Equivalence of deterministic DAG automata for picture language recognition

Determine whether the equivalence problem for deterministic DAG automata with respect to picture language recognition under a specified picture-to-DAG encoding is decidable: given two deterministic DAG automata and a fixed picture-to-DAG encoding as in Definition 3.2, decide whether the sets of encoded pictures they accept are identical, even though the automata may accept different DAGs that are not valid picture encodings.

Background

Blum and Drewes showed that for DAG languages, deterministic DAG automata admit polynomial-time equivalence checking. However, when DAG automata are used as recognizers for picture languages via picture-to-DAG encodings, the underlying automata may also accept DAGs that do not encode pictures, complicating direct equivalence transfer.

The paper formalizes picture-equivalence (Definition 3.5) by comparing automata only on the subset of DAGs that encode pictures. The authors note that two deterministic DAG automata can accept the same encodings of pictures yet differ on non-picture DAGs, leaving open whether equivalence with respect to picture language recognition is decidable.