Fuzzy Deterministic Top-down Tree Automata

In this paper we introduce and study fuzzy deterministic top-down (DT) tree automata over a lattice L. The L-fuzzy tree languages recognized by these automata are said to be DT-recognizable, and they form a proper subfamily $DRec_L$ of the family of $Rec_L$ of all regular L-fuzzy tree languages. We prove a Pumping Lemma for $DRec_L$ from which several decidability results follow. The closure properties of $DRec_L$ under various operations are established. We also characterize DT-recognizability in terms of L-fuzzy path languages, and prove that the path closure of any regular L-fuzzy tree language is DT-recognizable, and that it is decidable whether a regular L-fuzzy tree language is DT-recognizable. In most of the paper, L is just any nontrivial bounded lattice, but sometimes it is assumed to be distributive or even a bounded chain.