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We show that it is decidable in time complexity 222O(n)2^{2^{2^{O{(n)}}}} whether the language accepted by an n-state non-deterministic automaton is of star height one, which is the first ever complexity result for the star height one problem. To achieve this, we introduce distance desert automata as a joint generalization of distance automata and desert automata, and show the decidability of its limitedness problem by solving the underlying Burnside problem.

Distance Desert Automata and the Star Height One Problem