插入、洗牌和交叉语言操作的相邻相等性的不可判定性

中文摘要 AI 辅助

我们研究一组基于插入、洗牌和交叉的语言操作，探讨相邻相等性以及有限收敛和相关谱问题的不可判定性。形式语言上的插入和洗牌操作出现在形式语言理论、并发模型和生物启发计算中。本文研究与常规闭包问题不同的问题：即通过重复插入生成的语言递增序列，或通过增加有界洗牌的允许度生成的语言递增序列，是否会在有限阶段后达到相邻相等性实例。我们证明多个此类相邻相等性问题是不可判定的，具体而言，对于以下各类情况，达到相邻相等性事件是不可判定的：正则语言对上下文无关语言的迭代插入；正则语言与上下文无关语言的有界洗牌（随界限增大）；以及上下文无关语言对应的自插入和自有界洗牌层次结构。新的归约直接源自上下文无关语言普遍性的不可判定性，采用分隔符分隔的块结构，对于自操作则采用违规防护的吸收正则语言。早期基于迹的证明依赖于死亡率和通用停机问题。更广泛地说，我们研究插入和有界洗牌生成的层次结构中的有限阶段相等性和稳定化（持久相等性）。除了为早期不可判定性结果提供大幅简化的证明外，我们还获得了一步相等性的通用准则，开发了自插入的新归约，并确定了若干开放问题，包括关于插入深度和度的结构问题，其解决将决定相邻相等性是否必然意味着永久稳定化。

英文摘要

We study a family of language operations based on insertion, shuffle, and crossover and investigate the undecidability of adjacent equality together with finite convergence and associated spectrum questions. Insertion and shuffle operations on formal languages arise in formal language theory, models of concurrency, and biologically inspired computation. This paper studies a different question from the usual closure problem, specifically whether an increasing sequence of languages generated by repeated insertion, or by increasing the permitted degree of bounded shuffle, reaches an instance of adjacent equality after finitely many stages. We show that several such adjacent equality questions are undecidable. In particular, reaching such an adjacent equality event is undecidable for each of the following: iterated insertion of a regular language into a context-free language; bounded shuffle of a regular language with a context-free language as the bound increases; and the corresponding self-insertion and self-bounded-shuffle hierarchies for context-free languages. The new reductions proceed directly from the undecidability of context-free-language universality, using separator-delimited block constructions and, for self-operations, an absorbing regular language of guard violations. Earlier trace-based proofs relied on mortality and uniform halting. More generally, we investigate finite-stage equality and stabilization (persistent equality) in hierarchies generated by insertion and bounded shuffle. In addition to giving substantially simpler proofs of earlier undecidability results, we obtain general criteria for one-step equality, develop new reductions for self-insertion, and identify several open problems, including structural questions concerning insertion depth and degree whose resolution determines whether adjacent equality necessarily implies permanent stabilization.

发表机构

• University of Central Florida（中佛罗里达大学）

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