Orbits of Bernoulli Measures in Cellular Automata

Abstract

Glossary Configuration space Set of all bisequences of symbols from the alphabet A of N symbols, A = {0, 1, . . . , N − 1}, denoted by A Z . Elements of A Z are called configurations and denoted by bold lowercase letters: x, y, etc. Block or word A finite sequence of symbols of the alphabet A . Set of all blocks of length n is denoted by A n , while the sent of all possible blocks of all lengths by A . Blocks are denoted by bold lowercase letters a, b, c, etc. Individual symbols of the block b are denoted by indexed italic form of the same letter, b = b 1 , b 2 , . . . , b n . To make formulae more compact, commas are sometimes dropped (if no confusion arises), and we simply write b = b 1 b 2 . . . b n . Cylinder set For a block b of length n, the cylinder set generated by b and anchored at i is the subset of configurations such that symbols at positions from i to i + n − 1 are fixed and equal to symbols in the block b, while the remaining symbols are arbitrary. Denoted by [b] i = {x ∈ A Z : x [i,i+n) = b}. Cellular automaton In this article, cellular automaton is understood as a map F in the space of shift-invariant probability measures over the configuration space A Z . To define F, two ingredients are needed, a positive integer r called radius and a function w : A × A 2r+1 → [0, 1], whose values are called transition probabilities. The image of a measure µ under the action of F is then defined by probabilities of cylinder sets, (F µ)([a] i ) = ∑ b∈A |a|+2r w(a|b)µ([b] i−r

Evaluating the Quality of Local Structure Approximation Using Elementary Rule 14

“…Note that although the above proposition provides probabilities of P n (0), P n (00), P n (000) and P n (010) only, the remaining probabilities of blocks of length up to 3 can be easily computed using eqs. (2).…”

“…In recent years, however, partial orbits of Bernoulli measures have been computed for some selected elementary CA [2], making a somewhat more rigorous approach possible. The goal of this paper is to provide an example of a CA rule for which some block probabilities are known exactly, and for which local structure equations can be analyzed rigorously, without relying exclusively on numerical iterations.…”

A decomposition theorem for number-conserving multi-state cellular automata on triangular grids

“…In order to accommodate for conservation laws -one of the most important concepts in physics -usually number-conserving CAs (NCCAs) are used, for which the sum of the states of all the cells is preserved at every update. Unfortunately, although there is an extensive literature on one-dimensional NCCAs (see, for example, [4,5,12,13,16,21,22,26]), the two-dimensional ones have not been studied in a satisfactory manner and most of the results are not applicable globally. For example, thanks to existing necessary and sufficient conditions it is possible (at least theoretically) to check whether a given CA is number conserving or not [9], or it is possible to design a number-conserving CA [28], but usually enumeration of all number-conserving CAs with a given state set is not feasible.…”

An Example of Computation of the Density of Ones in Probabilistic Cellular Automata by Direct Recursion

“…This approach has been successfully used by the author for a number of deterministic CA rules, such as elementary rules 172, 142, 130 (references [4], [3], and [6] respectively), and several others. It has also been used for a special class of PCA known as single-transition α-asynchronous rules [7].…”