170 research outputs found
Aperiodic tilings and entropy
In this paper we present a construction of Kari-Culik aperiodic tile set -
the smallest known until now. With the help of this construction, we prove that
this tileset has positive entropy. We also explain why this result was not
expected
Compressing Random Microstructures via Stochastic Wang Tilings
This paper presents a stochastic Wang tiling based technique to compress or
reconstruct disordered microstructures on the basis of given spatial
statistics. Unlike the existing approaches based on a single unit cell, it
utilizes a finite set of tiles assembled by a stochastic tiling algorithm,
thereby allowing to accurately reproduce long-range orientation orders in a
computationally efficient manner. Although the basic features of the method are
demonstrated for a two-dimensional particulate suspension, the present
framework is fully extensible to generic multi-dimensional media.Comment: 4 pages, 6 figures, v2: minor changes as suggested by reviewers, v3:
corrected two typos in the revised versio
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Quasiperiodicity and non-computability in tilings
We study tilings of the plane that combine strong properties of different
nature: combinatorial and algorithmic. We prove existence of a tile set that
accepts only quasiperiodic and non-recursive tilings. Our construction is based
on the fixed point construction; we improve this general technique and make it
enforce the property of local regularity of tilings needed for
quasiperiodicity. We prove also a stronger result: any effectively closed set
can be recursively transformed into a tile set so that the Turing degrees of
the resulted tilings consists exactly of the upper cone based on the Turing
degrees of the later.Comment: v3: the version accepted to MFCS 201
Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
Let T be a triangulation of a simple polygon. A flip in T is the operation of
removing one diagonal of T and adding a different one such that the resulting
graph is again a triangulation. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
The Equivalence Problem for Deterministic MSO Tree Transducers is Decidable
It is decidable for deterministic MSO definable graph-to-string or
graph-to-tree transducers whether they are equivalent on a context-free set of
graphs
Coarse-graining of cellular automata, emergence, and the predictability of complex systems
We study the predictability of emergent phenomena in complex systems. Using
nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show
how to construct local coarse-grained descriptions of CA in all classes of
Wolfram's classification. The resulting coarse-grained CA that we construct are
capable of emulating the large-scale behavior of the original systems without
accounting for small-scale details. Several CA that can be coarse-grained by
this construction are known to be universal Turing machines; they can emulate
any CA or other computing devices and are therefore undecidable. We thus show
that because in practice one only seeks coarse-grained information, complex
physical systems can be predictable and even decidable at some level of
description. The renormalization group flows that we construct induce a
hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity
and is therefore a good candidate for a complexity measure and a classification
method. Finally we argue that the large scale dynamics of CA can be very
simple, at least when measured by the Kolmogorov complexity of the large scale
update rule, and moreover exhibits a novel scaling law. We show that because of
this large-scale simplicity, the probability of finding a coarse-grained
description of CA approaches unity as one goes to increasingly coarser scales.
We interpret this large scale simplicity as a pattern formation mechanism in
which large scale patterns are forced upon the system by the simplicity of the
rules that govern the large scale dynamics.Comment: 18 pages, 9 figure
A Full Computation-relevant Topological Dynamics Classification of Elementary Cellular Automata
Cellular automata are both computational and dynamical systems. We give a
complete classification of the dynamic behaviour of elementary cellular
automata (ECA) in terms of fundamental dynamic system notions such as
sensitivity and chaoticity. The "complex" ECA emerge to be sensitive, but not
chaotic and not eventually weakly periodic. Based on this classification, we
conjecture that elementary cellular automata capable of carrying out complex
computations, such as needed for Turing-universality, are at the "edge of
chaos"
How to Tackle Integer Weighted Automata Positivity
International audienceThis paper is dedicated to candidate abstractions to capture relevant aspects of the integer weighted automata. The expected effect of applying these abstractions is studied to build the deterministic reachability graphs allowing us to semi-decide the positivity problem on these automata. Moreover, the papers reports on the implementations and experimental results, and discusses other encodings
A Review of Flight-Initiation Distances and their Application to Managing Disturbance to Australian Birds
Disturbance – the response of birds to a stimulus such as the presence of a person – is considered a conservation threat for some Australian birds. The distance at which a bird flees from perceived danger is defined as the flight-initiation distance (FID), and could be used to designate separation distances between birds and stimuli that might cause disturbance. We review the known FIDs for Australian birds, and report FIDs for 250 species. Most FIDs are from south-eastern Australia, and almost all refer to a single walker as the stimulus. Several prominent factors correlated with FID are discussed (e.g. body mass and the distance at which an approach begins). FIDs have not been used extensively in the management of disturbance, for a variety of reasons including lack and inaccessibility of available data. We call for standardised data collection and greater application of available data to the management of disturbance
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