New decidable upper bound of the second level in the Straubing-Therien concatenation hierarchy of star-free languages

Ondrej Klima,Jorge Almeida

doi:10.46298/dmtcs.490

Abstract

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.

Highlights

A well-known result due to Schutzenberger [22] gives a syntactic characterization of star-free regular languages
The hierarchies were later refined by Pin [14] by introducing intermediate levels whose syntactic characterization depends on a stable quasiorder rather than just a congruence
Starting from the trivial variety of languages, the levels of the refined Straubing-Therien hierarchy are defined inductively by alternately taking polynomial and Boolean closures. While it is decidable whether a given regular language belongs to each of the levels 0, 1/2, 1 and 3/2, decidability remains an open problem for level 2 or higher

Summary

Introduction

A well-known result due to Schutzenberger [22] gives a syntactic characterization of star-free regular languages. Starting from the trivial variety of languages, the levels of the refined Straubing-Therien hierarchy are defined inductively by alternately taking polynomial and Boolean closures. While it is decidable whether a given regular language belongs to each of the levels 0, 1/2, 1 and 3/2, decidability remains an open problem for level 2 or higher. Via Eilenberg’s correspondence, for the class V2 of all languages from the second level, the problem translates to decidability of membership of an arbitrary given finite monoid in. The new pseudoidentities are all those of the form uω = uωvuω where u and v are pseudowords such that V3/2 satisfies the inequality u ≤ v; here V3/2 is the pseudovariety of ordered monoids corresponding to level 3/2 in the Straubing-Therien hierarchy.

Stable quasiorders

Free profinite monoid

Regular languages and syntactic quasiorder

Some known results on the Straubing-Therien hierarchy

New upper bound

Application of the Factorization Forest Theorem

The new versus the old upper bound