On minimal elements of upward-closed sets

Upward-closed sets of integer vectors enjoy the merit of having a finite number of minimal elements, which is behind the decidability of a number of Petri net related problems. In general, however, such a finite set of minimal elements may not be effectively computable. In this paper, we develop a unified strategy for computing the sizes of the minimal elements of certain upward-closed sets associated with Petri nets. Our approach can be regarded as a refinement of a previous work by Valk and Jantzen (in which a necessary and sufficient condition for effective computability of the set was given), in the sense that complexity bounds now become available provided that a bound can be placed on the size of a witness for a key query. The sizes of several upward-closed sets that arise in the theory of Petri nets as well as in backward-reachability analysis in automated verification are derived in this paper, improving upon previous decidability results shown in the literature.