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Concurrency, σ-Algebras, and Probabilistic Fairness

  • Samy Abbes
  • Albert Benveniste
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5504)

Abstract

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 Borelσ-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 theσ-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”.

Keywords

Probabilistic Petri nets probabilistic event structures true-concurrency probabilistic fairness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Samy Abbes
    • 1
  • Albert Benveniste
    • 2
  1. 1.PPS/Université Paris 7 Denis Diderot.ParisFrance
  2. 2.INRIA/IRISARennes CedexFrance

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