802 research outputs found
Asymptotic properties of free monoid morphisms
Motivated by applications in the theory of numeration systems and
recognizable sets of integers, this paper deals with morphic words when erasing
morphisms are taken into account. Cobham showed that if an infinite word is the image of a fixed point of a morphism under another
morphism , then there exist a non-erasing morphism and a coding
such that .
Based on the Perron theorem about asymptotic properties of powers of
non-negative matrices, our main contribution is an in-depth study of the growth
type of iterated morphisms when one replaces erasing morphisms with non-erasing
ones. We also explicitly provide an algorithm computing and
from and .Comment: 25 page
Helly meets Garside and Artin
A graph is Helly if every family of pairwise intersecting combinatorial balls
has a nonempty intersection. We show that weak Garside groups of finite type
and FC-type Artin groups are Helly, that is, they act geometrically on Helly
graphs. In particular, such groups act geometrically on spaces with convex
geodesic bicombing, equipping them with a nonpositive-curvature-like structure.
That structure has many properties of a CAT(0) structure and, additionally, it
has a combinatorial flavor implying biautomaticity. As immediate consequences
we obtain new results for FC-type Artin groups (in particular braid groups and
spherical Artin groups) and weak Garside groups, including e.g.\ fundamental
groups of the complements of complexified finite simplicial arrangements of
hyperplanes, braid groups of well-generated complex reflection groups, and
one-relator groups with non-trivial center. Among the results are:
biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones
conjecture, the coarse Baum-Connes conjecture, and a description of higher
order homological and homotopical Dehn functions. As a mean of proving the
Helly property we introduce and use the notion of a (generalized) cell Helly
complex.Comment: Small modifications according to the referee report, updated
references. Final accepted versio
Profinite Techniques for Probabilistic Automata and the Markov Monoid Algorithm
We consider the value 1 problem for probabilistic automata over finite words:
it asks whether a given probabilistic automaton accepts words with probability
arbitrarily close to 1. This problem is known to be undecidable. However,
different algorithms have been proposed to partially solve it; it has been
recently shown that the Markov Monoid algorithm, based on algebra, is the most
correct algorithm so far. The first contribution of this paper is to give a
characterisation of the Markov Monoid algorithm. The second contribution is to
develop a profinite theory for probabilistic automata, called the prostochastic
theory. This new framework gives a topological account of the value 1 problem,
which in this context is cast as an emptiness problem. The above
characterisation is reformulated using the prostochastic theory, allowing us to
give a simple and modular proof.Comment: Conference version: STACS'2016, Symposium on Theoretical Aspects of
Computer Science Journal version: TCS'2017, Theoretical Computer Scienc
On a conjecture by Pierre Cartier about a group of associators
In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative
formal power series on with coefficients in a
\Q-extension, , subjected to some suitable conditions, there exists an
unique algebra homomorphism from the \Q-algebra generated by the
convergent polyz\^etas to such that is computed from
Drinfel'd associator by applying to each coefficient. We prove
exists and it is a free Lie exponential over . Moreover, we give a
complete description of the kernel of polyz\^eta and draw some consequences
about a structure of the algebra of convergent polyz\^etas and about the
arithmetical nature of the Euler constant
Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
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