8,684 research outputs found
Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata
A discretized time evolution of the wave function for a Dirac particle on a
cubic lattice is represented by a very simple quantum cellular automaton. In
each evolution step the updated value of the wave function at a given site
depends only on the values at the nearest sites, the evolution is unitary and
preserves chiral symmetry. Moreover, it is shown that the relationship between
Dirac particles and cellular automata operating on two component objects on a
lattice is indeed very close. Every local and unitary automaton on a cubic
lattice, under some natural assumptions, leads in the continuum limit to the
Weyl equation. The sum over histories is evaluated and its connection with path
integrals and theories of fermions on a lattice is outlined.Comment: 6, RevTe
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Quantum Walks and Reversible Cellular Automata
We investigate a connection between a property of the distribution and a
conserved quantity for the reversible cellular automaton derived from a
discrete-time quantum walk in one dimension. As a corollary, we give a detailed
information of the quantum walk.Comment: 15 pages, minor corrections, some references adde
Density Classification Quality of the Traffic-majority Rules
The density classification task is a famous problem in the theory of cellular
automata. It is unsolvable for deterministic automata, but recently solutions
for stochastic cellular automata have been found. One of them is a set of
stochastic transition rules depending on a parameter , the
traffic-majority rules.
Here I derive a simplified model for these cellular automata. It is valid for
a subset of the initial configurations and uses random walks and generating
functions. I compare its prediction with computer simulations and show that it
expresses recognition quality and time correctly for a large range of
values.Comment: 40 pages, 9 figures. Accepted by the Journal of Cellular Automata.
(Some typos corrected; the numbers for theorems, lemmas and definitions have
changed with respect to version 1.
Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration
The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have been for a long time a central focus of complexity science, and physics.
Here, we introduce group-theoretic concepts to identify and enumerate the symmetric inputs, which result in irreversible system behaviors with undesired effects on many computational tasks. The concept of so-called configuration shift-symmetry is applied on two-dimensional cellular automata as an ideal model of computation. The results show the universal insolvability of “non-symmetric” tasks regardless of the transition function. By using a compact enumeration formula and bounding the number of shift-symmetric configurations for a given lattice size, we efficiently calculate how likely a configuration randomly generated from a uniform or density-uniform distribution turns shift-symmetric. Further, we devise an algorithm detecting the presence of shift-symmetry in a configuration.
The enumeration and probability formulas can directly help to lower the minimal expected error for many crucial (non-symmetric) distributed problems, such as leader election, edge detection, pattern recognition, convex hull/minimum bounding rectangle, and encryption. Besides cellular automata, the shift-symmetry analysis can be used to study the non-linear behavior in various synchronous rule-based systems that include inference engines, Boolean networks, neural networks, and systolic arrays
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