Optimal Label Splitting for Embedding an LTS into an arbitrary Petri Net Reachability Graph is NP-complete

For a given labelled transition system (LTS), synthesis is the task to find an unlabelled Petri net with an isomorphic reachability graph. Even when just demanding an embedding into a reachability graph instead of an isomorphism, a solution is not guaranteed. In such a case, label splitting is an option, i.e. relabelling edges of the LTS such that differently labelled edges remain different. With an appropriate label splitting, we can always obtain a solution for the synthesis or embedding problem. Using the label splitting, we can construct a labelled Petri net with the intended bahaviour (e.g. embedding the given LTS in its reachability graph). As the labelled Petri net can have a large number of transitions, an optimisation may be desired, limiting the number of labels produced by the label splitting. We show that such a limitation will turn the problem from being solvable in polynomial time into an NP-complete problem.