Finite and Infinite Model Theory and Applications

该项目集中在涉及有限结构的类以及o-极小的变体和扩展的问题上。大多数致力于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(可测量的结构)和稳健的有限结构类。广义地说，这项研究的目的是发展一种类似于无限结构的主流模型理论的有限结构模型理论。其他正在进行的涉及有限模型的工作将解决关于两个逻辑的相对表达能力的猜想，这特别将证明其中一个逻辑不能捕获多项式时间计算复杂性类。致力于o-极小扩张问题的计划研究集中在继续发展一阶以上的有序结构的模型理论。已经取得了重要的基础性成果，拟议的调查建立在这些成果的基础上。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，可能还会应用到经济理论的其他方面。关于o-极小的扩展的研究的另一个方面涉及到以前关于o-极小结构的扩展的问题，其可定义开集形成原始结构的o-极小约简。上面概述的研究以数理逻辑的主要子领域之一的模型理论为基础。模型理论家研究可以用谓词逻辑等形式数学语言表达的熟悉的数学结构的性质。这种独特的观点可以提供对这种结构的理解和洞察，否则就不容易获得这些结构。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。有限结构通常是计算机科学的中心：任何数据库都可以被解释为在这里研究它们的意义上的有限结构，而被称为有限域的一类特定的有限结构在密码学中尤其重要。这个项目的第二个主要方面集中在包括并在重要方面表现的结构，如实数的有序域，即在微积分中研究的实数以及多项式和代数函数，并描述物理和生命科学以及更定量的社会科学中的广泛现象。从模型理论观点出发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，如实函数的分析和几何、神经网络、关系数据库理论和统计学中的估计理论。例如，拟议研究的一些最有趣的应用涉及并统一了经济学中的新古典主义效用理论和在环境经济学中已变得突出的条件估值理论。在获奖期间，首席研究员还打算继续指导Vassar科学学者计划，这是一个学年科学和数学推广计划，面向具有市中心人口统计学特征的当地高中的学生，他发起并自成立以来一直指导该计划。

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.