Splitting of Actions, Higher-Dimensional Automata, and Net Synthesis

The behaviour of pure Petri nets (i.e. without side condition) is given by ordinary automata because all information about concurrency is encoded in the structure of the marking graphs. By contrast, the behaviour of (possibly) impure nets requires higher-dimensional automata: independence should in that case be made explicit. These higher-dimensional automata are step transition systems in the case of general Petri nets and asynchronous transition systems if we restrict to safe Petri nets. The aim of this report is to show that the synthesis problem for nets can reduce to the synthesis problem of pure nets. For that purpose, we discretize an higher-dimensional automaton by splitting its actions and we prove that it is the behaviour of some Petri net if and only if its discretized automaton is the marking graph of some pure Petri net.