Filtrations of Formal Languages by Arithmetic Progressions

TL;DR

This paper investigates how filtering formal languages by arithmetic progressions affects their properties, showing regular languages produce finitely many results while some context-free languages produce infinitely many, and analyzing the diag operation's effects.

Contribution

It introduces a novel technique to analyze language filtrations by arithmetic progressions and demonstrates how certain operations preserve or break language class properties.

Findings

Filtering regular languages yields finitely many languages.

Some context-free languages produce infinitely many filtered languages.

The diag operation preserves regularity but not context-freeness.

Abstract

A filtration of a formal language L by a sequence s maps L to the set of words formed by taking the letters of words of L indexed only by s. We consider the languages resulting from filtering by all arithmetic progressions. If L is regular, it is easy to see that only finitely many distinct languages result. By contrast, there exist CFL's that give infinitely many distinct languages as a result. We use our technique to show that the operation diag, which extracts the diagonal of words of square length arranged in a square array, preserves regularity but does not preserve context-freeness.