Concurrency, sigma-algebras and probabilistic fairness

International audienceWe give an interpretation through sigma-algebras of phenomena encountered in concurrency theory when dealing with "infinite confusion"--the extreme opposite of confusion-free event structures. The set of runs of a safe Petri net is equipped with its Borel sigma-algebra F. The fine structure of F describes the complexity of choices along runs, and we show that a transfinite induction of finite degree is needed to explore all choices of runs in general. The degree is minimal (zero) when confusion is bounded, corresponding to the classes of confusion free and locally finite event structures. We relate this construction to probabilistic fairness by showing how to randomize the net equipped with its Borel sigma-algebra by using only the first step of our decomposition, and making it thus more effective. Hence the serious difficulty brought by the above transfiniteness in the application of Kolmogorov extension theorem is bypassed thanks to probabilistic fairnes