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At least 199 records · Page 11Linked to original sources

arXiv2013

Solving the Parity Problem with Rule 60 in Array Size of the Power of Two

In the parity problem, a given cellular automaton has to classify any initial configuration into two classes according to its parity. Elementary cellular automaton rule 60 can solve the parity problem in periodic boundary conditions with array size of the power of two. The spectral analysis of the configurations of rule 60 at each time step in the evolution reveals that spatial periodicity emerges as the evolution proceeds and the patterns with longer period split into the ones with shorter period. This phenomenon is analogous to the cascade process in which large scale eddies split into smaller ones in turbulence. By measuring the Lempel-Ziv complexity of configuration, we found the stepping decrease of the complexity during the evolution. This result might imply that a decision problem solving process is accompanied with the decline of complexity of configuration.

cs.OH↗

arXiv2013

Computational Universality and 1/f Noise in Elementary Cellular Automata

It is speculated that there is a relationship between 1/f noise and computational universality in cellular automata. We use genetic algorithms to search for one-dimensional and two-state, five-neighbor cellular automata which have 1/f-type spectrum. A power spectrum is calculated from the evolution starting from a random initial configuration. The fitness is estimated from the power spectrum in consideration of the similarity to 1/f-type spectrum. The result shows that the rule with the highest average fitness has a propagating structure like other computationally universal cellular automata, although computational universality of the rule has not been proved yet.

nlin.CG↗

arXiv2013

Physical improvements to a mesoscopic cellular automaton model for three-dimensional snow crystal growth

We motivate and derive the dynamical rules for a computationally feasible three-dimensional cellular automaton model of snow crystal growth. The model improves upon points of weak physical connections identified in other similar models which have produced morphological features observed in many snow crystal photographs. A systematic survey of the morphologies resulting from our model illustrates the degree to which these features persist in our results, and the trends that appear as model parameters are varied.

cond-mat.mtrl-sci↗

arXiv2013

Exact Complexity: The Spectral Decomposition of Intrinsic Computation

We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.

cond-mat.stat-mech↗

arXiv2015

Substitution Systems and Nonextensive Statistics

Substitution systems evolve in time by generating sequences of symbols from a finite alphabet: At a certain iteration step, the existing symbols are systematically replaced by blocks of $N_{k}$ symbols also within the alphabet (with $N_{k}$, a natural number, being the length of the $k$-th block of the substitution). The dynamics of these systems leads naturally to fractals and self-similarity. By using $\mathcal{B}$-calculus [V. Garcia-Morales, Phys. Lett. A 376 (2012) 2645] universal maps for deterministic substitution systems both of constant and non-constant length, are formulated in 1D. It is then shown how these systems can be put in direct correspondence with Tsallis entropy. A `Second Law of Thermodynamics' is also proved for these systems in the asymptotic limit of large words.

math-ph↗

arXiv2015

$μ$-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $μ$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $μ$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $μ$-limit sets can have a $Σ\_3^0$-hard language, second, they can contain only $α$-complex configurations, third, any non-trivial property concerning them is at least $Π\_3^0$-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.

cs.DM↗

arXiv2013

Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata

Cellular automata (CA) have been utilized for decades as discrete models of many physical, mathematical, chemical, biological, and computing systems. The most widely known form of CA, the elementary cellular automaton (ECA), has been studied in particular due to its simple form and versatility. However, these dynamic computation systems possess evolutionary rules dependent on a neighborhood of adjacent cells, which limits their sampling radius and the environments that they can be used in. The purpose of this study was to explore the complex nature of one-dimensional CA in configurations other than that of the standard ECA. Namely, "long-distance cellular automata" (LDCA), a construct that had been described in the past, but never studied. I experimented with a class of LDCA that used spaced sample cells unlike ECA, and were described by the notation LDCA-x-y-n, where x and y represented the amount of spacing between the cell and its left and right neighbors, and n denoted the length of the initial tape for tapes of finite size. Some basic characteristics of ECA are explored in this paper, such as seemingly universal behavior, the prevalence of complexity with varying neighborhoods, and qualitative behavior as a function of x and y spacing. Focusing mainly on purely Class 4 behavior in LDCA-1-2, I found that Rule 73 could potentially be Turing universal through the emulation of a cyclic tag system, and revealed a connection between the mathematics of binary trees and Eulerian numbers that might provide insight into unsolved problems in both fields.

