AI 中文总结
研究可计算结构中斯科特分析相关概念的复杂度问题，开发好战对定理和好战跳跃反转定理，确定可计算斯科特句子等在句法与预言机复杂度间的最优作用，解决了相关来回类复杂度问题。
AI 中文摘要
斯科特分析为研究可数结构提供了两个基本工具：用于刻画同构结构的斯科特句子，以及衡量结构相似性的来回关系。在可计算结构理论中，一个反复出现的现象是，许多与斯科特分析的α层级自然相关的概念，其有效复杂度大约在2α跳跃。这种差异在来回关系的复杂度以及从任意无穷公式到可计算无穷公式的转换中都有体现。我们开发了两个新的编码工具，即好战对定理和好战跳跃反转定理，它们能让复杂度为2α的信息反映在可计算结构中，其显著特征已出现在α层级。这些结果将哈里森 - 特雷纳的有限不友好跳跃反转统一扩展到整个可计算序数。作为应用，我们确定了可计算斯科特句子和区分可计算结构的公式在句法复杂度和预言机复杂度之间的最优相互作用。对于每个可计算无穷序数α以及有Πα斯科特句子的可计算结构，我们确定了计算其Πα斯科特句子所需的预言机。任何有Πα斯科特句子 的可计算结构都有一个可计算的Π2α斯科特句子，且我们证明这两个界限都是精确的。我们还为见证α来回关系失败的公式证明了类似的最优结果，并获得了进一步的应用，包括解决了陈、冈萨雷斯和哈里森 - 特雷纳关于来回类复杂度的一个问题。
英文摘要
Scott analysis provides two fundamental tools for studying countable structures: Scott sentences, which characterize structures up to isomorphism, and back-and-forth relations, which measure structural similarity. A recurring phenomenon in computable structure theory is that many notions naturally associated with level $α$ of Scott analysis have effective complexity at approximately $2α$ jumps. This discrepancy appears both in the complexity of the back-and-forth relations and in the passage from arbitrary infinitary formulas to computable infinitary formulas. We develop two new coding tools, the Belligerent Pairs Theorem and Belligerent Jump Inversion Theorem, which allow information at complexity level $2α$ to be reflected in computable structures whose distinguishing features already appear at level $α$. These results extend Harrison-Trainor's finite unfriendly jump inversion uniformly throughout the computable ordinals. As applications, we determine the optimal interaction between syntactic complexity and oracle complexity for computable Scott sentences and for formulas distinguishing computable structures. For every computable infinite ordinal $α$, we determine the oracle needed to compute a $Π_α$ Scott sentence for a computable structure which has a $Π_α$ Scott sentence. Any computable structure with a $Π_α$ Scott sentence has a computable $Π_{2α}$ Scott sentence. We show that both of these bounds are sharp. We prove analogous optimal results for formulas witnessing failure of the $α$-back-and-forth relation. We also obtain further applications, including a resolution of a question of Chen, Gonzalez, and Harrison-Trainor concerning the complexity of back-and-forth classes.
Comments31 pages
