The Monadic Second Order Theory of Grid-Free 1-Safe Petri Nets is Decidable

Finite 1-safe Petri nets, also called \emph{net systems}, are natural models of asynchronous concurrency. The event structure of a net system describes all its possible executions and their concurrent nature: two events may be causally ordered, occur in parallel or be conflicting. Monadic second order logic (MSO) can be used to specify behavioural properties of net systems. Thiagarajan's conjecture states that MSO is decidable if and only if the net system is grid-free. The present paper gives a positive answer to this conjecture.