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Dot-depth three, return of the J-class

Thomas Place, Marc Zeitoun

TL;DR

The paper tackles the decidability of level membership in concatenation hierarchies, with a focus on the operator $\textup{BPol}$. It develops a generic algebraic characterization of $\textup{BPol}(\mathscr{C})$ via two equations tied to $\mathscr{C}$-pairs and $\mathscr{C}$-swaps, enabling a reduction from $\textup{BPol}(\mathscr{C})$-membership to $\mathscr{C}$-covering in key cases. A specialized variant handles when $\mathscr{C}$ itself is $\textup{BPol}(\mathscr{D})$, reducing $\textup{BPol}(\mathscr{C})$-membership to $\mathscr{C}$-covering and yielding new decidability results (e.g., level 2 for modulo/group hierarchies and level 3 for dot-depth and Straubing–Thérien). The framework connects algebraic recognizers (syntactic monoids) with logical definability (quantifier blocks), and provides concrete tools to decide higher-level membership where covering is decidable. Overall, the work advances the dot-depth problem and offers a reusable method to propagate decidability up concatenation hierarchies through a careful interplay of syntactic covers, patterns, and morphism constructions.

Abstract

We look at concatenation hierarchies of classes of regular languages. Each such hierarchy is determined by a single class, its basis: level $n$ is built by applying the Boolean polynomial closure operator (BPol), $n$ times to the basis. A prominent and difficult open question in automata theory is to decide membership of a regular language in a given level. For instance, for the historical dot-depth hierarchy, the decidability of membership is only known at levels one and two. We give a generic algebraic characterization of the operator BPol. This characterization implies that for any concatenation hierarchy, if $n$ is at least two, membership at level $n$ reduces to a more complex problem, called covering, for the previous level, $n-1$. Combined with earlier results on covering, this implies that membership is decidable for dot-depth three and for level two in most of the prominent hierarchies in the literature. For instance, we obtain that the levels two in both the modulo hierarchy and the group hierarchy have decidable membership.

Dot-depth three, return of the J-class

TL;DR

The paper tackles the decidability of level membership in concatenation hierarchies, with a focus on the operator . It develops a generic algebraic characterization of via two equations tied to -pairs and -swaps, enabling a reduction from -membership to -covering in key cases. A specialized variant handles when itself is , reducing -membership to -covering and yielding new decidability results (e.g., level 2 for modulo/group hierarchies and level 3 for dot-depth and Straubing–Thérien). The framework connects algebraic recognizers (syntactic monoids) with logical definability (quantifier blocks), and provides concrete tools to decide higher-level membership where covering is decidable. Overall, the work advances the dot-depth problem and offers a reusable method to propagate decidability up concatenation hierarchies through a careful interplay of syntactic covers, patterns, and morphism constructions.

Abstract

We look at concatenation hierarchies of classes of regular languages. Each such hierarchy is determined by a single class, its basis: level is built by applying the Boolean polynomial closure operator (BPol), times to the basis. A prominent and difficult open question in automata theory is to decide membership of a regular language in a given level. For instance, for the historical dot-depth hierarchy, the decidability of membership is only known at levels one and two. We give a generic algebraic characterization of the operator BPol. This characterization implies that for any concatenation hierarchy, if is at least two, membership at level reduces to a more complex problem, called covering, for the previous level, . Combined with earlier results on covering, this implies that membership is decidable for dot-depth three and for level two in most of the prominent hierarchies in the literature. For instance, we obtain that the levels two in both the modulo hierarchy and the group hierarchy have decidable membership.
Paper Structure (25 sections, 47 theorems, 53 equations)

This paper contains 25 sections, 47 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words, classes and regular languages
  4. Membership, separation and covering
  5. $\mathscr{C}$-morphisms
  6. Concatenation hierarchies
  7. Polynomial closure and Boolean closure
  8. Concatenation hierarchies
  9. C-pairs, C-sets and C-swaps
  10. The C-pairs associated to a morphism
  11. The C-sets associated to a morphism
  12. The C-swaps associated to a morphism
  13. General characterization
  14. A first application: the class $\textup{BPol}(\textup{AT}\xspace)$
  15. Proof of Theorem \ref{['thm:cargen']}: "only if" direction
  16. ...and 10 more sections

Key Result

Lemma 0

Let $M$ be a finite monoid and $s,t \in M$ such that $t \leqslant_{\mathscr{J}}\xspace s$. If $s \leqslant_{\mathscr{R}}\xspace t$, then $s \mathrel{\mathscr{R}}\xspace t$. Moreover, if $s \leqslant_{\mathscr{L}}\xspace t$ then $s \mathrel{\mathscr{L}}\xspace t$.

Theorems & Definitions (85)

  • Lemma 0
  • Lemma 0
  • Remark 0
  • Remark 0
  • Remark 0
  • Remark 0
  • Lemma 0
  • Lemma 0
  • Remark 0
  • Proposition 0
  • ...and 75 more