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Decidability of membership in the Boolean closure of DTDA-recognizable tree languages

Determine the decidability status of the membership problem for the Boolean closure of the class of tree languages recognized by deterministic top-down tree automata (DTDA); that is, given a regular tree language, decide whether it belongs to Bool(T(DTDA)), or prove that this membership problem is undecidable.

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Background

Deterministic top-down tree automata (DTDA) recognize a strict subclass of the regular tree languages, characterized by closure under inclusion of their path sets. While membership in T(DTDA) is decidable (by Virágh’s theorem), the status for the Boolean closure Bool(T(DTDA)) has remained unresolved for decades.

This paper provides partial progress: it introduces DTDA with set acceptance, which exactly characterizes Bool(T(DTDA)), and establishes decidability for the subclass consisting of Boolean combinations of at most k DTDA-recognizable languages for fixed k. However, these advances do not settle the general membership problem for Bool(T(DTDA)).

References

A major problem, open now for several decades, is to show the analogue of Vir{a}gh's result mentioned above, i.e., to show that membership of a regular tree language in ${\rm Bool}({\cal T}({\rm DTDA}))$ is decidable (or to show the unlikely opposite -- that it is undecidable).

— On the Boolean Closure of Deterministic Top-Down Tree Automata (2401.06596 - Löding et al., 12 Jan 2024) in Introduction (Section 1)