Cellular automata complexity trade-offs

Since Richard Feynman introduced the notion of computation in 1982, various models of quantum computers have been proposed (R. Feynman, 1992). These models include Turing machines and circuits. We define another computational model, one dimensional cellular automata, and demonstrate that any Turing machine can be efficiently simulated by a one dimensional cellular automaton with constant slowdown. This can be accomplished by consideration of a restricted class of one dimensional cellular automata called one dimensional partitioned cellular automata. We also show that any one dimensional partitioned cellular automaton can be simulated by a Turing machine with linear slowdown, but the problem of efficiently simulating an arbitrary one dimensional cellular automaton with a Turing machine is left open. From this discussion, some interesting facts concerning these models are easily deduced.

We study the influence of the dimension of cellular automata (CA) for real time language recognition of one-dimensional languages with parallel input. Specifically, we focus on the question of determining whether every language that can be recognized in real time on a 2-dimensional CA working on the Moore neighborhood can also be recognized in real time by a 1-dimensional CA working on the standard two-way neighborhood. We show that 2-dimensional CA in real time can perform a linear number of simulations of a 1-dimensional real time CA. If the two classes are equal then the number of simulated instances can be polynomial.

Matrix algebraic formulae concerning some exceptional rules of two-dimensional cellular automata

This study investigates the effects of sample size and sample prevalence on cellular automata (CA) simulation of urban growth. We take the CA models based on an artificial neural network (ANN), logistic regression (LR), and support vector machine (SVM) as examples, to simulate the urban growth of Wuhan city in China and the Wuhan Metropolitan Area under different sampling schemes. The results of the CA models based on the ANN, LR, and SVM methods are generally consistent. The sampling scheme with a small sample size and a low sample prevalence should be discarded because of the high uncertainty. Sample size determines the robustness of a CA model, whereas sample prevalence affects the performance of a CA model when there are sufficient samples. In particular, the closer the sample prevalence is to the population prevalence, the higher the simulation accuracy and the lower the shape complexity and fragmentation of the simulated urban patterns. We suggest that the optimal sampling scheme has a sample rate of 1% and a sample prevalence that is the same as the population prevalence. The selection of the optimal sampling scheme is independent of the population sizes represented by different study areas.

A Random NP-complete problem for inversion of 2D cellular automata

In this paper a fast and space efficient method for simulating a d′-dimensional cellular automaton (CA) on a d-dimensional CA (d<d′) is introduced. For d′=2 and d=1 this method is optimal (under certain assumptions) with respect to time as well as space complexity. Let in this case t(n) be the time complexity and r(n) the side length of the smallest square enclosing all used cells. Then the simulation does not need more than O(r 2) cells and has a running time of d′ = d+1 a version with the time and space complexity of O(t 2) will be presented. Finally it will be shown, how it is possible to simulate a 2d-dimensional CA on a d-dimensional CA in a similarly efficient way.KeywordsTime ComplexityCellular AutomatonSpace ComplexityQuiescent StateComputation StepThese 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.

