On the Home-Space Problem for Petri Nets

In a recent paper (at Concur 2023) we answered a former question by D. de Frutos Escrig and C. Johnen, by showing the decidability of the "semilinear home-space problem" for Petri nets (that asks if a given semilinear set H is a home-space for a given semilinear set X of markings of a Petri net). We used an approach constructing semilinear "non-reachability cores" for linear sets. This was sufficient for a decision algorithm, a detailed analysis of which even showed that the problem is Ackermann-complete, but it has remained unclear if such semilinear cores can be constructed for general semilinear sets. Here we give a positive answer to this question; this also yields a conceptually simpler decision algorithm for the mentioned home-space problem, though with no complexity bound.