The Dissecting Power of Regular Languages

TL;DR

This paper explores how regular languages can dissect infinite languages, revealing new structural properties and demonstrating that various classes of languages are REG-dissectible, expanding understanding of language behaviors.

Contribution

It introduces the concept of REG-dissectibility for regular languages and shows its applicability to diverse language classes, including semi-linear and bounded languages.

Findings

Every context-free language is REG-dissectible.

Constantly-growing and semi-linear languages are REG-dissectible.

Intersections of bounded context-free languages are REG-dissectible.

Abstract

A recent study on structural properties of regular and context-free languages has greatly promoted our basic understandings of the complex behaviors of those languages. We continue the study to examine how regular languages behave when they need to cut numerous infinite languages. A particular interest rests on a situation in which a regular language needs to "dissect" a given infinite language into two subsets of infinite size. Every context-free language is dissected by carefully chosen regular languages (or it is REG-dissectible). In a larger picture, we show that constantly-growing languages and semi-linear languages are REG-dissectible. Under certain natural conditions, complements and finite intersections of semi-linear languages also become REG-dissectible. Restricted to bounded languages, the intersections of finitely many context-free languages and, more surprisingly, the entire Boolean hierarchy over bounded context-free languages are REG-dissectible. As an immediate application of the REG-dissectibility, we show another structural property, in which an appropriate bounded context-free language can "separate with infinite margins" two given nested infinite bounded context-free languages.