A Generalization of the Stochastic-Tarski Preservation Theorem -论文摘要
A Generalization of the Łoś-Tarski Preservation Theorem - Dissertation Summary
摘要：
本文简要介绍了作者的博士学位。论文（arXiv：1609.06297）。除了概述的概念和结果，它还提供了各种证明和简化介绍的论文，涉及树表示的结构的某些抽象结果的草图。此外，本文的一些结果的推广。其中包括论文中引入的模型理论概念与稀疏图类结构理论中的固定参数易处理性和概念之间的联系。论文的模型理论结果的证明的建设性方面被用来获得（算法）元内核的各种密集的图形，如有界的树的宽度和子类的这些像$m$ -particular上图和图形类的有界的灌木深度的图。最后，本文给出了关于逻辑分形概念的更新定义和结果，逻辑分形是本文中等价有界子结构性质的推广。特别是，我们的研究结果表明，（自然有限适应）的向上和向下版本的Löwenheim-Skolem定理从经典模型理论可以恢复在各种算法有趣的设置，并进一步在大多数情况下，在有效的形式，甚至超过一阶逻辑。
英文摘要：
This article gives a summary of the author's Ph.D. dissertation (arXiv:1609.06297). In addition to an overview of notions and results, it also provides sketches of various proofs and simplified presentations of certain abstract results of the dissertation, that concern tree representations of structures. Further, some extensions of the dissertation results are presented. These include the connections of the model-theoretic notions introduced in the thesis with fixed parameter tractability and notions in the structure theory of sparse graph classes. The constructive aspects of the proofs of the model-theoretic results of the dissertation are used to obtain (algorithmic) meta-kernels for various dense graphs such as graphs of bounded clique-width and subclasses of these like $m$-partite cographs and graph classes of bounded shrub-depth. Finally, the article presents updated definitions and results concerning the notion of logical fractals which is a generalization of the Equivalent Bounded Substructure Property from the dissertation. In particular, our results show that (natural finitary adaptations of) both the upward and downward versions of the Löwenheim-Skolem theorem from classical model theory can be recovered in a variety of algorithmically interesting settings, and further in most cases, in effective form and even for logics beyond first order logic.
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