Mathematical Sciences: Research in Model Theory

首席研究员将继续研究有序结构的模型理论和一元二阶逻辑的模型理论这两个大标题。在几年前开始的研究中，他研究了o-极小结构的概念。这项工作统一了几类特定的线性有序结构的模型理论，并得到了以前已知的仅在特殊情况下的一般结果。他希望考虑有关o-极小结构的几个问题。最重要的是，他计划尝试开发一种类似于线性有序结构稳定性理论的方法，并研究以一阶方式给出自然拓扑的结构类的一般模型理论的可能性。首席研究员目前还在与Y.Gurevich一起为《数理逻辑透视》系列撰写一元二阶逻辑专著。这本书将概述已经被用来证明各种结构的一元二阶理论的可判定性或不可判定性的丰富的各种方法，以及使用这些方法所获得的结果。这本书的写作将使逻辑学家和理论计算机科学家都感兴趣。

The principal investigator will continue research in model theory that falls under the two general headings of model theory for ordered structures and monadic second-order logic. In research begun several years ago, he studied the notion of an o-minimal structure. This work has unified the model theory for several specific classes of linearly ordered structures, and has yielded general results known before only in special cases. He wishes to consider several questions concerning o-minimal structures. Most importantly, he plans to attempt to develop an analogue to stability theory for linearly ordered structures, and to investigate the possibilities for a general model theory for classes of structures whose natural topology is given in a first-order way. The principal investigator also is currently engaged in writing a monograph with Y. Gurevich on monadic second-order logic for the "Perspectives in Mathematical Logic" series. This book will survey both the rich variety of methods that have been employed to prove the decidability or undecidability of the monadic second-order theory of various structures, and also the results that have been obtained using these methods. The book will be written so as to be of interest to both logicians and theoretical computer scientists.