Decidability and computability of the C-swap relation underlying the generic BPol(C) characterization

Determine whether the ad hoc problem defining the C-swap relation used in the second equation of the generic characterization of BPol(C) is decidable, and establish an effective procedure to compute C-swaps for a given class C and morphism α (or provide a reduction to standard decision problems such as C-separation or C-covering).

Background

The paper gives a generic algebraic characterization of BPol(C) using two parameterized equations. The first equation depends on C-pairs and reduces to C-separation, which is often decidable. The second equation relies on a new relation derived from an ad hoc problem for C (later formalized as C-swaps), whose general decidability status is unknown.

Because the computability of C-swaps is not established in general, the characterization does not immediately yield a membership decision procedure for BPol(C) for arbitrary classes C, motivating the need to resolve the decidability and effective computation of this relation.