Characterization and Decidability of FC-Definable Regular Languages

TL;DR

This paper investigates the expressive power of FC, a first-order logic over factors of words, showing some regular languages are not FC-definable and providing a decidable characterization of FC-definable regular languages.

Contribution

It establishes that not all regular languages are FC-definable and offers a decidable characterization using algebra, automata, and a natural class of regular expressions.

Findings

Some regular languages are not FC-definable.

Decidable criteria for FC-definability are provided.

Characterization involves algebra, automata, and extended regular expressions.

Abstract

FC is a first-order logic that reasons over all factors of a finite word using concatenation, and can define non-regular languages like that of all squares (ww). In this paper, we establish that there are regular languages that are not FC-definable. Moreover, we give a decidable characterization of the FC-definable regular languages in terms of algebra, automata, and regular expressions. The latter of which is natural and concise: Star-free generalized regular expressions extended with the Kleene star of terminal words.