Structure Theorems in Model Theory

模型理论是数理逻辑的一个分支，本研究的重点是研究模型理论中的结构定理。在许多情况下，一个理论（即形式语言的一组句子）足够强大，可以在它的模型类上允许一个结构定理（即，代数结构满足理论的每个句子）。已知，如果一个稳定理论不允许对二阶信息进行编码，那么该理论的充分饱和模型类就存在一个强结构定理。将这样的结构定理推广到饱和度较低的模型类需要理解组合现象，通常是在可定义群的存在下。在许多情况下，描述集合论的技术是有益的。这项研究将有助于代数结构的分类。本文所研究的结构定理中的信息给出了某些神经网络复杂性的边界。

The primary emphasis of this research is the study structure theorems in model theory, which is a branch of mathematical logic. In many instances, a theory (i.e., a set of sentences in a formal language) is strong enough to allow for a structure theorem on the class of its models (i.e, the algebraic structures satisfying every sentence of the theory). It is known that if a stable theory forbids the encoding of second-order information, then there is a strong structure theorem for the class of sufficiently saturated models of the theory.Extending such a structure theorem to classes of models with less saturation requires understanding combinatorial phenomena, frequently in the presence of a definable group. In many instances, techniques from Descriptive Set Theory are beneficial.This research will contribute to the taxonomy of algebraic structures. Information present in structure theorems investigated here yield bounds on the complexity of certain neural networks.