A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices

We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.