Block cellular automaton

block cellular automatablockMargolus
The ease of designing reversible block cellular automata, and of testing block cellular automata for reversibility, is in strong contrast to cellular automata with other non-block neighborhood structures, for which it is undecidable whether the automaton is reversible and for which the reverse dynamics may require much larger neighborhoods than the forward dynamics.

Tommaso Toffoli

. * Cellular Automata Machines: A New Environment for Modeling, MIT Press (1987), with Norman Margolus. ISBN: 0-262-20060-0. *CAM-6 Homepage. List of published articles.

Norman Margolus

Margolus (born 1955) is a Canadian-American physicist and computer scientist, known for his work on cellular automata and reversible computing. He is a research affiliate with the Computer Science and Artificial Intelligence Laboratory at the Massachusetts Institute of Technology. Margolus was one of the organizers of a seminal research meeting on the connections between physics and computation theory, held on Mosquito Island in 1982. He is known for inventing the block cellular automaton and the Margolus neighborhood for block cellular automata, which he used to develop cellular automaton simulations of billiard-ball computers.

Stochastic cellular automaton

probabilistic cellular automataStochastic cellular automataProbabilistic cellular automaton
Stochastic cellular automata or 'probabilistic cellular automata' (PCA) or 'random cellular automata' or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete. The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously. Stochastic Cellular Automata are CA whose updating rule is a stochastic one, which means the new entities' states are chosen according to some probability distributions. It is a discrete-time random dynamical system.

Jarkko Kari

Kari, Jarkko
Kari has also used the Wang tiling problem as the basis of proofs that several algorithmic problems in the theory of cellular automata are undecidable. In particular, in his thesis research, he showed that it is undecidable to determine whether a given cellular automaton rule in two or more dimensions is reversible. For one-dimensional cellular automata, reversibility is known to be decidable, and Kari has provided tight bounds on the size of the neighborhood needed to simulate the reverse dynamics of reversible one-dimensional automata. * Jarkko Kari's personal homepage

Continuous spatial automaton

Continuous spatial automata
Continuous spatial automata, unlike cellular automata, have a continuum of locations, while the state of a location still is any of a finite number of real numbers. Time can also be continuous, and in this case the state evolves according to differential equations. One important example is reaction-diffusion textures, differential equations proposed by Alan Turing to explain how chemical reactions could create the stripes on zebras and spots on leopards. When these are approximated by CA, such CAs often yield similar patterns. Another important example is neural fields, continuum limit neural networks where average firing rates evolve based on integro-differential equations.

Continuous automaton

continuous automata
A continuous automaton can be described as a cellular automaton extended so the valid states a cell can take are not just discrete (for example, the states consist of integers between 0 and 3), but continuous, for example, the real number range [0,1]. The cells however remain discretely separated from each other. One example is called computational verb cellular network (CVCN) ., of which the states of cells are in the region of [0,1]. Such automata can be used to model certain physical reactions more closely, such as diffusion. One such diffusion model could conceivably consist of a transition function based on the average values of the neighbourhood of the cell.

Gustav A. Hedlund

Gustav HedlundG. A. HedlundHedlund, Gustav A.
The Curtis–Hedlund–Lyndon theorem, a topological characterization of cellular automata, is named after Hedlund. Hedlund first published this theorem in 1969, crediting Morton L. Curtis and Roger Lyndon as co-discoverers. Hedlund was the co-author of the book Topological Dynamics (with Walter Gottschalk, American Mathematical Society, 1955). Hedlund was elected to Sigma Xi in 1943. In 1972, a conference on topological dynamics was held to honor Hedlund on the occasion of his retirement from Yale. The editor of the festschrift from the conference, Anatole Beck, wrote that it was "our token of respect to the man who did so much to foster and build this field". *

Rule 110

Rule 110 cellular automaton
Rule 110 in Wolfram's atlas of cellular automata. Rule 110 repository.

Konrad Zuse

ZuseZuse, KonradZuse KG
In 1967, Zuse also suggested that the universe itself is running on a cellular automaton or similar computational structure (digital physics); in 1969, he published the book Rechnender Raum (translated into English as Calculating Space). This idea has attracted a lot of attention, since there is no physical evidence against Zuse's thesis. Edward Fredkin (1980s), Jürgen Schmidhuber (1990s), and others have expanded on it. Zuse received several awards for his work: The Zuse Institute Berlin is named in his honour.

