直在执导。