Procedures for calculating reversible one-dimensional cellular automata

TLDR: Two algorithms for calculating reversible one-dimensional cellular automata of neighborhood size 2 using two basic properties of reversible automata such as uniform multiplicity of ancestors and Welch indices are described.

About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 2005-03-01 and is currently open access. It has received 31 citations till now. The article focuses on the topics: Mobile automaton & Reversible cellular automaton.

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Table 2: Instances of valid connectivity matrices for reversible one-dimensional cellular automata of 4 states, neighborhood size 2 and Welch indices L = 2 and R = 2.

Table 3: Some forbidden connectivity matrices for reversible automata of 4 states, neighborhood size 2 and Welch indices L = 2 and R = 2.

Table 4: Results of the processes for calculating reversible automata.

Table 1: Reversible one-dimensional cellular automata of 3 states, neighborhood size 2, and Welch indices L = 1 and R = 3. The state-pair matrix and its transitive closure are presented, note that the only connected component is composed by {(0, 0), (1, 1), (2, 2)}.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Procedures for calculating reversible one-dimensional cellular automata" ?

In this paper, two algorithms for computing reversible one-dimensional cellular automata of neighborhood size 2 are presented. 

Q2. How long does it take to search for a state?

For 5 states the search takes thirty seconds by the algorithm using connectivity matrices and forty-five seconds by the algorithm using the state-pair matrix. 

Q3. What is the evolution rule for the state-pair matrix?

If a given connected component contains an ordered pair where both elements are different one another, then there is a sequence of states with two different cyclic ancestors and the automaton is not reversible. 

Q4. How many blocks of cells are ordered in the case of a cellular automata?

For automata of 5 states, the first ordered neighborhood is not grouped and the rest is grouped in blocks of two cells, numbering each block with a number in base 25. 

Q5. What is the procedure for checking the connected components?

The procedure for checking the connectedcomponents is O(k4) so the algorithm is polynomial with regard to the number of states, but with a very big exponent. 

Q6. How many steps to form the state-pair matrix?

The number of necessary steps to form the state-pair matrix is O(k4) and the transitive closure of each matrix is calculated by Warshall’s algorithm [13] which is O(k6). 

Q7. What is the definition of cellular automata?

Reversible cellular automata are systems where the information is conserved during its evolution, for this reason they present a very interesting mathematical theory, and have been used as models for data ciphering, information coding [8] and simulation of reversible physical phenomena [14], [16] among other applications. 

Q8. What is the meaning of cellular automata?

The concept of cellular automata began with the work of John von Neumann [15] for proving the existence of self-reproductive system. 

Q9. What is the definition of a state-pair matrix?

Given an evolution rule ϕ and two ordered pairs (i1, j1), (i2, j2) where im, jm ∈ K, m = 1, 2, the entry ((i1, j1), (i2, j2)) at the state-pair matrix is defined as follows:((i1, j1), (i2, j2)) = 1 if ϕ(i1, i2) = ϕ(j1, j2)0 in other case (1) The state-pair matrix is a 0− 1 matrix and shows what pairs of states evolve in the same state. 

Q10. What is the order of the state-pair matrix?

For constructing the state-pair matrix take all the ordered pairs of states in the automaton, these pairs will be the indices by rows and columns of the state-pair matrix and its order is k2.