Around Dot-depth One

TL;DR

This paper explores the structure of certain language classes within the dot-depth hierarchy, providing algebraic characterizations and decidability results for fragments related to logical quantifier restrictions.

Contribution

It introduces effective algebraic characterizations for language fragments with restricted dot-depth and proves decidability of membership problems for these classes.

Findings

Effective algebraic characterizations for language fragments with dot-depth 1/2 and 1.

Decidability of membership problem for these fragments.

New combinatorial proofs for the decidability results.

Abstract

The dot-depth hierarchy is a classification of star-free languages. It is related to the quantifier alternation hierarchy of first-order logic over finite words. We consider fragments of languages with dot-depth 1/2 and dot-depth 1 obtained by prohibiting the specification of prefixes or suffixes. As it turns out, these language classes are in one-to-one correspondence with fragments of existential first-order logic without min- or max-predicate. For all fragments, we obtain effective algebraic characterizations. Moreover, we give new combinatorial proofs for the decidability of the membership problem for dot-depth 1/2 and dot-depth 1.