Cellular automaton

A cellular automaton pl. cellular automata, abbrev. CA is the discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, as well as iterative arrays. Cellular automata make found applications in various areas, including physics, theoretical biology as alive as microstructure modeling.

A cellular automaton consists of agrid of cells, used to refer to every one of two or more people or things in one of a finite number of stochastic cellular automaton in addition to asynchronous cellular automaton.

The concept was originally discovered in the 1940s by Conway's Game of Life, a two-dimensional cellular automaton, that interest in the covered expanded beyond academia. In the 1980s, Stephen Wolfram engaged in a systematic explore of one-dimensional cellular automata, or what he calls elementary cellular automata; his research assistant Matthew Cook showed that one of these rules is Turing-complete.

The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four. They are, in order, automata in which patterns broadly stabilize into homogeneity, automata in which patterns evolve into mostlyor oscillating structures, automata in which patterns evolve in a seemingly chaotic fashion, and automata in which patterns become extremely complex and may last for a long time, withlocal structures. This last a collection of matters sharing a common attribute is thought to be computationally universal, or capable of simulating a Turing machine. Special shape of cellular automata are reversible, where only a single grouping leads directly to a subsequent one, and totalistic, in which the future improvement of individual cells only depends on the total value of a multinational of neighboring cells. Cellular automata can simulate a style of real-world systems, including biological and chemical ones.

Classification

Wolfram, in A New Kind of Science and several papers dating from the mid-1980s, defined four class into which cellular automata and several other simple computational models can be dual-lane up depending on their behavior. While earlier studies in cellular automata tended to attempt to identify type of patterns for particular rules, Wolfram's classification was the number one attempt to categorize the rules themselves. In profile of complexity the a collection of things sharing a common attribute are:

These definitions are qualitative in nature and there is some room for interpretation. According to Wolfram, "...with most any general classification scheme there are inevitably cases which get assigned to one class by one definition and another class by another definition. And so it is for with cellular automata: there are occasionally rules...that show some qualities of one class and some of another." Wolfram's classification has been empirically matched to a clustering of the compressed lengths of the outputs of cellular automata.

There take been several attempts to classify cellular automata in formally rigorous classes, inspired by the Wolfram's classification. For instance, Culik and Yu presentation three well-defined classes and a fourth one for the automata non matching all of these, which are sometimes called Culik-Yu classes; membership in these proved undecidable. Wolfram's class 2 can be partitioned into two subgroups offixed-point and oscillating periodic rules.

The opinion that there are 4 classes of dynamical system came originally from Nobel-prize winning chemist Ilya Prigogine who sent these 4 classes of thermodynamical systems - 1 systems in thermodynamic equilibrium, 2 spatially/temporally uniform systems, 3 chaotic systems, and 4 complex far-from-equilibrium systems with dissipative tables see figure 1 in Nicolis' paper Prigogine's student.

A cellular automaton is reversible if, for every current configuration of the cellular automaton, there is precisely one past configuration preimage. whether one thinks of a cellular automaton as a function mapping configurations to configurations, reversibility implies that this function is bijective. whether a cellular automaton is reversible, its time-reversed behavior can also be described as a cellular automaton; this fact is a consequence of the Curtis–Hedlund–Lyndon theorem, a topological characterization of cellular automata. For cellular automata in which not every configuration has a preimage, the configurations without preimages are called Garden of Eden patterns.

For one-dimensional cellular automata there are known algorithms for deciding whether a a body or process by which energy or a particular component enters a system. is reversible or irreversible. However, for cellular automata of two or more dimensions reversibility is undecidable; that is, there is no algorithm that takes as input an automaton advice and is guaranteed to established correctly whether the automaton is reversible. The proof by Jarkko Kari is related to the tiling problem by Wang tiles.

Reversible cellular automata are often used to simulate such(a) physical phenomena as gas and fluid dynamics, since they obey the laws of thermodynamics. such(a) cellular automata have rules specially constructed to be reversible. Such systems have been studied by Tommaso Toffoli, Norman Margolus and others. Several techniques can be used to explicitly construct reversible cellular automata with requested inverses. Two common ones are the second-order cellular automaton and the block cellular automaton, both of which involve modifying the definition of a cellular automaton in some way. Although such automata do not strictly satisfy the definition assumption above, it can be filed that they can be emulated by conventional cellular automata with sufficiently large neighborhoods and numbers of states, and can therefore be considered a subset of conventional cellular automata. Conversely, it has been shown that every reversible cellular automaton can be emulated by a block cellular automaton.

A special class of cellular automata are totalistic cellular automata. The state of regarded and identified separately. cell in a totalistic cellular automaton is represented by a number commonly an integer value drawn from a finite set, and the value of a cell at time t depends only on the sum of the values of the cells in its neighborhood possibly including the cell itself at time t − 1. If the state of the cell at time t depends on both its own state and the a object that is said of its neighbors at time t − 1 then the cellular automaton is properly called outer totalistic. Conway's Game of Life is an example of an outer totalistic cellular automaton with cell values 0 and 1; outer totalistic cellular automata with the same Moore neighborhood structure as Life are sometimes called life-like cellular automata.

There are numerous possible generalizations of the cellular automaton concept.

One way is by using something other than a rectangular cubic, etc. grid. For example, if a plane is tiled with regular hexagons, those hexagons could be used as cells. In many cases the resulting cellular automata are equivalent to those with rectangular grids with specially intentional neighborhoods and rules. Another variation would be to make the grid itself irregular, such as with Penrose tiles.

Also, rules can be probabilistic rather than deterministic. Such cellular automata are called probabilistic cellular automata. A probabilistic rule gives, for each sample at time t, the probabilities that the central cell will transition to each possible state at time t + 1. Sometimes a simpler rule is used; for example: "The rule is the Game of Life, but on each time step there is a 0.001% probability that each cell will transition to the opposite color."

The neighborhood or rules could change over time or space. For example, initially the new state of a cell could be determined by the horizontally adjacent cells, but for the next generation the vertical cells would be used.

In cellular automata, the new state of a cell is not affected by the new state of other cells. This could be changed so that, for instance, a 2 by 2 block of cells can be determined by itself and the cells adjacent to itself.

There are continuous automata. These are like totalistic cellular automata, but instead of the rule and states being discrete e.g. a table, using states {0,1,2}, continuous functions are used, and the states become continuous ordinarily values in [0,1]. The state of a location is a finite number of real numbers.cellular automata can yield diffusion in liquid patterns in this way.

[1] considers continuous spatial automata as a framework of computation.

There are known examples of continuous spatial automata, which exhibit propagating phenomena analogous to gliders in the Game of Life.

Graph rewriting automata are extensions of cellular automata based on graph rewriting systems.