Over the years, bisimulations have emerged as a pervasive paradigm, finding applications in numerous areas, including concurrency theory, model checking, automata theory, logic, programming languages and category theory. In this paper, we establish a connection between bisimulations and data compression. More precisely, we study the relationship between bisimulations and Wheeler automata (Alanko et al., SODA 2020), a class of automata that has received considerable attention in recent years. The standard notion of bisimulation is not appropriate, so we introduce Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata. We show that Wheeler bisimilarity induces a unique minimal Wheeler NFA (analogously to standard bisimulations). In particular, in the deterministic case, we retrieve the minimal 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 partition refinement algorithm. Compared to previous state-reduction techniques, our bisimulation-induced construction is the first for which (i) we obtain a canonical Wheeler NFA and (ii) the resulting Wheeler NFA can be built in linear time.

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