Lookahead elimination for deterministic top-down tree transducers

Determine the decidability of the lookahead-elimination problem for deterministic top-down tree transducers with regular lookahead: given a deterministic top-down tree transducer equipped with regular lookahead, decide whether there exists an equivalent deterministic top-down tree transducer without regular lookahead that computes the same tree-to-tree function.

Background

Deterministic top-down tree transducers can be extended with regular lookahead, allowing transitions to depend on regular properties of subtrees recognized by an auxiliary deterministic bottom-up automaton. From a bottom-up perspective, such lookahead corresponds to adding finite control states, and this perspective connects top-down transducers to multi bottom-up tree transducers.

For deterministic macro tree transducers (MTTs), regular lookahead can always be eliminated: any deterministic MTT with regular lookahead can be transformed into an equivalent deterministic MTT without lookahead. In contrast, for deterministic top-down tree transducers, whether one can decide if a given lookahead-using transducer computes a function realizable without lookahead remains unresolved; only partial results are known.