Fusion-based additive manufacturing of metal is particularly challenging to simulate due to the broad range of length/time scales of multiphysics phenomena to bridge. Integrated Computational Materials Engineering (ICME) offers a promising route to accelerate the design of new materials, optimize manufacturing processes, and thus improve material properties and performance. ICME heavily relies on linking multiple modelling techniques relevant to distinct length/time scales and/or different physics. While a wide range of physics-based and predictive models exist, the efficient linking of these models remains challenging. In the context of additive manufacturing, interdependent phenomena across scales include: multicomponent alloys with complex phase diagrams, multiple phase transformations and temperature-dependent properties; complex thermal exchanges in the vicinity of the melt pool; and the development of nonequilibrium grain microstructures within this melt pool. A multiscale modeling strategy (i.e. micro and macro) has been developed to simulate powder-bed fusion of metallic alloys, which combines: (1) Temperaturedependent alloy properties and phase diagrams calculated by CalPhaD, (2) macroscale thermal and thermo-mechanical simulations of the material addition and fusion using finite elements (FE), (3) Phase-field (PF) simulations of the melt pool’s solidification at the microscopic scale, and (4) Simulations of the microscopic solidification of the melt pool using cellular automaton (CA). Realistic processing parameters are used in the methodology to simulate selective laser melting (SLM) of an Inconel 718 alloy. In the first step, three-dimensional thermal and thermo-mechanical frameworks were developed – using an in-house finite element code (IRIS) and a material library (MUESLI) implemented in C++ programming language – by considering these four factors: (1) Representing the thermal input from the scanning laser, (2) A material model to specify the temperature-dependent material property in the cyclic heating and cooling environment, (3) A physics-based method to explain the layer build-up process, and (4) A laser material interaction model to account for transient thermo-mechanical phenomena. Then, the thermal model was coupled with CalPhaD and PF methods to predict the formation of microstructures at the scale of the entire melt pool. In this part, the effect of temperature-dependent properties and the necessity of taking into account differences in properties between the powder bed and the dense material have been discussed. We perform an appropriately-converged PF solidification simulation at the scale of the entire melt pool using a two-dimensional longitudinal slice of the thermal field calculated through FE simulations. This calculation has over one billion grid points, yet is performed on a single cluster node with eight graphics processing units (GPUs). These microscale simulations, with a level of detail down to individual dendrites, offer new insight into the selection of grain texture via polycrystalline growth competition under realistic SLM conditions. Finally, comparisons between phase-field (PF) and cellular automaton (CA) simulations of polycrystalline growth in a two-dimensional melt pool under additive manufacturing (powder-bed fusion) relevant conditions have been performed. The generated grain structures from local (point-by-point) measurements and grain orientation distributions that were averaged over various simulations have been compared. We investigated how the melt pool aspect ratio and the CA spatial discretization level affected the selected grain texture. Our simulations demonstrate that only PF simulations are capable of capturing the fine-grained microscopic features related to transient growth conditions and solid-liquid interface stability, such as the initial planar growth stage before its cellular/dendritic destabilization or the early elimination of unfavorably oriented grains due to neighbor grain sidebranching. CA grid refinement can only partially address the resulting disagreement between PF and CA predictions. In addition, with some variability on the CA grid and melt pool shape, overall grain distributions averaged throughout the entire melt pools of several simulations appear to lead to a noticeably better agreement between PF and CA. This research offers a helpful step in that direction by quantitatively comparing both approaches at process-relevant length and time scales, even though more effort is still needed, especially to identify the appropriate selection of CA spatial discretization and its link to characteristic microstructural length scales.

Cellular automata (CA) with given evolution rules have been widely investigated, but the inverse problem of extracting CA rules from observed data is less studied. Current CA rule extraction approaches are both time consuming and inefficient when selecting neighborhoods. We give a novel approach to identifying CA rules from observed data and selecting CA neighborhoods based on the identified CA model. Our identification algorithm uses a model linear in its parameters and gives a unified framework for representing the identification problem for both deterministic and probabilistic CA. Parameters are estimated based on a minimum variance criterion. An incremental procedure is applied during CA identification to select an initial coarse neighborhood. Redundant cells in the neighborhood are then removed based on parameter estimates, and the neighborhood size is determined using the Bayesian information criterion. Experimental results show the effectiveness of our algorithm and that it outperforms other leading CA identification algorithms.

The purpose of this article is to describe a way to simulate any 3-dimensional cellular automaton with a 2-dimensional cellular automaton. We will present the problems that arise when changing the dimension, propose solutions and discuss the main properties of the obtained simulation.

