Uniform and Bernoulli measures on the boundary of trace monoids

Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace combinatorics with the M\"obius polynomial in the key role. These new measures provide a theoretical foundation for the probabilistic study of concurrent systems.Comment: 34 pages, 5 figures, 27 reference

Computing the average parallelism in trace monoids

Le lien vers arXiv : arXiv:cs/0112012v2The height of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let h(t) and |t| stand respectively for the height and the length of a trace t. Roughly speaking, |t| is the `sequential' execution time and h(t) is the `parallel' execution time. We prove that the bivariate commutative series \sum_t x^{h(t)}y^{|t|} is rational, and we give a finite representation of it. We use the rationality to obtain precise information on the asymptotics of the number of traces of a given height or length. Then, we study the average height of a trace for various probability distributions on traces. For the uniform probability distribution on traces of the same length (resp. of the same height), the asymptotic average height (resp. length) exists and is an algebraic number. To illustrate our results and methods, we consider a couple of examples: the free commutative monoid and the trace monoid whose independence graph is the ladder graph