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All about unambiguous polynomial closure

Thomas Place, Marc Zeitoun

TL;DR

The paper develops a generic algebraic framework for unambiguous polynomial closure (UPol) built from a base class C of regular languages. It proves that, for prevarieties, membership in UPol(C) reduces to membership in C, and shows UPol(C) coincides with Pol(C) ∩ co-Pol(C), with APOL and WAPOL also aligning under these conditions. It provides decidability results for separation and covering when C is finite, and delivers broad logical characterizations linking UPol to unary temporal logic and two-variable FO logic, including Δ2-type hierarchies. The results unify and extend classical analyses of concatenation hierarchies, giving general reduction techniques and enabling computation of optimal imprints via rating maps, thereby enabling algorithmic decision procedures for coverage/separation. The findings have broad implications for the decidability of language classes defined via unambiguous concatenation and their logical counterparts, clarifying the connections between algebraic, logical, and automata-theoretic perspectives.

Abstract

We study a standard operator on classes of languages: unambiguous polynomial closure. We prove that for every class C of regular languages satisfying mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Moreover, we prove that if additionally C is finite, the separation and covering problems are decidable for UPol(C). Finally, we present an overview of the generic logical characterizations of the classes built using unambiguous polynomial closure.

All about unambiguous polynomial closure

TL;DR

The paper develops a generic algebraic framework for unambiguous polynomial closure (UPol) built from a base class C of regular languages. It proves that, for prevarieties, membership in UPol(C) reduces to membership in C, and shows UPol(C) coincides with Pol(C) ∩ co-Pol(C), with APOL and WAPOL also aligning under these conditions. It provides decidability results for separation and covering when C is finite, and delivers broad logical characterizations linking UPol to unary temporal logic and two-variable FO logic, including Δ2-type hierarchies. The results unify and extend classical analyses of concatenation hierarchies, giving general reduction techniques and enabling computation of optimal imprints via rating maps, thereby enabling algorithmic decision procedures for coverage/separation. The findings have broad implications for the decidability of language classes defined via unambiguous concatenation and their logical counterparts, clarifying the connections between algebraic, logical, and automata-theoretic perspectives.

Abstract

We study a standard operator on classes of languages: unambiguous polynomial closure. We prove that for every class C of regular languages satisfying mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Moreover, we prove that if additionally C is finite, the separation and covering problems are decidable for UPol(C). Finally, we present an overview of the generic logical characterizations of the classes built using unambiguous polynomial closure.
Paper Structure (51 sections, 66 theorems, 29 equations)

This paper contains 51 sections, 66 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Unambiguous closure.
  3. Contributions.
  4. Organization of the paper.
  5. Preliminaries
  6. Classes of regular languages
  7. Classes.
  8. Finite monoids and morphisms.
  9. Regular languages and syntactic morphisms.
  10. Decision problems
  11. Group languages
  12. Well-suited extensions.
  13. $\mathcal{C}$-morphisms
  14. Fragments of first-order logic
  15. Positions.
  16. ...and 36 more sections

Key Result

Lemma 2.1

Consider a finite monoid $M$ and $s,t \in M$ such that $s \mathrel{\mathscr{J}}\xspace t$. Then, $s \leqslant_{\mathscr{R}}\xspace t$ implies $s \mathrel{\mathscr{R}}\xspace t$. Symmetrically, $s \leqslant_{\mathscr{L}}\xspace t$ implies $s \mathrel{\mathscr{L}}\xspace t$.

Theorems & Definitions (105)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 95 more