Decidability Border for Petri Nets with Data: WQO Dichotomy Conjecture

In Petri nets with data, every token carries a data value, and executability of a transition is conditioned by a relation between data values involved. Decidability status of various decision problems for Petri nets with data may depend on the structure of data domain. For instance, if data values are only tested for equality, decidability status of the reachability problem is unknown (but decidability is conjectured). On the other hand, the reachability problem is undecidable if data values are additionally equipped with a total ordering. We investigate the frontiers of decidability for Petri nets with various data, and formulate the WQO Dichotomy Conjecture: under a mild assumption, either a data domain exhibits a well quasi-order (in which case one can apply the general setting of well-structured transition systems to solve problems like coverability or boundedness), or essentially all the decision problems are undecidable for Petri nets over that data domain.