nlin.CG↗

arXiv2013

Self-controlled growth, coherent shrinkage, eternal life in a self-bounded space and other amazing evolutionary dynamics of stochastic pattern formation and growth models inspired by Conways Game of Life

Results of experimental investigation are presented of evolutionary dynamics of several stochastic pattern formation and growth models designed by modifications of the famous mathematical Game of Life. The modifications are two-fold: Game of Life rules are made stochastic and mutual influence of neighboring cells is made non-uniform. The results reveal a number of new phenomena in the evolutionary dynamics of the models: - Ordering of chaos to maze-like patterns: evolutionary formation, from arbitrary seed patterns, of stable maze-like patterns with chaotic dislocations that resemble natural patterns frequently found in the nature, such as skin patterns of some animals. The remarkable property of these patterns is their capability of unlimited growth, self-healing and transplantation. - Self-controlled growth of chaotic live formations into communities bounded, depending on the model, by a square, hexagon or octagon, until they reach a certain critical size, after which the growth stops. - Coherent shrinkage of mature, after reaching a certain size, communities into one of stable or oscillating patterns preserving in this process isomorphism of their bounding shapes until the very end. - Eternal life in a self-bounded space of communities: seemingly permanent birth/death activity of communities after they reach a certain size and shape.

nlin.CG↗

arXiv2013

A Comparison of Tram Priority at Signalized Intersections

We study tram priority at signalized intersections using a stochastic cellular automaton model for multimodal traffic flow. We simulate realistic traffic signal systems, which include signal linking and adaptive cycle lengths and split plans, with different levels of tram priority. We find that tram priority can improve service performance in terms of both average travel time and travel time variability. We consider two main types of tram priority, which we refer to as full and partial priority. Full tram priority is able to guarantee service quality even when traffic is saturated, however, it results in significant costs to other road users. Partial tram priority significantly reduces tram delays while having limited impact on other traffic, and therefore achieves a better result in terms of the overall network performance. We also study variations in which the tram priority is only enforced when trams are running behind schedule, and we find that those variations retain almost all of the benefit for tram operations but with reduced negative impact on the network.

nlin.CG↗

arXiv2014

On generalization of reversible second-order cellular automata

A cellular automaton with $n$ states may be used for construction of reversible second-order cellular automaton with $n^2$ states. Reversible cellular automata with hidden parameters discussed in this paper are generalization of such construction and may have number of states $N=n m$ with arbitrary $m$. Further modification produces reversible cellular automata with reduced number of states $N' < N = n m$.

nlin.CG↗

arXiv2013

Analysis and synthesis of nonlinear reversible cellular automata in linear time

Cellular automata (CA) have been found as an attractive modeling tool for various applications, such as, pattern recognition, image processing, data compression, encryption, and specially for VLSI design & test. For such applications, mostly a special class of CA, called as linear/additive CA, have been utilized. Since linear/additive CA refer a limited number of candidate CA, while searching for solution to a problem, the best result may not be expected. The nonlinear CA can be a better alternative to linear/additive CA for achieving desired solutions in different applications. However, the nonlinear CA are yet to be characterized to fit the design for modeling an application. This work targets characterization of the nonlinear CA to utilize the huge search space of nonlinear CA while developing applications in VLSI domain. An analytical framework is developed to explore the properties of CA rules. The characterization is directed to deal with the reversibility, as the reversible CA are primarily targeted for VLSI applications. The reported characterization enables us to design two algorithms of linear time complexities -- one for identification and nother for synthesis of nonlinear reversible CA. Finally, the CA rules are classified into 6 classes for developing further efficient synthesis algorithm.

