"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)
"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)
"Cellular automata may be considered as discrete dynamical systems. In almost all cases, cellular automaton evolution is irreversible. Trajectories in the configuration space for cellular automata therefore merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto 'attractors'. These attractors typically contain only a very small fraction of possible states. Evolution to attractors from arbitrary initial states allows for 'self-organizing' behaviour, in which structure may evolve at large times from structureless initial states. The nature of the attractors determines the form and extent of such structures." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)
"The problem is that, in many cases for many systems, there may simply be no such thing as a truly independent variable. Out here in the real world, the phrase “independent” may often be oxymoronic. The very notion of independence reflects the world of Platonic idealism, a pure world populated by separate discrete things each with its essence, floating suspended in a sea of laws, rules governing the relations between these autonomous entities." (Jon Koerner, "Nontrimialtiy of Nonlinear Dynamics in Psychology of Learning", 1989)
"The essential idea of semantic networks is that the graph-theoretic structure of relations and abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks", Computers & Mathematics with Applications Vol. 23 (2-5), 1992)
"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods."
"At the other far extreme, we find many systems ordered as a patchwork of parallel operations, very much as in the neural network of a brain or in a colony of ants. Action in these systems proceeds in a messy cascade of interdependent events. Instead of the discrete ticks of cause and effect that run a clock, a thousand clock springs try to simultaneously run a parallel system. Since there is no chain of command, the particular action of any single spring diffuses into the whole, making it easier for the sum of the whole to overwhelm the parts of the whole. What emerges from the collective is not a series of critical individual actions but a multitude of simultaneous actions whose collective pattern is far more important. This is the swarm model." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)
"Systems thinking means the ability to see the synergy of the whole rather than just the separate elements of a system and to learn to reinforce or change whole system patterns. Many people have been trained to solve problems by breaking a complex system, such as an organization, into discrete parts and working to make each part perform as well as possible. However, the success of each piece does not add up to the success of the whole. to the success of the whole. In fact, sometimes changing one part to make it better actually makes the whole system function less effectively." (Richard L Daft, "The Leadership Experience", 2002)
"Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell." (Ramon Alonso-Sanz, "Cellular Automata with Memory", 2009)
"Cellular automata (CA) are idealizations of physical systems in which both space and time are assumed to be discrete and each of the interacting units can have only a finite number of discrete states." (Andreas Schadschneider et al, "Vehicular Traffic II: The Nagel–Schreckenberg Model" , 2011)
"Discrete dynamic systems that evolve in space and time. A cellular automaton is composed of a set of discrete elements - the cells - connected with other cells of the automaton, and in each time unit each cell receives information about the current state of the cells to which it is connected. The cellular automaton evolve according a transition rule that specifies the current possible states of each cell as a function of the preceding state of the cell and the states of the connected cells." (Francesc S. Beltran et al, "A Language Shift Simulation Based on Cellular Automata", 2011)
"The idea of natural computation has grown into a new scientific paradigm and has proved to be a rich source of new insights about nature. Many processes in nature exhibit key characteristics of computation, especially discrete units or steps and repetition according to a fixed set of rules. Although the processes may be highly complex, their regularity makes them highly amenable to simulation [...]." (David G Green & Tania Leishman, "Computing and Complexity: Networks, Nature and Virtual Worlds, Philosophy of Complex Systems, 2011)
"In the context of a zero-sum game, opposing tendencies are formulated in two distinct ways. First, conflicting tendencies are conceptualized as two mutually exclusive, discrete entities. The conflicts are treated as dichotomies that are usually expressed as X or NX. If X is right then NX has to be wrong. This represents an or relationship, a win/lose struggle with a moral obligation to win. The loser, usually declared wrong, is eliminated. Second, opposing tendencies are formulated in such a way that they can be represented by a continuum. Between black and white are a thousand shades of gray."
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