While finite-state tree automata correspond to a single concept, and share all the nice properties of string automata (Comon et al., 2007), the situation with tree-to-tree or tree-to-string transducers is more complicated (Maletti, 2010;Graehl et al., 2008): several variants exist, only some of which support the operations that our conceptual model requires (composition with the string transducer σ and intersection with the tree automaton ω).
the "linear non-deleting topdown tree transducers" defined in (Maletti, 2010) 9 have the requisite properties.