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A cellular automaton (plural: cellular automata) is a computational model studied in mathematics, physics, computability theory and theoretical biology. It consists of a regular grid of cells (e.g. a checkerboard), everyone of which is assigned an integer number from a finite set. This simple system evolves during time according to a local rule which assigns synchronously (i.e at the same time) a new number to every cell depending on the numbers of cells in a neighborhood around it.

Overview

One way to simulate a two-dimensional cellular automaton is with an infinite sheet of graph paper along with a set of rules for the cells to follow. Each square is called a "cell" and each cell has two possible states, black and white. The "neighbors" of a cell are the 8 squares touching it. For such a cell and its neighbors, there are 512 (= 2⁹) possible patterns. For each of the 512 possible patterns, the rule table would state whether the center cell will be black or white on the next time interval. Conway's Game of Life is a popular version of this model.

It is usually assumed that every cell in the universe starts in the same state, except for a finite number of cells in other states, often called a configuration. More generally, it is sometimes assumed that the universe starts out covered with a periodic pattern, and only a finite number of cells violate that pattern. The latter assumption is common in one-dimensional cellular automata.

A torus, a toroidal shape.

Cellular automata are often simulated on a finite grid rather than an infinite one. In two dimensions, the universe would be a rectangle instead of an infinite plane. The obvious problem with finite grids is how to handle the cells on the edges. How they are handled will affect the values of all the cells in the grid. One possible method is to allow the values in those cells to remain constant. Another method is to define neighbourhoods differently for these cells. One could say that they have fewer neighbours, but then one would also have to define new rules for the cells located on the edges. These cells are usually handled with a toroidal arrangement: when one goes off the top, one comes in at the corresponding position on the bottom, and when one goes off the left, one comes in on the right. (This essentially simulates an infinite periodic tiling, and in the field of Partial Differential Equations is sometimes referred to as periodic boundary conditions.) This can be visualized as taping the left and right edges of the rectangle to form a tube, then taping the top and bottom edges of the tube to form a torus (doughnut shape). Universes of other dimensions are handled similarly. This is done in order to solve boundary problems with neighborhoods. For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell x_i^t—where t is the time step (vertical), and i is the index (horizontal) in one generation—is {x_i−1^t−1, x_i^t−1, x_i+1^t−1}. There will obviously be problems when a neighbourhood on a left border references its upper left cell, which is not in the cellular space, as part of its neighborhood.

The Gnostics in the Early Christian Era

See also: Samuel Angus

In the formation of early Christianity, various sectarian groups, labeled "gnostics" by their opponents, emphasised spiritual knowledge (gnosis) over faith (pistis) in the teachings and traditions of the established community of Christians. These sectarians considered the most essential part of the process of salvation to be this personal knowledge, in contrast to faith as an outlook in their world view along with faith in the ecclesiastical authority. These break away groups were branded minuth by Hebrews (see the Notzrim) and heretics by the Fathers of the early church due to teaching this type of authority rejection referred to as antinomianism. The knowledge of these sectarian groups is contested by orthodox Christian theology as speculative knowledge derived from religio-philosophical systems rather than knowledge derived from revelation coming from faith. Gnosis itself is and was obtained through understanding, arrived at via inner experience or contemplation such as an internal epiphany for example. Their systems of thought were pagan (folk) in origin and syncretic in nature. The gnostic sectarians vilified the concepts of an subjective creator God (Plato's demiurge) and objective creator God (one that creates ex-nihilo) as in the Judeo-Christian God (creator) and sought to reconcile the individual to their own personal deification (henosis), making of each individual God. As such the gnostic sects made a duality out of the difference between the activities of the spirit (nous), called noesis (insight), and those of faith.

During the early formation of Christianity, church authorities (Fathers of the Church) exerted considerable amounts of energy attempting to weed out what were considered to be false doctrines (e.g. Irenaeus' On the Detection and Overthrow of False Gnosis). The gnostics (as one sectarian group) held views which were incompatible with the emerging Ante-Nicene community. Among Christian heresiologists, the concept of false gnosis was used to denote different Pagan, Jewish or Christian belief systems (e.g. the Eleusinian Mysteries or Glycon) and their various teachings of what was deemed religio-philosophical systems of knowledge, as opposed to authentic gnosis (see below, Gnosis among the Greek Fathers). The sectarians used gnosis or secret knowledge to reject the traditions of the established community or church. The authorities throughout the community criticized this antinomianism as inconsistent with the communities teachings. Hence sectarians and followers of gnosticism were first rejected by the Jewish communities of the Mediterranean (see the Notzrim 139–67 BCE), then by the Christian communities and finally by the late Hellenistic philosophical communities (see Neoplatonism and Gnosticism).