Characterizing level one in group-based concatenation hierarchies

TL;DR

This paper characterizes the first level of concatenation hierarchies of regular languages using algebraic methods, enabling decidability of language membership based on separation problems.

Contribution

It provides generic algebraic characterizations of classes formed by polynomial and Boolean closures of group languages, advancing understanding of concatenation hierarchies.

Findings

Algebraic characterizations of classes Bool(Pol(G)) and Bool(Pol(G+))

Decidability of language membership based on separation problem

Constructive proofs using language and automata theory

Abstract

We investigate two operators on classes of regular languages: polynomial closure (Pol) and Boolean closure (Bool). We apply these operators to classes of group languages G and to their well-suited extensions G+, which is the least Boolean algebra containing G and the singleton language containing the empty word. This yields the classes Bool(Pol(G)) and Bool(Pol(G+)). These classes form the first level in important classifications of classes of regular languages, called concatenation hierarchies, which admit natural logical characterizations. We present generic algebraic characterizations of these classes. They imply that one may decide whether a regular language belongs to such a class, provided that a more general problem called separation is decidable for the input class G. The proofs are constructive and rely exclusively on notions from language and automata theory.