A counterexample to Thiagarajan's conjecture on regular event structures

Abstract

We provide a counterexample to a conjecture by Thiagarajan (1996, 2002) that regular event structures correspond to event structures obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999). Using that domains of events structures are CAT(0) cube complexes, we construct our counterexample from an example by Wise (1996, 2007) of a nonpositively curved square complex whose universal cover contains an aperiodic plane. We prove that other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009). On the positive side, using breakthrough results by Agol (2013) and Haglund and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's conjecture holds for strongly hyperbolic regular event structures.