发表机构
University of Bremen; RWTH Aachen University（不来梅大学; 亚琛工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明每一种正则语言都存在二分仙人掌自动机，其转移图为二分仙人掌结构，比汤普森构造的串并联结构更简单，证明基于奇偶类型正则表达式的归纳及仙人掌图的基本闭包性质。
AI 中文摘要
汤普森的经典构造将每一个正则表达式转化为等价的ε-NFA，其转移图为串并联结构。该构造使用ε-转移，而通过常规捷径构造消除这些转移可能无法保持平面性或转移图树宽不超过2的界限。不过，布克和钱德拉证明，每一种正则语言都存在无ε-转移且转移图为平面结构的NFA。本文进一步证明，该转移图实际上可选择为二分仙人掌结构：每个环都是偶环，且任意两个环最多仅有一个公共顶点。特别地，该结构是外平面图，且树宽不超过2，比汤普森构造所保证的一般串并联结构简单得多。证明过程是对奇偶类型正则表达式的直接归纳，仅使用仙人掌图的基本闭包性质。
英文摘要
Thompson's classical construction turns every regular expression into an equivalent epsilon-NFA whose transition graph is series-parallel. The construction uses epsilon-transitions, and eliminating them by the usual shortcut construction need not preserve planarity or the bound of two on the treewidth of the transition graph. Book and Chandra proved that every regular language nevertheless has an epsilon-free NFA with a planar transition graph. We show that the transition graph can in fact be chosen to be a bipartite cactus: every cycle is even, and any two cycles have at most one vertex in common. In particular, it is outerplanar and has treewidth at most two. This is a considerably more simple structure than the general series-parallel guarantee supplied by Thompson's construction. The proof is a direct induction on parity-typed regular expressions and uses only elementary closure properties of cactus graphs.
CommentsIn Proceedings GandALF 2026, arXiv:2610.08898
Journal refEPTCS 454, 2026, pp. 117-129
