Three-dimensional Automata Research Articles

Some Accepting Powers of Bottom-Up Pyramid Cellular Acceptors with n-dimensional Layers

In theoretical computer science, the Turing machine was introduced as a simple mathematical model of computers in 1936, and has played a number of important roles in understanding and exploiting basic concepts and mechanisms in computing and information processing. After that, the development of the processing of pictorial information by computer was rapid in those days. Therefore, the problem of computational complexity was also arisen in the two-dimensional information processing. M.Blum and C.Hewitt first proposed two-dimensional automata as a computational model of two-dimensional pattern processing in 1967[1]. Since then, many researchers in this field have been investigating many properties of two- or three-dimensional automata. In 1997, C.R.Dyer and A.Rosenfeld introduced an acceptor on a two-dimensional pattern (or tape), called the pyramid cellular acceptor, and demonstrated that many useful recognition tasks are executed by pyramid cellular acceptors in time proportional to the logarithm of the diameter of the input. They also introduced a bottom-up pyramid cellular acceptor which is a restricted version of the pyramid cellular acceptor, and proposed some interesting open problems about bottom-up pyramid cellular acceptors. On the other hand, we think that the study of n-dimensional automata has been mean- ingful as the computational model of n-dimensional information processing[9]. In this paper, we investigate about bottom-up pyramid cellular accptors with n-dimensional layers, and show their some accepting powers.

Simulation of Grain Coarsening Using One-Dimensional Cellular Automaton

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.

Bottom-up pyramid cellular acceptors with four-dimensional layers

In 1967, M. Blum and C. Hewitt first proposed two-dimensional automata as a computational model of two-dimensional pattern processing. Since then, many researchers in this field have been investigating the many properties of two- or three-dimensional automata. In 1977, C.R. Dyer and A. Rosenfeld introduced an acceptor on a two-dimensional pattern (or tape) called the pyramid cellular acceptor, and demonstrated that many useful recognition tasks are executed by pyramid cellular acceptors in a time which is proportional to the logarithm of the diameter of the input. They also introduced a bottom-up pyramid cellular acceptor, which is a restricted version of the pyramid cellular acceptor, and proposed some interesting open problems about bottom-up pyramid cellular acceptors. On the other hand, we think that the study of four-dimensional automata has been meaningful as the computational model of four-dimensional information processing such as computer animation, moving picture processing, and so forth. In this article, we investigate bottom-up pyramid cellular acceptors with four-dimensional layers, and show some of their accepting powers.

Hierarchies based on the number of cooperating systems of three-dimensional finite automata

The question of whether processing three-dimensional digital patterns is much more difficult than two-dimensional ones is of great interest from both theoretical and practical standpoints. Recently, owing to advances in many application areas, such as computer vision, robotics, and so forth, it has become increasingly apparent that the study of three-dimensional pattern processing is of crucial importance. Thus, the study of three-dimensional automata as a computational model of three-dimensional pattern processing has become meaningful. This article introduces a cooperating system of three-dimensional finite automata as one model of three-dimensional automata. A cooperating system of three-dimensional finite automata consists of a finite number of three-dimensional finite automata and a three-dimensional input tape where these finite automata work independently (in parallel). Those finite automata whose input heads scan the same cell of the input tape can communicate with each other, i.e., every finite automaton is allowed to know the internal states of other finite automata on the cell it is scanning at the moment. In this article, we continue the study of cooperating systems of three-dimensional finite automata, and mainly investigate hierarchies based on the number of their cooperating systems.

Remarks on the recognizability of topological components by three-dimensional automata

It is conjectured that the three-dimensional pattern processing has its own difficulties not arising in two-dimensional case. One of these difficulties occurs in recognizing topological properties of three-dimensional patterns because the three-dimensional neighborhood is more complicated than two-dimensional case. Generally a property or relationship is topological only if it is preserved when an arbitrary “rubber-sheet” distortion is applied to the pictures. For example, adjacency and connectedness are topological; area, elongatedness, convexity, and straightness are not. In recent years, there have been many interesting papers on digital topological properties. For example, an interlocking component was defined as a new topological property in three-dimensional digital pictures, and it was proved that no one marker automaton can recognize interlocking components in a three-dimensional digital picture. In this paper, we deal with recognizability of topological components by three-dimensional Turing machines, and investigate some properties.

Three-dimensional multiinkdot automata

Recently, related to the open problem whether deterministic and nondeterministic space (especially lower-level) complexity classes are separated, inkdot Turing machines were introduced. On the other hand, due to the advances in many application areas about three-dimensional information processing, the research on three-dimensional automata as the computation of three-dimensional pattern processing has been meaningful. From this viewpoint, we propose a three-dimensional multiinkdot automata, and investigate some properties of three-dimensional multiinkdot automata.

A Lattice Gas Automaton Capable of Modeling Three-Dimensional Electromagnetic Fields

A lattice gas automaton (LGA) capable of modeling Maxwell's equations in three dimensions is described. The automaton is a three-dimensional interconnection of two-dimensional LGA cells, with appropriate operations at the junctions between cells to include the properties of polarization. A homogeneous mathematical description of the heterogeneous three-dimensional automaton is provided in terms of the underlying binary variables. The dynamics of the automaton conserve the scalar components of the electric and magnetic fields. The implementation of the automaton on the CAM-8 cellular automata machine is described. The LGA has been validated through calculation of resonant frequencies within various cavities. The numerical results indicate the success of the automaton in analyzing three-dimensional EM field problems. We have not proven analytically that this model reproduces Maxwell's equations in the macroscopic limit, as this is a topic of future study.

A note on two-dimensional finite automata

During the past about thirty-five years, many types of twoor three-dimensional automata have been proposed and investigated the properties of them as the computational model of pattern processing. On the other hand, recently, due to the advances in many application areas such as computer animation, motion image processing, and so on, the study of three-dimensional pattern processing with the time axis has been of crucial importance. Thus, we think that it is very useful for analyzing computation of three-dimensional pattern processing with the time axis to explicate the properties of four-dimensional automata. In this paper, we propose a four-dimensional Turing machine and a four-dimensional finite automaton, and show the space complexities necessary and sufficient for seven-way four-dimensional Turing machines to simulate four-dimensional finite automata, where each sidelength of each input tape of these automata is equivalent. Key-Words: Turing machine, finite automaton, on-line tessellation acceptor, four-dimensional input tape, determinism, nondeterminism, space-bound, configuration, computation, space complexity