Wheeler Bisimulations

TL;DR

This paper introduces Wheeler bisimulations, a new concept that respects the structure of Wheeler automata, enabling efficient construction of minimal Wheeler NFAs and linking automata theory with data compression.

Contribution

It defines Wheeler bisimulations, shows they induce minimal Wheeler NFAs, and provides a linear-time algorithm for their construction, advancing understanding of Wheeler automata.

Findings

Wheeler bisimulations induce a unique minimal Wheeler NFA.

Minimal Wheeler NFA can be built in linear time.

Wheeler bisimulations differ from standard bisimulations in properties.

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.