发表机构
Paisii Hilendarski University of Plovdiv（保加利亚普罗夫迪夫帕伊西·希伦达斯基大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究受保护幂语言算子的不动点与Picard迭代，证明上下文无关种子语言唯一确定不动点并给出文法构造，且在特例中实现初始语言恢复与轨道分类。
AI 中文摘要
我们研究了受保护的q-幂语言算子的不动点与有限Picard迭代的语言理论结构。对于一般算子，我们证明了上下文无关的种子语言唯一确定一个上下文无关的不动点，并给出了生成该不动点的上下文无关文法的有效构造，该构造独立于初始语言。随后，我们考虑一个带标记的单保护特例，其中两个新符号将递归贡献与种子贡献分开。在此设置下，初始语言可以通过正则切片和固定词商从每个有限Picard迭代中精确追踪并恢复。这产生了有限时间映射的单射性以及一个抽象的有限时间类保持原理。作为推论，我们根据初始语言在乔姆斯基层级中的确切位置，对有限Picard迭代进行了分类。因此，精确的语言理论复杂度可能在每个有限阶段持续存在，而所有Picard轨道收敛到同一上下文无关不动点。
英文摘要
We study the language-theoretic structure of fixed points and finite Picard iterates for guarded q-power language operators. For the general operator, we prove that a context-free seeded language deter- mines a unique context-free fixed point and give an effective construction of a context-free grammar generating this fixed point, independently of the initial language.We then consider a marked single-guard special case in which two new symbols separate the recursive contribution from the the contribution of the seed. In this setting, the initial language can be traced and recovered exactly from every finite Picard iterate by means of a regular slice and fixed-word quotients. This yields injectivity of the finite-time maps and an abstract finite-time class-preservation prin- ciple. As a consequence, we obtain a classification of the finite Picard iterates according to the exact position of the initial language in the Chomsky hierarchy. Thus the exact language-theoretic complexity may persist at every finite stage, while all Picard orbits converge to the same context-free fixed point.
Journal refScientific Works of Paisii Hilendarski University of Plovdiv, Vol. 41, Book 3, 2026, Mathematics, pp. 34-56
