Shape Preserving Bottom-Up Tree Transducers

It is easy to observe that both top-down and bottom-up (finite state) relabelling tree transducers preserve the shape of the trees. It is known that every shape preserving top-down tree transducer is semantically equivalent to a top-down relabelling tree transducer. In this way, for top-down tree transducers, the semantic property "being shape preserving" is characterized by the syntactic property "being relabelling". In this paper we prove the analogous result for bottom-up tree transducers. Namely, we show that every shape preserving bottom-up tree transducer is equivalent to a bottom-up relabelling tree transducer. We also prove that it is decidable if two shape preserving bottom-up tree transducers are equivalent.