cs.FL↗

arXiv2014

Distance Closures on Complex Networks

To expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with widely different structural properties. For instance, the All Pairs Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent to a metric closure, which is only one of the possible ways to calculate shortest paths. Understanding and mapping this isomorphism is necessary to analyse models of complex networks based on weighted graphs. Any conclusions derived from such models should take into account the distortions imposed on graph topology when converting proximity/strength into distance graphs, to subsequently compute path length and shortest path measures. We characterise the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This allows us to: (1) study alternative distance closures, such as those based on diffusion, metric, and ultra-metric distances; (2) identify the operators closest to the metric closure of distance graphs (the APSP), but which are logically consistent; and (3) propose a simple method to compute alternative distance closures using existing algorithms for the APSP. In particular, we show that a specific diffusion distance is promising for community detection in complex networks, and is based on desirable axioms for logical inference or approximate reasoning on networks; it also provides a simple algebraic means to compute diffusion processes on networks. Based on these results, we argue that choosing different distance closures can lead to different conclusions about indirect associations on network data, as well as the structure of complex networks, and are thus important to consider.

cs.SI↗

arXiv2015

Nonlocal and global dynamics of cellular automata: A theoretical computer arithmetic for real continuous maps

A digit function is presented which provides the $i$th-digit in base $p$ of any real number $x$. By means of this function, formulated within $\mathcal{B}$-calculus, the local, nonlocal and global dynamical behaviors of cellular automata (CAs) are systematically explored and universal maps are derived for the three levels of description. None of the maps contain any freely adjustable parameter and they are valid for any number of symbols in the alphabet $p$ and neighborhood range $ρ$. A discrete general method to approximate any real continuous map in the unit interval by a CA on the rational numbers $\mathbb{Q}$ (Diophantine approximation) is presented. This result leads to establish a correspondence between the qualitative behavior found in bifurcation diagrams of real nonlinear maps and the Wolfram classes of CAs. The method is applied to the logistic map, for which a logistic CA is derived. The period doubling cascade into chaos is interpreted as a sequence of global cellular automata of Wolfram's class 2 leading to Class 3 aperiodic behavior. Class 4 behavior is also found close to the period-3 orbits.

math-ph↗

arXiv2014

On rotational invariance of lattice Boltzmann schemes

We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new "Berliner version" of the Taylor expansion method. The details of the implementation are presented. These ideas are applied for the D2Q9, D2Q13, D3Q19 and D3Q27 lattice Boltzmann schemes. Some limitations associated with necessary stability conditions are also presented.

math.NA↗

arXiv2014

Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes

We show that an hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes with the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two space dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and unstationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in unstationary situations.

math.NA↗

arXiv2014

A property of the rule 150 elementary cellular automaton

We studied the rule 150 elementary cellular automaton in terms of the distribution of the spacings of the singular values of the matieces obtained from proper time evolutions patterns. The distribution has strong resembrance to that of the random matrices which is derived from Painlevé V equation. Some analytic results for the relative period of the ECS are also presented.

nlin.CG↗

arXiv2014

Turing degrees of limit sets of cellular automata

Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a computable point and that any non-trivial property on them is undecidable. We go one step further in this article by giving a full characterization of the sets of Turing degrees of cellular automata: they are the same as the sets of Turing degrees of effectively closed sets containing a computable point.

cs.FL↗

arXiv2014

Flexible Time and Ether in One-dimensional Cellular Automata

Flexible Time is a new formalism for calculations about one-dimensional cellular automata. It unifies the states of a finite number of cells into a single object, even if they occur at different times. This gives greater flexibility to handle the structures that occur in the development of a cellular automaton. An ether is a periodic pattern of cells that arises in some cellular automata from almost all random initial configurations. In this thesis, the formalism is developed in detail and then applied to the problem of ether formation. For the elementary cellular automaton Rule 54, a partial result is proved: There is a fragment of the ether that arises with probability 1 from every random initial configuration and is propagated with probability 1 to any later time. This is a strong argument that the ether under Rule 54 indeed arises from almost all input configurations.

nlin.CG↗