AI 中文总结
研究自动机组合问题，引入Σ-链积这一受限变体，它能实现线性大小表示且比级联积简洁指数倍，证明其与特定自动机类表达等价，得出语言正则性与Σ-链识别的关系，并分析了重置自动机Σ-链的结构特性。
AI 中文摘要
级联积是自动机理论中的基本构造，能实现自动机的分层组合，但表示级联所需的指数大小限制了其应用。为此引入Σ-链积，它是受限变体，各组件仅依赖输入字母表和前一个组件。研究表明Σ-链积能实现线性大小表示，比级联积简洁指数倍。还证明了Σ-链积与级联积和特定自动机类在表达上等价，得出语言正则当且仅当能被置换重置自动机的Σ-链识别。最后分析了重置自动机Σ-链的结构特性。
英文摘要
The cascade product is a fundamental construction in automata theory, enabling hierarchical composition of automata and playing a central role in decomposition results such as the Krohn-Rhodes theorem. However, its use is limited by the exponential size required to represent cascades, which stems from the fact that each component may depend on all preceding ones, leading to exponentially large alphabets. To address this issue, we introduce the $Σ$-chain product, a restricted variant in which each component depends only on the input alphabet and the component immediately preceding it. We show that $Σ$-chains achieve linear-size representations and can be exponentially more succinct than cascades. We prove that $Σ$-chains and cascades are expressively equivalent to specific classes of automata, such as permutation-reset automata. As a consequence, we derive that a language is regular if and only if it is recognized by a $Σ$-chain of permutation-reset automata. Finally, we analyze structural properties of $Σ$-chains of reset automata, including a relation with well-known subclasses of star-free languages.
