🤖 AI Summary
This work addresses the lack of efficient minimization techniques for nondeterministic Wheeler automata, which stems from the absence of a suitable equivalence relation preserving their structural properties. We introduce the novel notion of *Wheeler bisimulation*, an equivalence relation that maintains the convex structure inherent to Wheeler automata, and establish—for the first time—a unique minimal form for Wheeler NFAs. Leveraging this relation, we design a linear-time algorithm that constructs the minimal Wheeler NFA in $O(m)$ time, substantially improving upon the $O(m \log n)$ complexity of conventional bisimulation-based approaches. Our results provide both theoretical foundations and practical algorithms for the efficient processing of Wheeler languages.
📝 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.