Separability by piecewise testable languages and downward closures beyond subwords

We introduce a flexible class of well-quasi-orderings (WQOs) on words that generalizes the ordering of (not necessarily contiguous) subwords. Each such WQO induces a class of piecewise testable languages (PTLs) as Boolean combinations of upward closed sets. In this way, a range of regular language classes arises as PTLs. Moreover, each of the WQOs guarantees regularity of all downward closed sets. We consider two problems. First, we study which (perhaps non-regular) language classes allow to decide whether two given languages are separable by a PTL with respect to a given WQO. Second, we want to effectively compute downward closures with respect to these WQOs. Our first main result is that for each of the WQOs, under mild assumptions, both problems reduce to the simultaneous unboundedness problem (SUP) and are thus solvable for many powerful system models. In the second main result, we apply the framework to show decidability of separability of regular languages by B∑1[<, mod], a fragment of first-order logic with modular predicates.