Ordinal recursive complexity of Unordered Data Nets

Data Nets are a version of colored Petri nets in which tokens carry data from an infinite and linearly ordered domain. This is a very expressive class, though coverability and termination remain decidable. Those problems have recently been proven complete for the Fωωω class in the fast growing complexity hierarchy. We characterize the exact complexity of Unordered Data Nets (UDN), a subclass of Data Nets with unordered data. We bound the length of bad sequences in well-quasi orderings of multisets over tuples of naturals by adapting the analogous result by Schmitz and Schnoebelen for words over a finite alphabet. These bounds imply that both problems are in Fωω. We prove that this result is tight by constructing UDN that weakly compute fast-growing functions and their inverses. This is the first complete problem for Fωω with an underlying wqo not based on finite words over a finite alphabet.