Dice Question Streamline Icon: https://streamlinehq.com

Decidability of membership for finite unions of DTDA-recognizable tree languages

Determine whether the membership problem for the class of finite unions of languages recognized by deterministic top-down tree automata is decidable; specifically, given a regular tree language, decide whether it can be expressed as a finite union of languages from T(DTDA).

Information Square Streamline Icon: https://streamlinehq.com

Background

A recent claim in the literature asserted decidability for the subclass of Bool(T(DTDA)) consisting of finite unions of DTDA-recognizable languages. The present paper points out that a key lemma used for that claim is invalid, and subsequent corrections in later work do not resolve this particular decidability question.

Consequently, the decidability status for the membership problem for finite unions of DTDA languages remains unsettled, standing as a distinct open problem separate from the general Bool(T(DTDA)) membership problem.

References

Since Lemma~5 of that paper is not valid (as a counter-example shows), this decidability claim seems open to us (parts of the results from have been fixed in , but this does not include the decidability of the class of finite unions of tree languages in ${\cal T}({\rm DTDA})$.)

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