Distance desert automata and the star height problem

Abstract

We show that it is decidable in time complexity \(2^{2^{2^{O{(n)}}}}\) 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.