Divisibility and cellular automata

We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries-termed "Pascal's triangle symmetries"-which provides a natural U(1) generalization of a spin-(1/2) system with Sierpinski triangle fractal symmetries introduced in Newman etal.,[Phys. Rev. E 60, 5068 (1999).PLEEE81063-651X10.1103/PhysRevE.60.5068]. The Pascal's triangle symmetry gives rise to exact degeneracies, as well as a manifold of low-energy states which are absent in the Sierpinski triangle model. Breaking the U(1) symmetry of this model to Z_{p}, with prime integer p, yields a lattice model with a unique fractal symmetry which is generated by an operator supported on a fractal subsystem with Hausdorff dimension d_{H}=ln(p(p+1)/2)/lnp. The Hausdorff dimension of the fractal can be probed through correlation functions at finite temperature. The phase diagram of these models at zero temperature in the presence of quantum fluctuations, as well as the potential physical construction of the U(1) model, is discussed.

An attractive approach to design antenna arrays with minimum sidelobe level is based on the binomial expansion, and they are referred to as binomial arrays. In fact, binomial arrays with element spacing equal or less than λ/2 have no minor lobes. However, there are no compact expressions available to obtain the directivity of binomial arrays with no restriction in the element spacing, except when the element spacing is λ/2. This letter proposes a compact expression for the directivity of the binomial array with no restriction in the element spacing and number of elements. The procedure is based on the properties of the Pascal's triangle. Then, the array factor and directivity of the binomial array are calculated by using the properties of the Fourier transform.

The mathematical characterization of patterns in Pascal’s triangle is useful and can be achieved by a variety of brute-force computations, but sometimes at a cost of the loss of an intuitive feeling for the regularities and irregularities in the structure. The graphic and mathematical approach described here reveals a visually striking and intricate class of patterns that make up a family of regular fractal networks known as Sierpinski gaskets. The figures indicate self-similarity of the gasket structures for several orders of dilational invariance.

We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius r. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form \( (0^{r+1}0^* + 1^{r+1}1^*)^* \), which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form \( s^* \) where s consists of \( O(r^2) \) consecutive sequences of cells with the same state, each such sequence is of length at most r, and the total length of s is \( O(r^2) \) as well. We show that an analogous result also holds for the minority rule.

In this paper an association between Petri nets (PN) and cellular automata (CA) is proposed to analyze the global dynamics of flexible manufacturing systems (FMS). This relation is carried out taking into account the discreteness in the dynamics of both PN and CA. In particular, generalized PN as well as one-dimensional CA are used. The work consists in modeling with PN both a single process with a shared resource and two parallel processes with several shared resources. The PN models are simplified by reduction rules and then the corresponding one-dimensional CA is obtained. Finally, the global dynamics of the FMS modeled is described by using the analysis methods of CA.

Digital audio scrambling is a process used in audio security applications. Scrambling of audio files breaks the correlation between adjacent samples in order to convert the original audio to an unintelligible format. Scrambling is used to protect the audio against wiretapping and illegal surveillance, in addition to being a step in security algorithms, such as watermarking and encryption algorithms. Cellular automata are models that are discrete in nature and depend on simple and local rules to achieve an interesting overall behavior. Two-dimensional cellular automata were previously proposed as a key generation mechanism to scramble audio files. The mechanism was built upon be researchers in the multimedia security field. This paper explores the use of one-dimensional cellular automata in audio scrambling, which simplifies the process as deploying two-dimensional cellular automata requires changing the dimension of the audio file and the one-dimensional cellular automata does not, additionally, elementary one-dimensional cellular automata requires less parameters to configure. The scrambling degree is used to evaluate the model effectiveness in breaking the correlation of adjacent samples. In the experiments, different parameters are taken into account including the cellular automata class, the iterations needed and the method used to calculate the cells at the boundary. Experiments show that the one-dimensional cellular automata are capable of scrambling the audio file without any dimensional change and the chaotic rules tested give the highest scrambling degree.

