A robust class of languages of 2-nested words

TL;DR

This paper introduces 2-wave words, a subclass of 2-nested words, which extends nested words while maintaining closure properties, decidability, and logical characterization, addressing limitations of previous models.

Contribution

It identifies 2-wave words, enabling closure under determinization and logical characterization for 2-nested words, which was previously unresolved.

Findings

Closure under determinization of automata for 2-wave words

Logical characterization of recognized languages

Graphs have bounded treewidth

Abstract

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.