125 research outputs found
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
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 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
Finite transducers for divisibility monoids
Divisibility monoids are a natural lattice-theoretical generalization of
Mazurkiewicz trace monoids, namely monoids in which the distributivity of the
involved divisibility lattices is kept as an hypothesis, but the relations
between the generators are not supposed to necessarily be commutations. Here,
we show that every divisibility monoid admits an explicit finite transducer
which allows to compute normal forms in quadratic time. In addition, we prove
that every divisibility monoid is biautomatic.Comment: 20 page
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
Uniform generation in trace monoids
We consider the problem of random uniform generation of traces (the elements
of a free partially commutative monoid) in light of the uniform measure on the
boundary at infinity of the associated monoid. We obtain a product
decomposition of the uniform measure at infinity if the trace monoid has
several irreducible components-a case where other notions such as Parry
measures, are not defined. Random generation algorithms are then examined.Comment: Full version of the paper in MFCS 2015 with the same titl
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
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