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Separating Automatic Relations

Authors Pablo Barceló , Diego Figueira , Rémi Morvan

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Author Details

Pablo Barceló
  • Institute for Mathematical and Computational Engineering, Universidad Católica de Chile & CENIA & IMFD, Santiago, Chile
Diego Figueira
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
Rémi Morvan
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France

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Pablo Barceló, Diego Figueira, and Rémi Morvan. Separating Automatic Relations. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 17:1-17:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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 ⩾ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Automatic relations
  • recognizable relations
  • separability
  • finite colorability


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