Asymptotic Approximation by Regular Languages

TL;DR

This paper introduces the concept of REG-measurability for formal languages, exploring which languages can be asymptotically approximated by regular languages and identifying classes that are or aren't REG-measurable.

Contribution

It defines REG-measurability, analyzes its properties, and characterizes which context-free languages are REG-measurable versus REG-immeasurable.

Findings

Several context-free languages are REG-measurable, including those with transcendental generating functions.

Simple deterministic context-free languages are REG-immeasurable.

The set of primitive words is strongly REG-immeasurable.

Abstract

This paper investigates a new property of formal languages called REG-measurability where REG is the class of regular languages. Intuitively, a language \(L\) is REG-measurable if there exists an infinite sequence of regular languages that "converges" to \(L\). A language without REG-measurability has a complex shape in some sense so that it can not be (asymptotically) approximated by regular languages. We show that several context-free languages are REG-measurable (including languages with transcendental generating function and transcendental density, in particular), while a certain simple deterministic context-free language and the set of primitive words are REG-immeasurable in a strong sense.