序列-创新可归约性与创新谱
Sequential-Innovation Reducibility and the Innovation Spectrum
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中文总结 AI 辅助
该研究定义了二元序列的创新谱，提出了一种精细化真值表可归约性的序列创新可归约性，揭示了序列度结构的新几何组织，为序列可计算性研究提供了新视角。
中文摘要 AI 辅助
序列预测自然会产生由因果预测器生成的预测误差构成的创新序列。我们利用所有此类创新序列的集合，定义了单个二元序列的\textit{创新谱}，并基于序列信息提取从中导出一种新的可归约性。我们证明，该可归约性是真值表可归约性的精细化版本，同时呈现出根本不同的几何结构。其度结构分解为两个典型区域：真值表主链，其诱导序与真值表度同构；以及互补的储备免疫区域，由无法从中提取任何无限可计算可预测储备的序列构成。我们建立了连接这两个区域的桥接构造，证明储备免疫性在序列创新下保持不变，并表明马丁-洛芙随机度构成储备免疫区域内一个真向下闭子结构。这些结果揭示了基于因果可预测性而非经典谕示计算的单个序列的新几何组织。
英文摘要
Sequential prediction naturally induces an innovation sequence consisting of the prediction errors produced by a causal predictor. We use the collection of all such innovation sequences to define the \emph{innovation spectrum} of an individual binary sequence and, from it, a new reducibility based on sequential information extraction. We show that this reducibility refines truth-table reducibility while exhibiting a fundamentally different geometry. The degree structure decomposes into two canonical regions: a truth-table spine, whose induced order is isomorphic to the truth-table degrees, and a complementary reservoir-immune region, consisting of sequences from which no infinite computable predictable reservoir can be extracted. We establish bridge constructions connecting the two regions, prove that reservoir immunity is preserved under sequential innovation, and show that the Martin--Löf-random degrees form a proper downward-closed substructure inside the reservoir-immune region. These results reveal a new geometric organization of individual sequences based on causal predictability rather than classical oracle computation.