We are living in an age where security of digital information like digital images is becoming more important in many aspects and the tranmission of digital images is increasing over the internal and international computer networks. The protection of digital images from unauthorized access is the main purpose of this paper.This paper introduced an approach to encrypt a digital image in frequency domain through DCT (Discrete Cosine Transform) and one dimenssional cellular automata. It is based on encryption of some quantified DCT coefficient in high frequencies. We applied our proposed model on one hundred numbers of JPEG grayscale images of size 800 × 800. The experimental results have indicated the robustness of our algorithm.KeywordsDigital Images EncryptionFrequency Domain EncryptionCellular Automata

In cellular automata, a single update rule is applied simultaneously to each cell on the lattice. Therefore, many approaches with parallel and distributed systems have been researched. In this paper, we propose a computation method of three dimensional cellular automata for small FPGA systems with limited memory bandwidth. We implemented the method on a FPGA board (Celoxica RC1000 with one Virtex-E XCV2000E). The speed gain for a 3D Life Game and a 3D Forest Fires model with 128 × 128 × 128 lattice is 386 and 79 times compared with Celeron Processor 1.20GHz.

Abstract. In a sequencing problem with linear time cost, Suijs (1996) proved that it is possible to achieve first best. By first best we mean that one can find mechanisms that satisfy efficiency of decision, dominant strategy incentive compatibility and budget balancedness. In this paper we show that among a more general and natural class of sequencing problems, sequencing problems with linear cost is the only class for which first best can be achieved.

A modelling method for a self-proliferation cellular automaton (CA) is proposed in this paper. Setting some attributes, the CA model can be used to solve some complicated problems. It is a useful model for some intelligent systems. Crystallization of hypereutectoid steel is a complex nonlinear process and is difficult to analyze by using the conventional numerical method to establish a complete and accurate mathematical model. Based on the principles of crystallization of hypereutectoid steel and the CA model, the author separated, in time and space, the process of crystallization into periodically growing arrays, so that the simulation of crystallization of hypereutectoid steel is implemented.

This study evaluates the effects of cellular automata (CA) with different neighborhood sizes on the predictive performance of the Land Transformation Model (LTM). Landsat images were used to extract urban footprints and the driving forces behind urban growth seen for the metropolitan areas of Tehran and Isfahan in Iran. LTM, which uses a back-propagation neural network, was applied to investigate the relationships between urban growth and the associated drivers, and to create the transition probability map. To simulate urban growth, the following two approaches were implemented: (a) the LTM using a top-down approach for cell allocation grounding on the highest values in the transition probability map and (b) a CA with varying spatial neighborhood sizes. The results show that using the LTM-CA approach increases the accuracy of the simulated land use maps when compared with the use of the LTM top-down approach. In particular, the LTM-CA with a 7 × 7 neighborhood size performed well and improved the accuracy. The level of agreement between simulated and actual urban growth increased from 58% to 61% for Tehran and from 39% to 43% for Isfahan. In conclusion, even though the LTM-CA outperforms the LTM with a top-down approach, more studies have to be carried out within other geographical settings to better evaluate the effect of CA on the allocation phase of the urban growth simulation.

Artificial genetic circuits have been recently proposed to realize unprecedented biological systems which do not exist in nature by combining several genes. For example, recombinant Escherichia coli provides novel artificial functions such as blinking periodically or invading cancer cells. In protein engineering, new functional proteins have been created by site-directed mutagenesis methods. The main purpose of our research is to implement the Cellular Automata (CA) in vivo employing these methods. Computational capacity of CA is equivalent to the universal Turing Machine. CA are multi-cellular systems in which uniform cells are allocated on lattice grid. Each cell has a finite number of states. The state transition is determined by the current state and the states of neighbor cells. Three mechanisms are required to implement CA in bacteria: (1) Sending and receiving signals between cells, (2) Sustaining the state, (3) Sensing input signals and changing the state following the state transition rules. To encode signals for the mechanism (1), we use small molecules inducing transcription. For the mechanism (2), the toggle switch circuit was employed to represent a finite number of states using gene expression. If one gene does express and the other does not, the state is 1, and vice versa, 0. For the mechanism (3), the state transition functions were implemented as logic gates using transcriptional regulatory proteins which bind to specific signal molecules (chemicals). In this paper, we applied this method to in vivo implementation of one-dimensional CA and corroborated whether it behaves correctly. In principle, this method can be applied to any dimensional CA by employing more signal molecules Permission to make digital or hard copies of all or part of this work for to construct more complex logic gate. Furthermore, we succeeded to execute one step calculation of one-dimensional CA in vivo in a laboratory experiment.