Coding tree languages based on lattice-valued logic

We consider tree automata based on complete residuated lattice-valued (for simplicity we write L-valued) logic. First, we define the concepts of response function and accessible states (with threshold c) of an L-valued tree automaton. Thereafter, we consider coding of trees and investigate the relation between response function on trees and their coding. Using the provided theorems, we give a pumping lemma for recognizable coding tree languages with threshold c. Moreover, we consider closure properties of recognizable coding tree languages. In this regard, we show that the class of recognizable coding tree languages with threshold c is closed under projection, intersection and union.