A Pumping Lemma and Decidability Problems for Recognizable Tree Series

In the present paper we show that given a tree series S, which is accepted by (a) a deterministic bottom-up finite state weighted tree automaton (for short: bu-w-fta) or (b) a non-deterministic bu-w-fta over a locally finite semiring, there exists for every input tree t ∈ supp(S) a decomposition t = C′[C[s]] into contexts C, C′ and an input tree s as well as there exist semiring elements a, a′, b, b′, c such that the equation (S, C′[Cn[s]]) = a′ a c b b′ holds for every non-negative integer n. In order to prove this pumping lemma we extend the power-set construction of classical theories and show that for every non-deterministic bu-w-fta over a locally finite semiring there exists an equivalent deterministic one. By applying the pumping lemma we prove the decidability of a tree series S being constant on its support, S being constant, S being boolean, the support of S being the empty set, and the support of S being a finite set provided that S is accepted by (a) a deterministic bu-wfta over a commutative semiring or (b) a non-deterministic bu-w-fta over a locally finite commutative semiring.