The Alternation Hierarchy of First-Order Logic on Words is Decidable

2025-01-24

AI Summary

This paper resolves a longstanding open problem—dating back to 1971—concerning the decidability of whether a given regular language belongs to the Σᵢ[<] level of the first-order logic FO[<] alternation hierarchy. Methodologically, it establishes, for the first time, a transfer theorem linking polynomial closure with the decidability of language separation, constructs a generic lifting framework tailored to positive varieties of languages, and integrates tools from algebraic language theory, polynomial closure constructions, and separation reduction techniques. The main contributions are: (1) the first decidable algorithm for Σᵢ[<]-expressibility, whose time complexity is exponential in that of the underlying separation problem; and (2) as corollaries, the decidability of both the dot-depth half-level hierarchy and the group-based concatenation hierarchy—resolving two long-standing open questions regarding the decidability of these fundamental hierarchies.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We show that for any $i>0$, it is decidable, given a regular language, whether it is expressible in the $Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the notion of polynomial closure of a class of languages $mathcal{V}$, that is, finite unions of languages of the form $L_0a_1L_1cdots a_nL_n$ where each $a_i$ is a letter and each $L_i$ a language of $mathcal{V}$. We show that if a class $mathcal{V}$ of regular languages with some closure properties (namely, a positive variety) has a decidable separation problem, then so does its polynomial closure Pol($mathcal{V}$). The resulting algorithm for Pol($mathcal{V}$) has time complexity that is exponential in the time complexity for $mathcal{V}$ and we propose a natural conjecture that would lead to a polynomial time blowup instead. Corollaries include the decidability of half levels of the dot-depth hierarchy and the group-based concatenation hierarchy.

Problem

• Regular Languages
• First-Order Logic
• Expressiveness

Innovation

• Regular Language
• First-Order Logic Representation
• Combination Ability