Wheeler Bisimulations

TL;DR

This paper introduces Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata, and shows that Wheeler bisimulations induce a unique minimal Wheeler NFA (analogously to standard bisimulations).

Abstract

Recently, a new paradigm was introduced in automata theory. The main idea is to classify regular languages according to their propensity to be sorted, establishing a deep connection between automata theory and data compression [J. ACM 2023]. This parameterization leads to two hierarchies of regular languages: a deterministic hierarchy and a non-deterministic hierarchy. While the deterministic hierarchy is well understood, the non-deterministic hierarchy appears much more complex. This is true even for the richest and most studied level of the hierarchies, corresponding to the class of Wheeler languages. In this paper, we study Wheeler language through the lens of bisimulations. We first show that the standard notion of bisimulation is not appropriate. Then, we introduce Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata. Although there are some differences between the properties of bisimulations and the properties of Wheeler bisimulations, we show that Wheeler bisimulations induce a unique minimal Wheeler NFA (analogously to standard bisimulations). In particular, in the deterministic case, we retrieve the minimum Wheeler deterministic automaton of a given language. We also show that the minimal Wheeler NFA induced by Wheeler bisimulations can be built in linear time. This is in contrast with standard bisimulations, for which the corresponding minimal NFA can be built in $ O(m \log n) $ time (where $ m $ is the number of edges and $ n $ is the number of states) by adapting Paige-Tarjan's partition refinement algorithm.

Figures (7)

• Figure 1: How to (partially) sort the states of an automaton based on the strings reaching each state.
• Figure 2: Left: A Wheeler NFA. The states are numbered following the Wheeler order. Right: Edges labeled with the same character (e.g., $a$) do not cross.
• Figure 3: Two non-isomorphic minimal Wheeler NFAs recogning $\mathcal{L} = a a^*$ (see Example \ref{['ex:nonisomorphicexample']}). The states of each NFA are numbered following the corresponding Wheeler order.
• Figure 4: The three NFAs used in Example \ref{['ex:bisimulationisnotenough']}. On the left, the Wheeler NFA $(\mathcal{A}, \le)$. In the center, the Wheeler NFA $(\mathcal{A}', \le')$. On the right, the NFA $\mathcal{A}"$. The states of $(\mathcal{A}, \le)$ and $(\mathcal{A}', \le')$ are numbered following the corresponding Wheeler orders.
• Figure 5: The Wheeler NFAs $(\mathcal{A}^{(3)}, \le^{(3)})$ and $(\mathcal{A}^{(4)}, \le^{(4)})$ used in Example \ref{['ex:canhaveinfiniteclasses']}.

Theorems & Definitions (40)

• Lemma 1
• Definition 2
• Lemma 3: alanko2021wheeler, Lemma 3.7
• Lemma 4: alanko2021wheeler, Lemma 3.2
• Example 5
• Definition 6
• Remark 7
• Example 8
• Definition 9
• Definition 10