Context-Free Fixed Points and Complete Classification of Orbits in Picard Iteration for Guarded Power Language Operators

2026-09-16

AI Summary

研究了守卫q幂语言算子的上下文无关固定点及有限Picard迭代，通过构造上下文无关文法生成固定点，并对特定情况下的迭代进行分类。

Abstract

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.

Problem

• context-free fixed points
• Picard iteration
• guarded power language operators

Innovation

• context-free fixed point
• guarded q-power language operators
• finite Picard iterates
• Chomsky hierarchy