思维链（Chain-of-Thought, CoT）展现了通往树的路径：实现分支复杂性

中文摘要 AI 辅助

思维链（CoT）提升了有界深度Transformer的表达上限，其特征将CoT步骤数与电路复杂性类关联起来。目前仍普遍缺乏带有显式有界深度构造的具体实例，以及此类特征所预设的遍历过程。我们针对分支复杂性填补了这一空白：通过至多两层的独特硬注意力解码器，给出了深度优先搜索（DFS）和迪杰斯特拉（Dijkstra）算法的CoT实现，其中迪杰斯特拉算法已包含广度优先搜索；并将它们用作共享计算基础：复用DFS解码器，可在4层结构下以2n-1步得到n顶点树的斯特拉勒数（Strahler number）；复用迪杰斯特拉解码器，可在3层结构下以n-1步得到其宽度。由于将项形式给出的二叉树的斯特拉勒数计算是\textsf{NC\textsuperscript{1}}完全问题，且我们的构造可处理任意n元树，无需层归一化或位置编码，这为CoT层次的线性步长机制提供了非平凡的见证。利用有序树与Dyck路径之间的经典双射（其本身由我们的DFS构造实现，DFS在遍历时输出路径），我们在路径表示上为这两个度量分别给出了独立构造。

英文摘要

Chain of Thought (CoT) lifts the expressive ceiling of bounded-depth Transformers, with characterizations tying the number of CoT steps to circuit complexity classes. What remains largely missing are concrete instantiations with explicit, depth-bounded constructions, and the traversal procedures such characterizations presuppose. We close this gap for branching complexity. We give CoT realizations of depth-first search (DFS) and of Dijkstra algorithm, the latter subsuming breadth-first search, by unique hard-attention decoders of at most two layers, and use them as a shared computational substrate: reusing the DFS decoder yields the Strahler number of an $n$-vertex tree in $2n-1$ steps with four layers, and reusing the Dijkstra decoder yields its width in $n-1$ steps with three. Since computing the Strahler number of a binary tree given as a term is \textsf{NC\textsuperscript{1}}-complete, and our constructions handle arbitrary $n$-ary trees without layer normalization or positional encodings, this is a non-trivial witness for the linear-step regime of the CoT hierarchy. Exploiting the classical bijection between ordered trees and Dyck paths, itself realized by our DFS construction, which emits the path as it traverses, we give independent constructions for both measures on the path representation.

发表机构

• Indian Statistical Institute（印度统计研究所）
• LTCI, Télécom Paris（法国电信巴黎高等学院 LTCI 实验室）

机构由 AI 辅助整理，请以论文原文为准。