62,791 research outputs found
Multidimensional extension of the Morse--Hedlund theorem
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a
sequence over a finite alphabet is ultimately periodic if and only if, for
some , the number of different factors of length appearing in is
less than . Attempts to extend this fundamental result, for example, to
higher dimensions, have been considered during the last fifteen years. Let
. A legitimate extension to a multidimensional setting of the notion of
periodicity is to consider sets of \ZZ^d definable by a first order formula
in the Presburger arithmetic . With this latter notion and using a
powerful criterion due to Muchnik, we exhibit a complete extension of the
Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of
$\ZZ^d$ definable in in terms of some functions counting recurrent
blocks, that is, blocks occurring infinitely often
Decidability and Periodicity of Low Complexity Tilings
International audienceIn this paper we study colorings (or tilings) of the two-dimensional grid Z 2. A coloring is said to be valid with respect to a set P of n × m rectangular patterns if all n × m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist m, n ∈ N and a set P of n × m rectangular patterns such that c is valid with respect to P and |P | ≤ nm. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, all valid colorings with respect to P such that |P | ≤ mn must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat's conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that the nm bound is multiplicatively optimal for the decidability of the domino problem, as for all ε > 0 it is undecidable to determine if there exists a valid coloring for a given m, n ∈ N and set of rectangular patterns P of size n×m such that |P | ≤ (1 + ε)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 2020)
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
Real-World Repetition Estimation by Div, Grad and Curl
We consider the problem of estimating repetition in video, such as performing
push-ups, cutting a melon or playing violin. Existing work shows good results
under the assumption of static and stationary periodicity. As realistic video
is rarely perfectly static and stationary, the often preferred Fourier-based
measurements is inapt. Instead, we adopt the wavelet transform to better handle
non-static and non-stationary video dynamics. From the flow field and its
differentials, we derive three fundamental motion types and three motion
continuities of intrinsic periodicity in 3D. On top of this, the 2D perception
of 3D periodicity considers two extreme viewpoints. What follows are 18
fundamental cases of recurrent perception in 2D. In practice, to deal with the
variety of repetitive appearance, our theory implies measuring time-varying
flow and its differentials (gradient, divergence and curl) over segmented
foreground motion. For experiments, we introduce the new QUVA Repetition
dataset, reflecting reality by including non-static and non-stationary videos.
On the task of counting repetitions in video, we obtain favorable results
compared to a deep learning alternative
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