Welch sets for random generation and representation of reversible one-dimensional cellular automata

doi:10.1016/j.ins.2016.12.009

Information Sciences

Volumes 382–383, March 2017, Pages 81-95

https://doi.org/10.1016/j.ins.2016.12.009 Get rights and content

Abstract

Reversible one-dimensional cellular automata are studied from the perspective of Welch Sets. This paper presents an algorithm to generate random Welch sets that define a reversible cellular automaton. Then, properties of Welch sets are used in order to establish two bipartite graphs describing the evolution rule of reversible cellular automata. The first graph gives an alternative representation for the dynamics of these systems as block mappings and shifts. The second graph offers a compact representation for the evolution rule of reversible cellular automata. Both graphs and their matrix representations are illustrated by the generation of random reversible cellular automata with 6 and 18 states.

Introduction

Cellular automata are discrete dynamical systems able to generate complex behavior based on simple local interaction among their components. Reversible cellular automata are characterized to produce an invertible global behavior provoked by a set of local mappings which are not reversible.

In the one-dimensional case, reversible cellular automata have been widely investigated in order to understand their topological, combinatorial and dynamical properties.

Due to their property of keeping the original information of the system, concepts and alternative definitions of reversible cellular automata have been employed in the specification of different cyphering systems [2], [8], [9], [11], [19], [20], [22], and error-correcting codes [13].

Recent works have presented different procedures to characterize reversible behavior in cellular automata. For instance: the generalization of reversibility defined by linear rules over the binary field case under null boundary conditions is explained in [21]. Reversibility of elementary cellular automata rule number 150 is described by circulant matrices in [14]. Reversibility in cellular automata with memory has been studied in [1] and [17].

Thus, there is a continuous investigation to characterize the properties of reversible cellular automata, both for theoretical research and practical application.

The first topological study of reversible cellular automata is developed by Hedlund in [10] establishing important concepts for the one-dimensional case such as the uniform multiplicity of ancestors and Welch sets. Several investigations followed this paper using Welch sets to define new graphs representing the local behavior inducing global reversibility [15], and using their vector representation to give a maximum bound for the minimum information needed to obtain global invertibility [7]. Another topic of interest has been to enumerate all the possible cases of reversible one-dimensional cellular automata [3], [4], [18], the last two references taking reversible automata as groupoids with algebraic properties. In this sense, references [6] and [5] establish the properties of these groupoids and their relation with isotopy and Cayley graphs.

Based on this previous work, the aim of this paper is to present an algorithm to create random Welch sets to define reversible one-dimensional cellular automata. From these Welch sets, we define two bipartite graphs, both for representing the forward and reverse local mappings of the associated reversible cellular automaton. We give examples showing that the former graph provides an alternative representation for the dynamics of reversible automata by block mappings and shifts firstly demonstrated by Kari in [12], and the latter needs less space than the original evolution rule.

The paper is organized as follows: Section 2 explains the basic concepts of reversible one-dimensional cellular automata and Welch sets. Section 3 defines the properties of Welch Sets for reversible automata with neighborhood size 2. Section 4 presents the algorithm generating random Welch sets and arbitrary reversible cellular automata. Section 5 exposes the construction of a bipartite graph based on Welch elements to calculate the evolution of reversible cellular automata and represent their dynamics by block mappings and shifts. Section 6 specifies a shorter version of this graph in order to represent the evolution of reversible automata with a reduced number of elements. The last section gives the final remarks of the study.

Section snippets

Basic concepts of Welch sets

One-dimensional cellular automata are defined by a discrete set of states S, a neighborhood radius $r \in N,$ an initial configuration $c_{0} : {1, 2, \dots, m} \to S$ for some $m \in N,$ and an evolution rule $φ : S^{2 r + 1} \to S$ . The dynamics is given applying the evolution rule over every cell in c₀, thus $c_{1} (i) = φ (c_{0} (i - r) \dots c_{0} (i + r))$ taking periodic boundary conditions. In general $c_{t + 1} (i) = φ (c_{t} (i - r) \dots c_{t} (i + r))$ .

For simplicity we shall represent $φ (c (i - r) \dots c (i + r))$ just as φ(c(i)). Thus, for a string v ∈ S^m with $m \geq 2 r + 1,$ we define φ(v) as

Properties of Welch sets

The previous properties give a particular structure for M_φ and $M_{φ^{- 1}}$ . Property 1 guarantees that there are |S| entries equal to s in both matrices for every s ∈ S. Property 2 establishes that if $M (s_{1}, s_{2}) = M (s_{3}, s_{4})$ then $M (s_{1}, s_{4}) = M (s_{3}, s_{2})$ for s_i ∈ S. Finally, Property 3 specifies that for every state s ∈ S there is a unique state a ∈ S such that $M_{φ} (a, a) = s$ . Thus, in M_φ and $M_{φ^{- 1}}$ every state has a rectangular arrangement, and both matrices define a rectangular groupoid [5].

For every reversible

Algorithm to generate random Welch sets

For U⊆S, we can represent the elements of U by a binary vector g_U: S → 0, 1 of size 1 × |S| such that $g_{U} (a) = 1$ iff a ∈ U; otherwise, $g_{U} (a) = 0$ . With this, we define two mappings $α (g_{U}) = U$ and $β (U) = g_{U}$ .

The next algorithm computes valid Welch sets for every state s ∈ S to fill up the matrix M_φ representing the evolution rule of a reversible cellular automaton. The algorithm applies the properties of Welch sets as vector and matrix operations in order to facilitate their computational implementation.

First representation: bipartite graph with Welch elements

For s ∈ S and its left Welch set L_s, a left Welch element is a pair (a, s) such that a ∈ L_s. Analogously, a right Welch element is a pair (s, a) such that a ∈ R_s.

Given the matrices G_L, G_R and M_φ, we can define a bipartite graph $P = N_{L} \times N_{R}$ where N_L is the set of left Welch elements and N_R is the set of right Welch elements for every state in S.

Every left Welch element (a, b) ∈ N_L holds that a ∈ L_b for a, b ∈ S. Analogously, every right Welch element (d, e) ∈ N_R holds that e ