On context-free languages of scattered words

TL;DR

This paper characterizes B"uchi context-free languages of scattered words, establishing conditions for their structure, operational characterizations, and expressive power, with applications to language generation and analysis.

Contribution

It provides a precise characterization of BCFLs of scattered words via rank bounds and operational constructs, advancing understanding of their structure and expressiveness.

Findings

BCFLs of scattered words have bounded Hausdorff rank.

An MCFL of scattered words is a BCFL iff ranks are uniformly bounded.

Operational characterizations for BCFLs of well-ordered and scattered words are established.

Abstract

It is known that if a B\"uchi context-free language (BCFL) consists of scattered words, then there is an integer $n$, depending only on the language, such that the Hausdorff rank of each word in the language is bounded by $n$. Every BCFL is a M\"uller context-free language (MCFL). In the first part of the paper, we prove that an MCFL of scattered words is a BCFL iff the rank of every word in the language is bounded by an integer depending only on the language. Then we establish operational characterizations of the BCFLs of well-ordered and scattered words. We prove that a language is a BCFL consisting of well-ordered words iff it can be generated from the singleton languages containing the letters of the alphabet by substitution into ordinary context-free languages and the $\omega$-power operation. We also establish a corresponding result for BCFLs of scattered words and define expressions denoting BCFLs of well-ordered and scattered words. In the final part of the paper we give some applications.