A Robust Class of Languages of 2-Nested Words

Authors Séverine Fratani, Guillaume Maurras, Pierre-Alain Reynier

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Séverine Fratani
  • LIS, Aix-Marseille Univ, CNRS, Marseille, France
Guillaume Maurras
  • LIS, Aix-Marseille Univ, CNRS, Marseille, France
Pierre-Alain Reynier
  • LIS, Aix-Marseille Univ, CNRS, Marseille, France

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Séverine Fratani, Guillaume Maurras, and Pierre-Alain Reynier. A Robust Class of Languages of 2-Nested Words. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Regular nested word languages (a.k.a. visibly pushdown languages) strictly extend regular word languages, while preserving their main closure and decidability properties. Previous works have shown that considering languages of 2-nested words, i.e. words enriched with two matchings (a.k.a. 2-visibly pushdown languages), is not as successful: the corresponding model of automata is not closed under determinization. In this work, inspired by homomorphic representations of indexed languages, we identify a subclass of 2-nested words, which we call 2-wave words. This class strictly extends the class of nested words, while preserving its main properties. More precisely, we prove closure under determinization of the corresponding automaton model, we provide a logical characterization of the recognized languages, and show that the corresponding graphs have bounded treewidth. As a consequence, we derive important closure and decidability properties. Last, we show that the word projections of the languages we define belong to the class of linear indexed languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Formal languages and automata theory
  • Nested word
  • Determinization
  • Indexed languages


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