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Markovian dynamics of concurrent systems

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Abstract

Monoid actions of trace monoids over finite sets are powerful models of concurrent systems—for instance they encompass the class of 1-safe Petri nets. We characterise Markov measures attached to concurrent systems by finitely many parameters with suitable normalisation conditions. These conditions involve polynomials related to the combinatorics of the monoid and of the monoid action. These parameters generalise to concurrent systems the coefficients of the transition matrix of a Markov chain. A natural problem is the existence of the uniform measure for every concurrent system. We prove this existence under an irreducibility condition. The uniform measure of a concurrent system is characterised by a real number, the characteristic root of the action, and a function of pairs of states, the Parry cocyle. A new combinatorial inversion formula allows to identify a polynomial of which the characteristic root is the smallest positive root. Examples based on simple combinatorial tilings are studied.

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Notes

  1. A note on terminology: we avoid the use of the word deterministic since it has different meanings according to the scientific community that uses it. In Probability theory, deterministic is opposed to probabilistic. On the contrary, in Computer science, deterministic is opposed to non-deterministic, regardless of the existence of a probabilistic context; and non-deterministic refers to the existence of a choice in the evolution of the system for a given action. In the Computer science language, all the systems considered in this paper are deterministic.

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  1. IRIF (UMR CNRS 8243)/Université Paris Diderot, Paris, France

    Samy Abbes

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  1. Samy Abbes
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Abbes, S. Markovian dynamics of concurrent systems. Discrete Event Dyn Syst 29, 527–566 (2019). https://doi.org/10.1007/s10626-019-00291-z

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