On the Home-Space Problem for Petri Nets and its Ackermannian Complexity

A set of configurations $H$ is a home-space for a set of configurations $X$ of aPetri net if every configuration reachable from (any configuration in) $X$ can reach (some configuration in) $H$. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations $X$, $H$, if $H$ is a home-space for $X$. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when $X$ is a singleton and $H$ is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any semilinear set of configurations $H$ we can effectively compute a semilinear set $C$ of configurations, called a non-reachability core for $H$, such that for every set $X$ the set $H$ is not a home-space for $X$ if, and only if, $C$ is reachable from $X$. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time.