Stanislaw Ulam

Stanisław UlamStan UlamUlam
He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures. Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months.

Asynchronous cellular automaton

Nehaniv in 1998) allows one to emulate exactly the behaviour of a synchronous cellular automaton via an asynchronous one constructed as a simple modification of the synchronous cellular automaton (Nehaniv 2002). Correctness of this method however has only more recently been rigorously proved (Nehaniv, 2004). As a consequence, it follows immediately from results on synchronous cellular automata that asynchronous cellular automata are capable of emulating, e.g., Conway's Game of Life, of universal computation, and of self-replication (e.g., as in a Von Neumann universal constructor).

Conway's Game of Life

Game of LifeConway's LifeConway’s Game of Life
Catagolue, an online database of objects in Conway's Game of Life and similar cellular automata. Cellular Automata FAQ – Conway's Game of Life.

John von Neumann

von NeumannJ. von NeumannNeumann, John von
The result was a universal copier and constructor working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only orthogonal cells), and with 29 states per cell. Von Neumann gave an existence proof that a particular pattern would make infinite copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so. Von Neumann addressed the evolutionary growth of complexity amongst his self-replicating machines.

A New Kind of Science

Principle of Computational EquivalencedisagreeNKS
Edward Fredkin and Konrad Zuse pioneered the idea of a computable universe, the former by writing a line in his book on how the world might be like a cellular automaton, and later further developed by Fredkin using a toy model called Salt. It has been claimed that NKS tries to take these ideas as its own, but Wolfram's model of the universe is a rewriting network, and not a cellular automaton, as Wolfram himself has suggested a cellular automaton cannot account for relativistic features such as no absolute time frame.

Turing completeness

Turing-completeTuring completeuniversal
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing complete or computationally universal if it can be used to simulate any Turing machine. This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing complete. The concept is named after English mathematician and computer scientist Alan Turing.

Complex system

complex systemscomplexity theorycomplexity science
One example of a complex system whose emergent properties have been studied extensively is cellular automata. In a cellular automaton, a grid of cells, each having one of the finitely many states, evolves according to a simple set of rules. These rules guide the "interactions" of each cell with its neighbors. Although the rules are only defined locally, they have been shown capable of producing globally interesting behavior, for example in Conway's Game of Life. When emergence describes the appearance of unplanned order, it is spontaneous order (in the social sciences) or self-organization (in physical sciences).

Matthew Cook

Cook, Matthew
Among other things, he developed a proof showing that the Rule 110 cellular automaton is Turing-complete. Cook presented his proof at the Santa Fe Institute conference CA98 before the publishing of Wolfram's book—an action that led Wolfram Research to accuse Cook of violating his NDA and resulted in the blocking of the publication of the proof in the conference proceedings. A New Kind of Science was released in 2002 with an outline of the proof. In 2004, Cook published his proof in Wolfram's journal Complex Systems. Personal web site. Site at INI Zurich.

Von Neumann neighborhood

orthogonally adjacentVon Neumann neighbourhoodneighborhood
In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells. The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it. It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the Moore neighborhood. This neighbourhood can be used to define the notion of 4-connected pixels in computer graphics. The von Neumann neighbourhood of a cell is the cell itself and the cells at a Manhattan distance of 1.

Moore neighborhood

neighbours8 connectivityinfluence its nearest neighbours
In cellular automata, the Moore neighborhood is defined on a two-dimensional square lattice and is composed of a central cell and the eight cells that surround it. The neighborhood is named after Edward F. Moore, a pioneer of cellular automata theory. It is one of the two most commonly used neighborhood types, the other one being the von Neumann neighborhood. The well known Conway's Game of Life, for example, uses the Moore neighborhood. It is similar to the notion of 8-connected pixels in computer graphics. The Moore neighbourhood of a cell is the cell itself and the cells at a Chebyshev distance of 1.

