Concurrency, σ -Algebras, and Probabilistic Fairness

• Authors: Samy Abbes;Albert Benveniste
• Venue: FOSSACS '09 Proceedings of the 12th International Conference on Foundations of Software Science and Computational Structures: Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009
• Year: 2009

We extend previous constructions of probabilities for a prime event structure E by allowing arbitrary confusion. Our study builds on results related to fairness in event structures that are of interest per se. Executions of E are captured by the set Ω of maximal configurations. We show that the information collected by observing only fair executions of E is confined in some σ -algebra $\mathfrak{F}_0$, contained in the Borel σ -algebra $\mathfrak{F}$ of Ω . Equality $\mathfrak{F}_0=\mathfrak{F}$ holds when confusion is finite (formally, for the class of locally finite event structures), but inclusion $\mathfrak{F}_0\subseteq\mathfrak{F}$ is strict in general. We show the existence of an increasing chain $\mathfrak{F}_0\subseteq\mathfrak{F}_1\subseteq\mathfrak{F}_2\subseteq\dots$ of sub-σ -algebra s of $\mathfrak{F}$ that capture the information collected when observing executions of increasing unfairness. We show that, if the event structure unfolds a 1-safe net, then unfairness remains quantitatively bounded, that is, the above chain reaches $\mathfrak{F}$ in finitely many steps. The construction of probabilities typically relies on a Kolmogorov extension argument. Such arguments can achieve the construction of probabilities on the σ -algebra $\mathfrak{F}_0$ only, while one is interested in probabilities defined on the entire Borel σ -algebra $\mathfrak{F}$. We prove that, when the event structure unfolds a 1-safe net, then unfair executions all belong to some set of $\mathfrak{F}_0$ of zero probability. Whence $\mathfrak{F}_0=\mathfrak{F}$ modulo 0 always holds, whereas $\mathfrak{F}_0\neq\mathfrak{F}$ in general. This yields a new construction of Markovian probabilistic nets, carrying a natural interpretation that "unfair executions possess zero probability".