A counterexample to Thiagarajan's conjecture.
arXiv: Formal Languages and Automata Theory(2016)
摘要
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event structures correspond exactly to finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded $natural$-cliques are recognizable by finite trace automata. A necessary condition for both conjectures to be true is that domains of respective regular event structures admit a regular nice labeling. Our counterexample of a regular event domain with bounded $natural$-cliques, and not admitting a regular nice labeling is based on (i) the bijection between event domains, median graphs, and CAT(0) cube complexes and is derived from (ii) an example by Wise (1996 and 2007) of a nonpositively curved square complex ${bf X}$ with six squares, whose edges are colored in five colors, and whose universal cover $widetilde{bf X}$ is a CAT(0) square complex containing a particular plane with an aperiodic tiling.
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