Concurrency, σ-Algebras, and Probabilistic Fairness View Full Text


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Chapter Info

DATE

2009

AUTHORS

Samy Abbes , Albert Benveniste

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 , contained in the Borelσ-algebra of Ω. Equality holds when confusion is finite (formally, for the class of locally finite event structures), but inclusion is strict in general. We show the existence of an increasing chain of sub-σ-algebra s of 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 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 only, while one is interested in probabilities defined on the entire Borel σ-algebra . We prove that, when the event structure unfolds a 1-safe net, then unfair executions all belong to some set of of zero probability. Whence modulo 0 always holds, whereas in general. This yields a new construction of Markovian probabilistic nets, carrying a natural interpretation that “unfair executions possess zero probability”. More... »

PAGES

380-394

References to SciGraph publications

  • 1990. Equivalences, congruences, and complete axiomatizations for probabilistic processes in CONCUR '90 THEORIES OF CONCURRENCY: UNIFICATION AND EXTENSION
  • 1994. Stationary regime and stability of free-choice Petri nets in 11TH INTERNATIONAL CONFERENCE ON ANALYSIS AND OPTIMIZATION OF SYSTEMS DISCRETE EVENT SYSTEMS
  • 2005-06-09. Multiprocessor and distributed system design: The integration of functional specification and performance analysis using Stochastic Process Algebras in PERFORMANCE EVALUATION OF COMPUTER AND COMMUNICATION SYSTEMS
  • 2005. Branching Cells as Local States for Event Structures and Nets: Probabilistic Applications in FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATIONAL STRUCTURES
  • 1981. Concurrency and automata on infinite sequences in THEORETICAL COMPUTER SCIENCE
  • 2000. Probabilistic Asynchronous π-Calculus in FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES
  • 2005. The (True) Concurrent Markov Property and Some Applications to Markov Nets in APPLICATIONS AND THEORY OF PETRI NETS 2005
  • Book

    TITLE

    Foundations of Software Science and Computational Structures

    ISBN

    978-3-642-00595-4
    978-3-642-00596-1

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-642-00596-1_27

    DOI

    http://dx.doi.org/10.1007/978-3-642-00596-1_27

    DIMENSIONS

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