Separating Automatic Relations
Abstract
We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations $R$ and $R'$, and asks if there exists a recognizable relation $S$ that contains $R$ and does not intersect $R'$. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation $S$ with at most $k$ products of regular languages that separates $R$ from $R'$ is undecidable, for each fixed $k \geq 2$. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.08727
 arXiv:
 arXiv:2305.08727
 Bibcode:
 2023arXiv230508727B
 Keywords:

 Computer Science  Formal Languages and Automata Theory
 EPrint:
 Long version of a paper accepted at MFCS 2023