Universally Wheeler Languages

2025-04-28

AI Summary

Wheeler languages fundamentally depend on a fixed alphabet ordering, severely limiting their applicability. Method: We introduce *Universally Wheeler (UW) languages*—regular languages that retain the Wheeler property under *every* possible alphabet order—and systematically characterize them by integrating finite automata theory, formal language classification, and fine-grained complexity analysis under SETH. Contributions: We prove UW strictly contains the class of strictly local testable languages; establish UW ∩ co-UW = Definite ∪ Reverse Definite; show UW membership is equivalent to closure under complementation for definite and reverse-definite languages; and provide a tight O(n²) optimal decision algorithm together with a matching quadratic-time lower bound. This work overcomes the inherent order-dependence of Wheeler structures and uncovers deep connections between Wheeler theory and classical subclasses of regular languages.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The notion of Wheeler languages is rooted in the Burrows-Wheeler transform (BWT), one of the most central concepts in data compression and indexing. The BWT has been generalized to finite automata, the so-called Wheeler automata, by Gagie et al. [Theor. Comput. Sci. 2017]. Wheeler languages have subsequently been defined as the class of regular languages for which there exists a Wheeler automaton accepting them. Besides their advantages in data indexing, these Wheelerlanguages also satisfy many interesting properties from a language theoretic point of view [Alanko et al., Inf. Comput. 2021]. A characteristic yet unsatisfying feature of Wheeler languages however is that their definition depends on a fixed order of the alphabet. In this paper we introduce the Universally Wheeler languages UW, i.e., the regular languages that are Wheeler with respect to all orders of a given alphabet. Our first main contribution is to relate UW to some very well known regular language classes. We first show that the Striclty Locally Testable languages are strictly included in UW. After noticing that UW is not closed under taking the complement, we prove that the class of languages for which both the language and its complement are in UW exactly coincides with those languages that are Definite or Reverse Definite. Secondly, we prove that deciding if a regular language given by a DFA is in UW can be done in quadratic time. We also show that this is optimal unless the Strong Exponential Time Hypothesis (SETH) fails.

Problem

• Defining Universally Wheeler languages independent of alphabet order
• Relating UW languages to known regular language classes
• Developing efficient algorithms for UW language recognition

Innovation

• Introduces Universally Wheeler languages for all alphabet orders
• Relates UW to Strictly Locally Testable language classes
• Quadratic time decision for regular languages in UW