From Regular to Strictly Locally Testable Languages

TL;DR

This paper explores the relationship between local testability, alphabet size, and automaton width, generalizing classical results to show how regular languages relate to strictly locally testable languages with optimized parameters.

Contribution

It extends Medvedev's classical result by establishing a quantitative relationship between alphabet size and width for representing regular languages as images of strictly locally testable languages.

Findings

Every regular language can be represented as an image of a k-slt language with doubled alphabet size.

The width of the k-slt language depends logarithmically on the automaton size.

Certain regular languages require a minimum alphabetic ratio, establishing a lower bound.

Abstract

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k=2 of the sliding window width in the McNaughton and Papert's infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedev's result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.