Decide eventual reducibility of monomials when Algorithm 3 does not terminate

Develop a criterion to decide, at a given point during the refined completion process (Algorithm 3), whether a given monomial will eventually become reducible in one of the infinitely many remaining iterations, thereby enabling a decision procedure even in cases where Algorithm 3 does not terminate.

Background

The authors propose a semi-decision procedure that interleaves construction of reduction systems with attempts to solve the target equation via reduction or degree bounds. However, when the completion process does not terminate, they lack a criterion to determine whether future reductions will make a monomial reducible.

This absence prevents a full decision procedure in non-terminating cases, making the development of such a criterion an explicit unresolved question.