Year: 2026
Type: preprint
Abstract: In 1980 Bucher asked whether, given context-free languages $L\subseteq U$ with $U\setminus L$ infinite, there must be a context-free language $K$ between them for which both $K\setminus L$ and $U\setminus K$ are infinite. We give a negative answer. We first construct an infinite language $D$ with context-free complement such that, for every regular language $R$, either $D\cap R$ or $D\setminus R$ is finite. The words of $D$ encode computations of factorials; repetition of letters ensures that each finite automaton either accepts all but finitely many words of $D$ or rejects all but finitely many words of $D$, while a one-counter automaton recognizes errors in the encodings. We then construct $L$ and $U$ from the complement of $D$. A grammar argument shows that any context-free intermediate language $K$ would divide $D$ in the same way as some regular language. This proves the required impossibility. Both $L$ and $U$ can be taken over a binary alphabet.
Source: arXiv (Cornell University)
Authors: Maslic, Rastko, Shallit, Jeffrey
Institution: University of Waterloo
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212038296
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