A cut-invariant law of large numbers for random heaps

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong law of large numbers is proved for heap monoids with Bernoulli measures. Finally, we study a sub-additive version of the Law of large numbers in this framework based on Kingman sub-additive Ergodic Theorem.Comment: 29 pages, 3 figures, 21 reference

Markovian dynamics of concurrent systems

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.Comment: 35 pages, 6 figures, 33 reference

Markov two-components processes

We propose Markov two-components processes (M2CP) as a probabilistic model of asynchronous systems based on the trace semantics for concurrency. Considering an asynchronous system distributed over two sites, we introduce concepts and tools to manipulate random trajectories in an asynchronous framework: stopping times, an Asynchronous Strong Markov property, recurrent and transient states and irreducible components of asynchronous probabilistic processes. The asynchrony assumption implies that there is no global totally ordered clock ruling the system. Instead, time appears as partially ordered and random. We construct and characterize M2CP through a finite family of transition matrices. M2CP have a local independence property that guarantees that local components are independent in the probabilistic sense, conditionally to their synchronization constraints. A synchronization product of two Markov chains is introduced, as a natural example of M2CP.Comment: 34 page

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

A local transform for trace monoids

10 pagesWe introduce a transformation for functions defined on the set of cliques of a trace monoid. We prove an inversion formula related to this transformation. It is applied in a probabilistic context in order to obtain a necessary normalization condition for the probabilistic parameters of invariant processes---a class of probabilistic processes introduced elsewhere, and intended to model an asynchronous and memoryless behavior

A method for designing asynchronous probabilistic processes

25 pagesWe present a method for constructing asynchronous probabilistic processes. The asynchronous probabilistic processes thus obtained are called invariant. They generalize the familiar independent and identically distributed sequences of random variables to an asynchronous framework. Invariant processes are shown to be characterised by a finite family of real numbers, their characteristic numbers. Our method provides first a way to obtaining necessary and sufficient normalization conditions for a finite family of real numbers to be the characteristic numbers of some invariant asynchronous probabilistic process; and second, a procedure for constructing new asynchronous probabilistic processes

Concurrency, sigma-algebras and probabilistic fairness

International audienceWe give an interpretation through sigma-algebras of phenomena encountered in concurrency theory when dealing with "infinite confusion"--the extreme opposite of confusion-free event structures. The set of runs of a safe Petri net is equipped with its Borel sigma-algebra F. The fine structure of F describes the complexity of choices along runs, and we show that a transfinite induction of finite degree is needed to explore all choices of runs in general. The degree is minimal (zero) when confusion is bounded, corresponding to the classes of confusion free and locally finite event structures. We relate this construction to probabilistic fairness by showing how to randomize the net equipped with its Borel sigma-algebra by using only the first step of our decomposition, and making it thus more effective. Hence the serious difficulty brought by the above transfiniteness in the application of Kolmogorov extension theorem is bypassed thanks to probabilistic fairnes