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

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

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

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

Möbius inversion formula for the trace group

A trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs of generators. We prove an analog for the trace group of the Möbius inversion formula for the trace monoid (Cartier and Foata, 1969)

Sampling solution traces for the problem of sorting permutations by signed reversals

International audienceBackgroundTraditional algorithms to solve the problem of sorting by signed reversals output just one optimal solution while the space of all optimal solutions can be huge. A so-called trace represents a group of solutions which share the same set of reversals that must be applied to sort the original permutation following a partial ordering. By using traces, we therefore can represent the set of optimal solutions in a more compact way. Algorithms for enumerating the complete set of traces of solutions were developed. However, due to their exponential complexity, their practical use is limited to small permutations. A partial enumeration of traces is a sampling of the complete set of traces and can be an alternative for the study of distinct evolutionary scenarios of big permutations. Ideally, the sampling should be done uniformly from the space of all optimal solutions. This is however conjectured to be ♯P-complete.ResultsWe propose and evaluate three algorithms for producing a sampling of the complete set of traces that instead can be shown in practice to preserve some of the characteristics of the space of all solutions. The first algorithm (RA) performs the construction of traces through a random selection of reversals on the list of optimal 1-sequences. The second algorithm (DFALT) consists in a slight modification of an algorithm that performs the complete enumeration of traces. Finally, the third algorithm (SWA) is based on a sliding window strategy to improve the enumeration of traces. All proposed algorithms were able to enumerate traces for permutations with up to 200 elements.ConclusionsWe analysed the distribution of the enumerated traces with respect to their height and average reversal length. Various works indicate that the reversal length can be an important aspect in genome rearrangements. The algorithms RA and SWA show a tendency to lose traces with high average reversal length. Such traces are however rare, and qualitatively our results show that, for testable-sized permutations, the algorithms DFALT and SWA produce distributions which approximate the reversal length distributions observed with a complete enumeration of the set of traces