Constructible functions in cellular automata and their applications to hierarchy results

In this paper, cellular automata (CA) are viewed as an abstract model for distributed computing. The author argues that the classical CA model must be modified in several important respects to become a relevant model for large-scale MAS. The paper first proposes sequential cellular automata (SCA) and formalizes deterministic and nondeterministic versions of SCA. The author then analyzes differences in possible dynamics between classical parallel CA and various SCA models. The analysis in this paper focuses on one-dimensional parallel and sequential CA with node update rules restricted to simple threshold functions, as arguably the simplest totalistic, yet non-linear (and non-affine) update rules. The author identifies properties of asymptotic dynamics that can be proven to be entirely due to the assumption of perfect synchrony in classical, parallel CA. Finally, the paper discusses what an appropriate CA-based abstraction would be for large-scale distributed computing, insofar as the inter-agent communication models. In that context, the author proposes genuinely asynchronous CA and discusses main differences between genuinely asynchronous CA and various weakly asynchronous sequential CA models found in the literature.

The density classification problem consists in using a binary cellular automaton (CA) to decide whether an initial configuration contains more 0s or 1s. This problem is known for having no exact solution in the case of binary, deterministic, one-dimensional CA. Stochastic cellular automata have been studied as an alternative for solving the problem. This paper is aimed at presenting techniques to analyse the behaviour of stochastic CA rules, seen as a ``blend'' of deterministic CA rules. Using analytical calculations and numerical simulations, we analyse two previously studied rules and present a new rule. We estimate their quality of classification and their average time of classification. We show that the new rule solves the problem with an arbitrary precision. From a practical point of view, this rule is effective and exhibits a high quality of classification, even when the simulation time is kept small.

Directional dynamics along arbitrary curves in cellular automata

An asynchronous solution to the synchronisation problem for binary one-dimensional cellular automata

Various investigations have been carried out on the computational power of cellular automata (CA), with concentrated efforts in the study of one-dimensional CA. One of the approaches is the use of genetic algorithms (GA) to look for CA with a predefined computational behavior. We have previously shown a set of parameters that can be effective in helping forecast CA dynamic behavior; here, they are used as an heuristic to guide the GA search, by biasing selection, mutation and crossover, in the context of the Grouping Task (GT) for one-dimensional CA. Since GT is a new task, no a priori knowledge about its solutions is presently available; even then, the incorporation of the parameter-based heuristic entails a significant improvement over the results achieved by the plain genetic search.KeywordsGenetic AlgorithmCellular AutomatonCellular AutomatonGroup TaskTransition RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular automaton analogous of localizations or quasi-local collective excitations traveling in a spatially extended nonlinear medium. They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyze what types of interaction occur between gliders traveling on a cellular automaton "cyclotron" and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in nonlinear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems.

A one-dimensional cellular automaton τ : A Z → A Z is a transformation of the full shift defined via a finite neighbourhood S ⊂ Z and a local function μ : A S → A . We study the family of cellular automata whose finite neighbourhood S is an interval containing 0, and there exists a pattern p ∈ A S satisfying that μ ( z ) = z ( 0 ) if and only if z ≠ p ; this means that these cellular automata have a unique active transition. Despite its simplicity, this family presents interesting and subtle problems, as the behaviour of the cellular automaton completely depends on the structure of p. We show that every cellular automaton τ with a unique active transition p ∈ A S is either idempotent or strictly almost equicontinuous, and we completely characterize each one of these situations in terms of p. In essence, the idempotence of τ depends on the existence of a certain subpattern of p with a translational symmetry.

Numerous literature [1,4,5] has reported on the effective use of cellular automaton method for the simulation of short-range diffusion. Using this model for the simulation of short-range diffusional phase transformations therefore is a resolved issue. It is proven that two- or three-dimensional automata can reflect the course of the abovementioned processes realistically. What our study demonstrates more than in the past [1] is that two-dimensional stochastic cellular automaton simulation already presented before has been simplified. This time our automaton operates in one dimension [2], which has consequently reduced running time, thus, made it possible to enhance the efficiency of the scaling of simulation. In our previous work the results of scaling of one-dimensional simulation of the recrystallization process [3] were demonstrated, in our current study fitting is performed for measurement results of grain coarsening using one-dimensional cellular automaton.