Markovian dynamics of concurrent systems

TL;DR

This paper characterizes Markov measures in concurrent systems modeled by monoid actions, providing conditions for their existence, and introduces combinatorial formulas and examples related to these measures.

Contribution

It offers a novel characterization of Markov measures for concurrent systems using finitely many parameters and introduces a new combinatorial inversion formula.

Findings

Existence of uniform measure under irreducibility condition

Characterization of uniform measure by characteristic root and Parry cocycle

Application to combinatorial tilings examples

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.