Generalization to nondeterministic macro tree transducers

Determine whether shape preservation is decidable for nondeterministic macro tree transducers. Concretely, given a nondeterministic macro tree transducer M over ranked input alphabet Σ and output alphabet Δ, ascertain whether there exists an effective procedure that decides if the tree relation realized by M preserves shape, i.e., for every (s, t) in the relation, sh_Δ(t) = sh_Σ(s).

Background

The paper proves that shape preservation is decidable for compositions of total deterministic macro tree transducers and provides a normalization that expresses shape-preserving macro tree transductions as a composition of a relabeling with a one-to-one macro tree transducer. These results rely on known decidability of equivalence in the deterministic MSO-definable framework and on decidability of linear size increase.

While these advances cover deterministic settings, nondeterministic macro tree transducers realize relations rather than functions, and standard determinization or equivalence techniques do not directly apply. The authors explicitly flag the extension of their decidability and normalization framework from deterministic to nondeterministic macro tree transducers as an open direction.