Golly (program)

Golly is a tool for the simulation of cellular automata. It is free open-source software written by Andrew Trevorrow and Tomas Rokicki; it can be scripted using Lua or Python. It includes a hashlife algorithm that can simulate the behavior of very large structured or repetitive patterns such as Paul Rendell's Life universal Turing machine, and that is fast enough to simulate some patterns for 2 32 or more time units. It also includes a large library of predefined patterns in Conway's Game of Life and other rules. *


There are many algorithms (based on arithmetics or cellular automaton) for generating pseudorandom numbers. The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are often quicker than getting "true" randomness from the environment. Aleatory. Chaitin's constant. Chance (disambiguation). Frequency probability. Indeterminism. Nonlinear system. Probability interpretations. Probability theory. Pseudorandomness.—generates random numbers using atmospheric noises. Sortition. Randomness by Deborah J. Bennett. Harvard University Press, 1998. ISBN: 0-674-10745-4. Random Measures, 4th ed. by Olav Kallenberg.

Edge of chaos

adaptation to the edge of chaosborder of chaosintermediate between "order" and "chaos
The phrase originally refers to an area in the range of a variable, λ (lambda), which was varied while examining the behavior of a cellular automaton (CA). As λ varied, the behavior of the CA went through a phase transition of behaviors. Langton found a small area conducive to produce CAs capable of universal computation. At around the same time physicist James P. Crutchfield and others used the phrase onset of chaos to describe more or less the same concept.

Majority problem (cellular automaton)

majority problemmajority cellular automatonrecognizing majorities
The majority problem, or density classification task is the problem of finding one-dimensional cellular automaton rules that accurately perform majority voting. Using local transition rules, cells cannot know the total count of all the ones in system. In order to count the number of ones (or, by symmetry, the number of zeros), the system requires a logarithmic number of bits in the total size of the system. It also requires the system send messages over a distance linear in the size of the system and for the system to recognize a non-regular language. Thus, this problem is an important test case in measuring the computational power of cellular automaton systems.

Wang tile

Domino ProblemWang tilesWang domino
Wang tiles have also been used in cellular automata theory decidability proofs. The short story Wang's Carpets, later expanded to the novel Diaspora, by Greg Egan, postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules. *. Edge-matching puzzle. Eternity II puzzle. Percy Alexander MacMahon. TetraVex. Steven Dutch's page including many pictures of aperiodic tilings. Animated demonstration of a naïve Wang tiling method - requires Javascript and HTML5.


Patterson (disambiguation)
Paterson (disambiguation). Pattersonville (disambiguation).

Discrete mathematics

discretediscrete mathdiscrete structure
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers.

Computer science

computer scientistcomputer sciencescomputer scientists
Computer science deals with the theoretical foundations of computation and practical techniques for their application.


Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). It has no generally accepted definition.


Physics (from, from φύσις phýsis 'nature') is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Paterson (surname)

Mike Paterson, Professor of Computer Science at the University of Warwick. Neil Paterson (disambiguation), multiple people. Owen Paterson, British politician. Owen Paterson, Australian production designer on The Matrix Series. Rex Paterson, British agriculturalist. Robert Paterson (disambiguation), multiple people. Stan Paterson (1924–2013), British glaciologist. Tania Paterson (born 1972), retired competitive diver from New Zealand. Tim Paterson, American computer programmer, author of MS-DOS. Tom Paterson, Scottish comics artist. Viola Paterson, Scottish artist. William Paterson (disambiguation), multiple people. Paterson (given name). Patterson (surname).


microstructuralMicrostructural Characterisationmicrostructures
Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers, ceramics or composites) can strongly influence physical properties such as strength, toughness, ductility, hardness, corrosion resistance, high/low temperature behaviour or wear resistance. These properties in turn govern the application of these materials in industrial practice.

State (computer science)

In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system.

Paterson (given name)

Paterson is a Scottish given name meaning "son of Patrick". It is more commonly used as a surname. People with the given name Paterson include: * Paterson (surname) Paterson Ewen (1925 - 2002), Canadian painter. Paterson Joseph (born 1964), British actor. T. Paterson Ross (d. 1957), American, San Francisco Bay Area architect.

Los Alamos National Laboratory

Los AlamosLos Alamos Scientific LaboratoryLANL
Los Alamos National Laboratory (Los Alamos or LANL for short) is a United States Department of Energy national laboratory initially organized during World War II for the design of nuclear weapons as part of the Manhattan Project. It is located a short distance northwest of Santa Fe, New Mexico in the southwestern US.