Finite and Infinite Model Theory and Applications

该项目的重点是处理有限结构类的问题，以及o-极小的变体和扩展。大部分的努力针对有限结构的研究涉及渐近类的有限结构，其无限类似物（可测结构），和强大的类的有限结构。从广义上讲，这项研究的目的是发展一个模型理论类的有限结构，是在类比与主流模型理论的无限结构。其他正在进行的工作，涉及有限的模型将解决一个猜想的相对表达能力的两个逻辑，特别是将证明，这些逻辑之一不能捕捉多项式时间的计算复杂性类。预计的研究致力于问题的扩展o-极小集中在继续发展的模型理论的有序结构的秩大于一。已经取得了重要的基础性成果，拟议的调查建立在这些基础上。其中一些工作似乎对数理经济学中的偏好和效用理论，以及经济理论的其他方面有着有趣的应用。另一个方面的研究要进行处理的扩展o-极小关注的问题所产生的问题，从以前的工作扩展o-极小结构的可定义的开放集形成一个o-极小约简的原始结构。上述研究概述了其基础模型理论，一个主要的子领域的数理逻辑。模型理论学家研究的是熟悉的数学结构的性质，这些结构可以用形式化的数学语言（如谓词逻辑）来表达。这种独特的观点可以提供对这种结构的理解和见解，否则就不容易获得。该项目的两个主要方面之一涉及有限结构类，即其域由有限集组成的数学结构类。有限结构通常是计算机科学的核心：任何数据库都可以被解释为有限结构，在这个意义上，它们在这里被研究，并且称为有限域的有限结构的特定类别在密码学中特别重要。这个项目的第二个主要方面集中在包括和表现在重要方面的结构，如真实的数的有序域，即，真实的数与在微积分中研究的多项式和代数函数一起，描述了物理和生命科学以及更定量的社会科学中的广泛现象。从模型理论的角度来看，研究加深了我们对熟悉的数学系统在不同领域的数学科学，如分析和几何的真实的功能，神经网络，关系数据库理论和估计理论的统计。一些最有趣的应用所提出的研究涉及并统一在一个单一的框架内的新古典主义理论的效用在经济学和条件估值理论，已成为突出的环境经济学，例如。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外展计划，面向当地一所高中的学生，该计划由他发起并自成立以来一直指导。

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.