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 $t \in \supp{S}$ 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 $(S,C'[C^n[s]]) = a'\odot a^n \odot c \odot b^n \odot b'$ 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-w-fta over a commutative semiring or (b) a non-deterministic bu-w-fta over a locally finite commutative semiring.

Available electronic editions: PDF.

DOI is not available for this article.

author = {Bj{\"o}rn Borchardt},

title = {A Pumping Lemma and Decidability Problems for Recognizable Tree Series},

journal = {Acta Cybernetica},

year = {2004},

volume = {16},

number = {4},

pages = {509--544},

abstract = {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 $t \in \supp{S}$ 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 $(S,C'[C^n[s]]) = a'\odot a^n \odot c \odot b^n \odot b'$ 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-w-fta over a commutative semiring or (b) a non-deterministic bu-w-fta over a locally finite commutative semiring.}

}