Von Neumann cellular automata are the original expression of cellular automata, the development of which were prompted by suggestions made to John von Neumann, by his close friend and mathematician Stanisław Ulam. These cellular automata are a subclass of cellular automata. The purpose for which von Neumann developed cellular automata was to provide insight into the logical requirements for machine self-replication. John von Neumann in the 1940s. ... StanisÅ‚aw Ulam in the 1950s. ... A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. ...

In general, cellular automata constitute an arrangement of finite state automata (FSA) that sit in positional relationships between one-another, each FSA exchanging information with those other FSAs to which it is positionally adjacent. The positional relationship of von Neumann cellular automata is rectilinear in two dimensions; i.e. a two-dimensional grid, as formed by the intersection of two sets of mutually perpendicular lines, producing cells within each of which resides an FSA. The set of FSAs define a cell space, this space being of infinite size. All FSAs are identical in terms of state-transition function, or rule-set. NB - the neighborhood (a grouping function) is part of the state-transition function, and defines for any cell, the set of other cells upon which the state of that cell depends. All cells transition state synchronously.

Each FSA of the von Neumann cell space can accept any of the 29 states of the rule-set. The rule-set includes 29 different states, these being grouped into five orthogonal subsets. They are

i) a ground state; ii) the transition states; iii) the confluent states; iv) the ordinary transmission state; and v) the special transmission states.

Elements of the last three subsets include an activity property. Elements of the last two subsets include a property of direction, as well. Activity represents data carriage, with data being communicated between states at the rate of one bit per state transition step. Confluent states also have the property of a one-cycle delay, thus holding two bits of data at any given time. The flow of bits between cells is indicated by the direction property. The two subsets of transmission states, ordinary and special, are mutually antagonistic, with mutually directed active cells of each causing mutual annihilation, yielding the ground state.

Data passes between ordinary transmission states, according to the direction property. The same is true between special transmission states. Confluent states do not pass data between each other. Confluent states take input from one or more ordinary transmission states, and deliver output to transmission states, ordinary and special, that are not directed toward the confluent state. Data are not transmitted against the transmission state direction property. Data held by a confluent state is lost if that state has no adjacent transmission state that is also not pointed at the confluent state. Transmission state apply the OR operator to inputs, while the confluent state applies the AND operator to inputs.

See also

University of Leeds Artificial Life Overview