Crisp-determinization of weighted tree automata over strong bimonoids

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics run semantics. There are wta for which these different; however, bottom-up deterministic semirings, the difference vanishes. A is crisp-deterministic if it each transition by one of unit elements bimonoid. prove that class languages recognized same as recognizable step mappings. Moreover, we investigate following two crisp-determinization problems: a given ${\cal A}$, (a) does there exist computes A}$ (b) A}$? show finiteness Nerode N}({\cal A})$ implies positive answer (a), finite order property (b). sufficient condition guarantees A}$. Also, provide an algorithm construction according to finite, similarly has property. undecidable whether arbitrary crisp-determinizable. also both, undecidable.