Closure of L(NIDPDAwtl) under union

Determine whether the language family L(NIDPDAwtl), consisting of all languages accepted by returning nondeterministic input-driven pushdown automata with translucent input letters, is closed under union when automata may have incompatible signatures (i.e., different partitions of the input alphabet into push, pop, and neutral classes), without assuming signature compatibility.

Background

Nondeterministic language families are often closed under union, but in this work the authors establish several non-closure results (e.g., non-closure under complementation for L(NIDPDAwtl) and L(nrNIDPDAwtl), and non-closure under intersection for L(NIDPDAwtl), even with regular languages). However, the union closure for the returning nondeterministic input-driven pushdown automata with translucent letters remains unresolved.

The authors note that union closure is immediate if the devices involved share compatible signatures. The open problem asks for a definitive resolution of union closure in the general setting where signatures may be incompatible.