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1-Safe Petri Nets and Special Cube Complexes

Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure EN. By a result of Thiagarajan [46], these unfoldings are exactly the trace-regular event structures. Thiagarajan [46] conjectured that regular event structures correspond exactly to trace-regular event structures. In a recent paper (Chalopin and Chepoi [12]), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. However, we proved that Thiagarajan’s conjecture is true for regular event structures whose domains are principal filters of universal covers of finite special cube complexes. In the current article, we prove the converse: To any finite 1-safe Petri net N, one can associate a finite special cube complex XN such that the domain of the event structure EN (obtained as the unfolding of N) is a principal filter of the universal cover X̃N of XN. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace-regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity), we disprove yet another conjecture by Thiagarajan (from the paper with Yang [48]) that the monadic second-order logic of a 1-safe Petri net (i.e., of its event structure unfolding) is decidable if and only if its unfolding is grid-free. It was proven by Thiagarajan and Yang [48] that the MSO logic is undecidable if the unfolding is not grid-free. Our counterexample is the trace-regular event structure that arises from a virtually special square complex Z. The domain of this event structure ĖZ is the principal filter of the universal cover Z̃ of Z in which to each vertex we added a pendant edge. The graph of the domain of ĖZ has bounded hyperbolicity (and, thus, the event structure ĖZ is grid-free) but has infinite treewidth. Using results of Seese, Courcelle, and Müller and Schupp, we show that this implies that the MSO theory of the event structure ĖZ is undecidable.