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.

该项目重点关注处理有限结构类以及 o-极小性的变体和扩展的问题。大多数针对有限结构研究的工作涉及有限结构的渐近类、它们的无限类似物（可测量结构）以及有限结构的鲁棒类。从广义上讲，这项研究的目标是开发一种类似于主流无限结构模型理论的有限结构类模型理论。其他正在进行的涉及有限模型的工作将解决关于两种逻辑的相对表达能力的猜想，这特别将证明这些逻辑之一无法捕获多项式时间计算复杂性类别。针对 o 极小性扩展问题的预计研究集中于秩大于 1 的有序结构模型理论的持续发展。重要的基础性结果已经获得，拟议的调查建立在这些结果的基础上。其中一些工作似乎对数理经济学中的偏好和效用理论有有趣的应用，也可能对经济理论的其他方面有有趣的应用。涉及 o-最小扩展的研究的另一个方面涉及先前关于 o-最小结构扩展的工作中产生的问题，这些结构的可定义开集形成了原始结构的 o-最小约简。上述研究已作为其基础模型理论，数理逻辑的主要子领域之一。模型理论家研究熟悉的数学结构的属性，这些结构可以用形式数学语言（例如谓词逻辑）来表达。这种独特的观点可以提供对此类结构的理解和见解，否则这些结构是不容易获得的。该项目的两个主要方面之一涉及有限结构的类，即其域由有限集组成的数学结构的类。一般来说，有限结构是计算机科学的核心：任何数据库都可以被解释为在这里研究的意义上的有限结构，并且一类称为有限域的特定有限结构在密码学中尤其重要。第二个主要方面该项目的重点是包含并在重要方面表现的结构，例如实数的有序域，即实数以及微积分中研究的多项式和代数函数，并描述了物理和物理领域的各种现象。生命科学，以及在更定量的社会科学中。从模型论角度进行的研究加深了我们对数学科学不同领域中熟悉的数学系统的理解，例如实函数的分析和几何、神经网络、关系数据库理论和统计中的估计理论。所提出的研究的一些最有趣的应用涉及经济学中的新古典效用理论和环境经济学中已变得突出的条件估值理论，并将其统一在一个框架内。在获奖期间，首席研究员还打算继续指导瓦萨科学学者计划，这是一个学年科学和数学推广计划，针对来自内城人口统计的当地高中的学生，该计划是他发起并自启动以来一直负责的。