181,819 research outputs found
Construction of µ-limit Sets
International audienceThe µ-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to infinity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as µ-limit set where µ is the uniform Bernoulli measure
-Limit Sets of Cellular Automata from a Computational Complexity Perspective
This paper concerns -limit sets of cellular automata: sets of
configurations made of words whose probability to appear does not vanish with
time, starting from an initial -random configuration. More precisely, we
investigate the computational complexity of these sets and of related decision
problems. Main results: first, -limit sets can have a -hard
language, second, they can contain only -complex configurations, third,
any non-trivial property concerning them is at least -hard. We prove
complexity upper bounds, study restrictions of these questions to particular
classes of CA, and different types of (non-)convergence of the measure of a
word during the evolution.Comment: 41 page
On the representation theory of Galois and Atomic Topoi
We elaborate on the representation theorems of topoi as topoi of discrete
actions of various kinds of localic groups and groupoids. We introduce the
concept of "proessential point" and use it to give a new characterization of
pointed Galois topoi. We establish a hierarchy of connected topoi:
[1. essentially pointed Atomic = locally simply connected],
[2. proessentially pointed Atomic = pointed Galois],
[3. pointed Atomic].
These topoi are the classifying topos of, respectively: 1. discrete groups,
2. prodiscrete localic groups, and 3. general localic groups.
We analyze also the unpoited version, and show that for a Galois topos, may
be pointless, the corresponding groupoid can also be considered, in a sense,
the groupoid of "points". In the unpointed theories, these topoi classify,
respectively: 1. connected discrete groupoids, 2. connected (may be pointless)
prodiscrete localic groupoids, and 3. connected groupoids with discrete space
of objects and general localic spaces of hom-sets, when the topos has points
(we do not know the class of localic groupoids that correspond to pointless
connected atomic topoi).
We comment and develop on Grothendieck's galois theory and its generalization
by Joyal-Tierney, and work by other authors on these theories.Comment: This is a revised version of arXiv.org/math.CT/02008222 to appear in
JPA
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