Mathematical and theoretical biology

mathematical biologytheoretical biologybiomathematics
Logical deterministic cellular automata – discrete time, discrete state space. See also: Cellular automaton. Non-Markovian processes – generalized master equation – continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur. Jump Markov process – master equation – continuous time with no memory of past events, discrete state space, waiting times between events discretely occur and are exponentially distributed. See also: Monte Carlo method for numerical simulation methods, specifically dynamic Monte Carlo method and Gillespie algorithm.

Elementary cellular automaton

elementary cellular automata
"Elementary Cellular Automata" at the Wolfram Atlas of Simple Programs. 32 bytes long MS-DOS executable drawing by cellular automaton (Rule 110 by default). A showcase of all the rules picked at random.

Coupled map lattice

Coupled Map Latticeslattice dynamical
Cellular automata. Lyapunov exponent. Stochastic cellular automata. Introduction to Chaos and Nonlinear Dynamics. Introduction to Chaos and Nonlinear Dynamics. Introduction to Chaos and Nonlinear Dynamics. Introduction to Chaos and Nonlinear Dynamics. Kaneko Laboratory. Institut Henri Poincaré, Paris, June 21 – July 2, 2004. Istituto dei Sistemi Complessi, Florence, Italy. Java CML/GML web-app. AnT 4.669 – A simulation and Analysis Tool for Dynamical Systems.

Von Neumann cellular automaton

von Neumann cellular automatavon Neumann's CAvon Neumann's cellular automata
Von Neumann cellular automata are the original expression of cellular automata, the development of which was prompted by suggestions made to John von Neumann by his close friend and fellow mathematician Stanislaw Ulam. Their original purpose was to provide insight into the logical requirements for machine self-replication, and they were used in von Neumann's universal constructor. Nobili's cellular automaton is a variation of von Neumann's cellular automaton, augmented with the ability for confluent cells to cross signals and store information. The former requires an extra three states, hence Nobili's cellular automaton has 32 states, rather than 29.

Von Neumann universal constructor

universal constructorself-replicating machinesconstruction
Codd's cellular automaton. Langton's loops. Nobili cellular automata. Quine, a program that produces itself as output. Santa Claus machine. Wireworld. Golly - the Cellular Automata Simulation Accelerator Very fast implementation of state transition and support for JvN, GoL, Wolfram, and other systems. von Neumann's Self-Reproducing Universal Constructor The original Nobili-Pesavento source code, animations and Golly files of the replicators. John von Neumann's 29 state Cellular Automata Implemented in OpenLaszlo by Don Hopkins. A Catalogue of Self-Replicating Cellular Automata. This catalogue complements the Proc. Automata 2008 volume.

Curtis–Hedlund–Lyndon theorem

translation-invariant and continuous
The counterexample given above for a continuous and shift-equivariant map which is not a classical cellular automaton, is an example of a generalized cellular automaton. When the alphabet is finite, the definition of generalized cellular automata coincides with the classical definition of cellular automata due to the compactness of the shift space. Generalized cellular automata were proposed by where it was proved they correspond to continuous shift-equivariant maps. * Surjunctive group

Stephen Wolfram

WolframS. WolframStephen
Cellular Automata and Complexity: Collected Papers (1994). Theory and Applications of Cellular Automata (1986). Wolfram Foundation. Wolfram Foundation.

Greenberg–Hastings cellular automaton

Greenberg-Hastings cellular automaton
The Greenberg–Hastings Cellular Automaton (abbrev. GH model) is a three state two dimensional cellular automaton (abbrev CA) named after James M. Greenberg and Stuart Hastings, designed to model excitable media, One advantage of a CA model is ease of computation. The model can be understood quite well using simple "hand" calculations, not involving a computer. Another advantage is that, at least in this case, one can prove a theorem characterizing those initial conditions which lead to repetitive behavior. As in a typical two dimensional cellular automaton, consider a rectangular grid, or checkerboard pattern, of "cells". It can be finite or infinite in extent.

