The Quantifier Alternation Hierarchy of Synchronous Relations

Authors Diego Figueira, Varun Ramanathan, Pascal Weil

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Author Details

Diego Figueira
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
  • CNRS, ReLaX, UMI2000, Siruseri, India
Varun Ramanathan
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
  • CNRS, ReLaX, UMI2000, Siruseri, India
Pascal Weil
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
  • CNRS, ReLaX, UMI2000, Siruseri, India

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Diego Figueira, Varun Ramanathan, and Pascal Weil. The Quantifier Alternation Hierarchy of Synchronous Relations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Transducers
  • synchronous relations
  • automatic relations
  • first-order logic
  • characterization
  • quantifier alternation


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