Topics in Model Theory

Laskowski继续他对模型理论的研究。 在第一组问题中，他研究了在不同背景下出现的几种不同的组合情况，包括递归模型理论和分类理论。 在每一种情况下，已知存在一个可定义的基团以及该基团对组合构型的作用。 Laskowski将继续研究这样一个群体必须拥有哪些具体属性。 当组作用于配置时，这样的结果将立即产生对配置的复杂性的限制。在许多不同的环境中，Laskowski已经能够应用描述集合论的一些工具来回答模型论中的问题，主要是可数语言的理论。 最近的发展给了他希望，欧米伽1饱和模型的主要差距是触手可及的。 第三个问题是继续他的调查代数基础的各种模糊逻辑。 具体来说，他的目标是在各种模糊逻辑系统中确定哪些句子在逻辑上是有效的计算复杂性上有一个明确的界限。通过建立在以前的结果，这个问题是铸造在某些有序交换半群之间的各种嵌入问题。模型理论关注的是理论之间的相互作用（即，非常正式的语言中的句子集）和代数结构类（模型）。 随着理论的加强，理论模型的种类减少。 特别是，有些理论强大到足以阻止特定的构型出现在理论的任何模型中。 到目前为止，Laskowski之前的大部分研究，以及大部分拟议的研究，都涉及理解这种消除可能发生的机制。 模糊逻辑在计算机科学和运筹学中已经有一段时间了，但直到最近才有重大的尝试为它们提供坚实的数学基础。 Laskowski和他的研究生们最近在这方面取得了进展，研究人员将继续在这一领域工作。

Laskowski is continuing his research into model theory. In the first set of problems he investigates several distinct combinatorial situations that arise in different contexts, including recursive model theory and classification theory. In each of these situations a definable group is known to be present as well as an action of the group on the combinatorial configuration. Laskowski will continue his investigations into what specific properties such a group must possess. As the group is acting on the configuration, such results would immediately yield restrictions on the complexity of the configuration. In many different settings, Laskowski has been able to apply some of the tools from Descriptive Set Theory to answer questions in model theory, primarily for theories in a countable language. Recent developments give him hope that the Main Gap for omega1-saturated models is within reach. A third problem is to continue his investigations into the algebraic underpinnings of various fuzzy logics. Specifically, he is aiming for a sharp bound on the computational complexity of determining which sentences are logically valid in various fuzzy logic systems. By building on previous results, this question is cast in terms of various embedding questions between certain ordered abelian semigroups.Model theory is concerned with the interplay between theories (i.e., sets of sentences in a very formal language) and classes of algebraic structures (models). As one strengthens the theory, the class of models of the theory decreases. In particular, some theories are strong enough to obviate specific configurations from appearing in any model of the theory. Much of Laskowski's prior research to date, as well as much of the proposed research, concerns understanding the mechanisms by which this elimination can occur. Fuzzy logics have been useful in computer science and operations research for some time, but only recently have significant attempts been made to provide a firm mathematical basis for them. Laskowski and his graduate students have recently made progress in this regard, and the investigator will continue working in this area.