John Horton Conway

John H. ConwayJohn ConwayConway
Nevertheless, the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with that of mathematics popularizer and Scientific American columnist Martin Gardner. When Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work.

Alvy Ray Smith

Dr. Alvy Ray Smith
In 1970, he received a Ph.D. in computer science from Stanford University, with a dissertation on cellular automata theory jointly supervised by Michael A. Arbib, Edward J. McCluskey, and Bernard Widrow. His first art show was at the Stanford Coffeehouse. From 1969 to 1973 he was an associate professor of Electrical Engineering and Computer Science at New York University, under chairman Herbert Freeman, one of the earliest computer graphics researchers. He taught briefly at the University of California, Berkeley in 1974. While at Xerox PARC in 1974, Smith worked with Richard Shoup on SuperPaint, one of the first computer raster graphics editor, or 'paint', programs.

Garden of Eden (cellular automaton)

Garden of EdenGarden of Eden theoremGarden of Eden pattern
There exist hyperbolic cellular automata that have twins but that do not have a Garden of Eden, and other hyperbolic cellular automata that have a Garden of Eden but do not have twins; these automata can be defined, for instance, in a rotation-invariant way on the uniform hyperbolic tilings in which three heptagons meet at each vertex, or in which four pentagons meet at each vertex. However, the Garden of Eden theorem can be generalized beyond Euclidean spaces, to cellular automata defined on the elements of an amenable group. A weaker form of the Garden of Eden theorem asserts that every injective cellular automaton is surjective.

Rule 30

The design was described by its architect as inspired by Conway's Game of Life, a different cellular automaton studied by Cambridge mathematician John Horton Conway, but is not actually based on Life. * Wolfram, Stephen, 1985, Cryptography with Cellular Automata, CRYPTO'85. * TED Talk from February 2010. Stephen Wolfram speaks about computing a theory of everything where he talks about rule 30 among other things. Rule 90. Rule 110. Rule 184. Rule 30 in Wolfram's atlas of cellular automata. Rule 30: Wolfram's Pseudo-random Bit Generator. Recipe 32 at David Griffeath's Primordial Soup Kitchen. Repeating Rule 30 patterns.

Digital physics

Pancomputationalismit from bitNaturalist computationalism
The computer could be, for example, a huge cellular automaton (Zuse 1967 ), or a universal Turing machine, as suggested by Schmidhuber (1997 ), who pointed out that there exists a short program that can compute all possible computable universes in an asymptotically optimal way. Loop quantum gravity could lend support to digital physics, in that it assumes space-time is quantized. Paola Zizzi has formulated a realization of this concept in what has come to be called "computational loop quantum gravity", or CLQG. Other theories that combine aspects of digital physics with loop quantum gravity are those of Marzuoli and Rasetti and Girelli and Livine.

Calculating Space

Rechnender Raum
Zuse proposed that the universe is being computed by some sort of cellular automaton or other discrete computing machinery, challenging the long-held view that some physical laws are continuous by nature. He focused on cellular automata as a possible substrate of the computation, and pointed out (among other things) that the classical notions of entropy and its growth do not make sense in deterministically computed universes. Bell's theorem is sometimes thought to contradict Zuse's hypothesis, but it is not applicable to deterministic universes, as Bell himself pointed out.

Lloyd A. Jeffress

JeffressJeffress, L.A.Hixon Symposium
The participants in the 1948 Hixon Symposium on neural mechanisms were noted academics from a number of disciplines: Pauling (chemistry), Heinrich Klüver (cybernetics), John von Neumann (cellular automata), Karl Lashley (behavior and learning), Ogden Lindsley (precision teaching), Rafael Lorente de Nó (neuroanatomy/neurophysiology), Warren McCulloch (neural network modeling), and W.C. Halstead (neuropsychological assessment). Many of the papers collected in Cerebral Mechanisms in Behavior: The Hixon Symposium, with Jeffress as editor, became classics cited in thousands of scientific articles.