Algorithmic realization and complexity of positionality criteria

Determine whether the positionality criteria for Dumont–Thomas complement numeration systems can be converted into an algorithmic procedure and, if so, determine the algorithmic complexity of that procedure.

Background

The paper gives necessary and sufficient conditions for a Dumont–Thomas complement numeration system associated with a substitution, a two-sided periodic point, and a residue class to be positional. These conditions involve constructing sets of letters and checking constancy of iterated image lengths, together with an additional boundary condition. The authors explicitly ask whether these theoretical criteria can be made into an effective algorithm and what resources such an algorithm would require.