Dissecting power of intersection of two context-free languages
Josef Rukavicka
TL;DR
The paper addresses the problem of dissecting infinite languages by leveraging intersections of two context-free languages to dissect geometrically growing languages. It introduces balanced extended non-associative words, obtained as $ENW\cap BAL$, and reduces geometric growth to a constant-growth framework via a homomorphic image, culminating in a theorem that there exist $M_1,M_2\subseteq \Theta^*$, an erasing homomorphism $\pi$, and a nonerasing homomorphism $\varphi$ such that for any geometrically growing $L\subseteq \Gamma^*$, a regular $R\subseteq \Theta^*$ yields $\varphi^{-1}(\pi(R\cap M_1\cap M_2))$ that dissects $L$. This generalizes REG-dissectibility to geometrically growing languages and ties dissectibility to i-separation, enabling a broader framework for studying $\mathrm{CFL}$-based dissections and suggesting further exploration of hiCFL and growth notions. The construction leverages Catalan-structure insights to relate word height to $z$-occurrence, enabling a controlled translation from geometric growth to regular witnesses.
Abstract
We say that a language $L$ is \emph{constantly growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is \emph{geometrically growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given two infinite languages $L_1,L_2$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_2\setminus L_1\vert=\infty$ and $\vert L_1\cap L_2\vert=\infty$. In 2013, it was shown that for every constantly growing language $L$ there is a regular language $R$ such that $R$ dissects $L$. In the current article we show how to dissect a geometrically growing language by a homomorphic image of intersection of two context-free languages. Consider three alphabets $Γ$, $Σ$, and $Θ$ such that $\vert Σ\vert=1$ and $\vert Θ\vert=4$. We prove that there are context-free languages $M_1,M_2\subseteq Θ^*$, an erasing alphabetical homomorphism $π:Θ^*\rightarrow Σ^*$, and a nonerasing alphabetical homomorphism $\varphi : Γ^*\rightarrow Σ^*$ such that: If $L\subseteq Γ^*$ is a geometrically growing language then there is a regular language $R\subseteq Θ^*$ such that $\varphi^{-1}\left(π\left(R\cap M_1\cap M_2\right)\right)$ dissects the language $L$.