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Computer Science > Formal Languages and Automata Theory

arXiv:2305.08727 (cs)
[Submitted on 15 May 2023 (v1), last revised 2 Aug 2023 (this version, v2)]

Title:Separating Automatic Relations

Authors:Pablo Barceló, Diego Figueira, Rémi Morvan
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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.
Comments: Long version of a paper accepted at MFCS 2023
Subjects: Formal Languages and Automata Theory (cs.FL)
Cite as: arXiv:2305.08727 [cs.FL]
  (or arXiv:2305.08727v2 [cs.FL] for this version)
  https://doi.org/10.48550/arXiv.2305.08727

Submission history

From: Rémi Morvan [view email]
[v1] Mon, 15 May 2023 15:41:53 UTC (250 KB)
[v2] Wed, 2 Aug 2023 09:38:12 UTC (250 KB)
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