Universally Wheeler Languages

TL;DR

The paper addresses the question of when a regular language is Wheeler in a universal sense, i.e., Wheeler with respect to all alphabet orders. It connects Universally Wheeler languages (UW) to well-known regular-language classes, showing SLT ⊊ UW and that a language and its complement both being in UW iff the language lies in DEF or RDEF. It then provides a quadratic-time algorithm to decide UW for a language given by a DFA, together with a fine-grained conditional lower bound based on the Orthogonal Vectors problem and SETH. The results advance both theoretical understanding of Wheeler properties across orders and practical feasibility for recognizing UW, with potential impact on automata-based indexing and language theory. The work also opens avenues for further study of Wheeler-like properties under fixed orders and co-lexicographic generalizations.

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.

Figures (3)

• Figure 1: A WDFA $\mathcal{D}$ recognizing the language $\mathcal{L}_d = ac^*\cup dc^+f$. The only order that makes $\mathcal{D}$ Wheeler is $s < q_1 < q_2 < q_3 < q_4 < q_5$.
• Figure 2: Left: The minimum complete DFA $\mathcal{D}^c_{\mathcal{L}}$ for a regular language $\mathcal{L}$, with both the absorbing state $\overline q$ for $\overline {\mathcal{L}}$ and the absorbing state $q$ for ${\mathcal{L}}$. The minimum complete DFA $\mathcal{D}^c_{\overline{\mathcal{L}}}$ for $\overline{\mathcal{L}}$ is obtained by switching final and non final states. Center: The minimum trimmed DFA $\mathcal{D}_{\mathcal{L}}$ for $\mathcal{L}$, in which the absorbing state for $\overline{\mathcal{L}}$ disappears. Right: The minimum trimmed DFA $\mathcal{D}_{\overline{\mathcal{L}}}$ for $\overline{\mathcal{L}}$, in which the absorbing state for ${\mathcal{L}}$ disappears.
• Figure 3: A language $\mathcal{L}$ (minimum trimmed DFA on the left) with $\operatorname{Pref}({\mathcal{L}})=\Sigma^*$ which is not in $\UW$ while $\overline \mathcal{L}\in \UW$ (minimum trimmed DFA on the right)

Theorems & Definitions (39)

• definition 1: Wheeler Automaton
• definition 2
• definition 3
• lemma 1: becker2023optimal
• definition 4
• lemma 2
• definition 5
• lemma 3: Caron
• lemma 4
• remark 1