20,806 research outputs found
The Spectra of Lamplighter Groups and Cayley Machines
We calculate the spectra and spectral measures associated to random walks on
restricted wreath products of finite groups with the infinite cyclic group, by
calculating the Kesten-von Neumann-Serre spectral measures for the random walks
on Schreier graphs of certain groups generated by automata. This generalises
the work of Grigorchuk and Zuk on the lamplighter group. In the process we
characterise when the usual spectral measure for a group generated by automata
coincides with the Kesten-von Neumann-Serre spectral measure.Comment: 36 pages, improved exposition, main results slightly strengthene
Procedures for calculating reversible one-dimensional cellular automata
We describe two algorithms for calculating reversible one-dimensional cellular automata of neighborhood size 2. We explain how this kind of automaton represents all the other cases. Using two basic properties of reversible automata such as uniform multiplicity of ancestors and Welch indices, these algorithms only require matrix products and transitive closures of binary relations to classify all the possible reversible automata of neighborhood size 2. We expose the features, advantages and differences with other well-known methods. Finally, we present results for reversible automata from three to six states and neighborhood size 2. © 2005 Elsevier B.V. All rights reserved
Automata in SageMath---Combinatorics meet Theoretical Computer Science
The new finite state machine package in the mathematics software system
SageMath is presented and illustrated by many examples. Several combinatorial
problems, in particular digit problems, are introduced, modeled by automata and
transducers and solved using SageMath. In particular, we compute the asymptotic
Hamming weight of a non-adjacent-form-like digit expansion, which was not known
before
Mean-Field Theory of Meta-Learning
We discuss here the mean-field theory for a cellular automata model of
meta-learning. The meta-learning is the process of combining outcomes of
individual learning procedures in order to determine the final decision with
higher accuracy than any single learning method. Our method is constructed from
an ensemble of interacting, learning agents, that acquire and process incoming
information using various types, or different versions of machine learning
algorithms. The abstract learning space, where all agents are located, is
constructed here using a fully connected model that couples all agents with
random strength values. The cellular automata network simulates the higher
level integration of information acquired from the independent learning trials.
The final classification of incoming input data is therefore defined as the
stationary state of the meta-learning system using simple majority rule, yet
the minority clusters that share opposite classification outcome can be
observed in the system. Therefore, the probability of selecting proper class
for a given input data, can be estimated even without the prior knowledge of
its affiliation. The fuzzy logic can be easily introduced into the system, even
if learning agents are build from simple binary classification machine learning
algorithms by calculating the percentage of agreeing agents.Comment: 23 page
The Kinetic Basis of Self-Organized Pattern Formation
In his seminal paper on morphogenesis (1952), Alan Turing demonstrated that
different spatio-temporal patterns can arise due to instability of the
homogeneous state in reaction-diffusion systems, but at least two species are
necessary to produce even the simplest stationary patterns. This paper is aimed
to propose a novel model of the analog (continuous state) kinetic automaton and
to show that stationary and dynamic patterns can arise in one-component
networks of kinetic automata. Possible applicability of kinetic networks to
modeling of real-world phenomena is also discussed.Comment: 8 pages, submitted to the 14th International Conference on the
Synthesis and Simulation of Living Systems (Alife 14) on 23.03.2014, accepted
09.05.201
Complex dynamics emerging in Rule 30 with majority memory
In cellular automata with memory, the unchanged maps of the conventional
cellular automata are applied to cells endowed with memory of their past states
in some specified interval. We implement Rule 30 automata with a majority
memory and show that using the memory function we can transform quasi-chaotic
dynamics of classical Rule 30 into domains of travelling structures with
predictable behaviour. We analyse morphological complexity of the automata and
classify dynamics of gliders (particles, self-localizations) in memory-enriched
Rule 30. We provide formal ways of encoding and classifying glider dynamics
using de Bruijn diagrams, soliton reactions and quasi-chemical representations
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