Distance Desert Automata and the Star Height One Problem

Abstract

Abstract. We show that it is decidable in time complexity 2whether 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.

Open Problems About Regular Languages, 35 Years Later

“…In 2004, Leung and Podolskiy [51] improved this bound to C(n) = 3n 3 + n ln n + n − 1 and proved that limitedness of distance automata is decidable in PSPACE. The problem is in fact PSPACE-complete, as shown by Kirsten [44].…”

“…Automata with outputs in the tropical semiring are sometimes called automata with distance [32,34,35,36] or distance automata [44,43,45].…”

“…A cost automaton is defined in a similar way, but the counters are not ordered and thus transitions can only increment or reset a given counter. These notions have been widely studied in the recent years, notably by Kirsten [44,43], Bojanczyk and Colcombet [5,6,16,17]. Very recently, Bojanczyk [5] proved that the limitedness problem for cost automata reduces to solving Gale-Stewart games with ω-regular winning conditions, which leads to an entirely new proof of the decidability of the restricted star-height problem.…”

Distance desert automata and the star height problem

“…The author acknowledges the discussions with Jean-Éric Pin on the paper. The author thanks the referees of the present paper and the conference version [23] for their very careful work and their useful comments which resulted in many detail improvements. The author thanks Igor Walukiewicz for the invitation to publish in this special issue.…”