Decidability and complexity of union equality for three GTESs

Determine whether, for arbitrary ground term equation systems D, E, and F over a fixed signature Σ, it is decidable that the congruence generated by D ∪ E ∪ F equals the union of the congruences generated by D, E, and F; if it is decidable, ascertain the time complexity of a decision algorithm for this problem.

Background

The paper establishes that for any two ground term equation systems (GTESs) E and F, one can decide in O(n^2) time whether the congruence generated by E ∪ F equals the union of the congruences generated by E and by F, and proves several equivalent characterizations of when this holds.

In the concluding discussion, the authors pose a more general question about extending these results from two to three GTESs. They note that a straightforward generalization may fail, providing an example to illustrate the difficulty, and thus highlight the need for a dedicated decidability and complexity analysis for the three-system case.