Desert Automata and the Finite Substitution Problem

Regular Cost Functions, Part I: Logic and Algebra over Words

“…The finite substitution problem takes as input two regular languages L, K, and asks whether it is possible to find a finite substitution σ (i.e., a morphism mapping each letter of the alphabet of L to a finite language over the alphabet of K) such that σ(L) = K. This problem was shown decidable independently by Bala and Kirsten by a reduction to the limitedness of desert automata (a form of automata weaker than nested distance desert automata, but incomparable to distance automata), and a proof of decidability of this latter problem [2,25].…”

“…It was raised by Brzozowski in 1966, and it took twelve years before being independently solved by Simon and Hashiguchi [40,16]. This problem is easily reduced to the limitedness problem for distance automata.The finite substitution problem takes as input two regular languages L, K, and asks whether it is possible to find a finite substitution σ (i.e., a morphism mapping each letter of the alphabet of L to a finite language over the alphabet of K) such that σ(L) = K. This problem was shown decidable independently by Bala and Kirsten by a reduction to the limitedness of desert automata (a form of automata weaker than nested distance desert automata, but incomparable to distance automata), and a proof of decidability of this latter problem [2,25].The relative inclusion star-height problem is an extension of the star height problem introduced and shown decidable by Hashiguchi using his techniques [22]. Still using nested distance desert automata, Kirsten gave another, more elegant proof of this result [29].The boundedness problem is a problem of model theory.…”

Distance desert automata and the star height problem

“…The subclass of 1-nested distance desert automata for which θ : E → { 0 , ∠ 1 } are exactly Hashiguchi's distance automata [10]. If we consider the subclass of 1-nested distance desert automata with the restriction θ : E → {∠ 0 , 0 }, then we recover the definition of desert automata due to Bala and the author [1,21,22].…”

“…Bala and the author introduced independently the notion of desert automata in [1,21,22]. Desert automata are nondeterministic finite automata with a set of marked transitions.…”

“…The weight of a word is the minimum of the weights of all successful paths of the word. The author showed that limitedness of desert automata is decidable [21,22]. As an application, Bala and the author solved the so-called finite substitution problem which was open for more than 10 years: given recognizable languages K and L, it is decidable whether there exists a finite substitution σ such that σ(K) = L [1,21,22].…”