# Regular Separability and Intersection Emptiness are Independent Problems

## TL;DR

This paper investigates the relationship between regular separability and intersection emptiness problems, showing through counterexamples that these problems are independent and cannot be reduced to each other under mild assumptions.

## Contribution

It provides counterexamples demonstrating that regular separability and intersection emptiness are independent problems, refuting the possibility of reducing one to the other.

## Key findings

- Intersection emptiness is decidable where regular separability is undecidable.
- Regular separability is decidable where intersection emptiness is undecidable.
- The two problems are fundamentally independent, with no reduction possible.

## Abstract

The problem of \emph{regular separability} asks, given two languages $K$ and $L$, whether there exists a regular language $S$ with $K\subseteq S$ and $S\cap L=\emptyset$. This problem has recently been studied for various classes of languages. All the results on regular separability obtained so far exhibited a noteworthy correspondence with the intersection emptiness problem: In eachcase, regular separability is decidable if and only if intersection emptiness is decidable. This raises the question whether under mild assumptions, regular separability can be reduced to intersection emptiness and vice-versa. We present counterexamples showing that none of the two problems can be reduced to the other. More specifically, we describe language classes $\mathcal{C_1}$, $\mathcal{D_1}$, $\mathcal{C_2}$, $\mathcal{D_2}$ such that (i)~intersection emptiness is decidable for $\mathcal{C_1}$ and $\mathcal{D_1}$, but regular separability is undecidable for $\mathcal{C_1}$ and $\mathcal{D_1}$ and (ii)~regular separability is decidable for $\mathcal{C_2}$ and $\mathcal{D_2}$, but intersection emptiness is undecidable for $\mathcal{C_2}$ and $\mathcal{D_2}$.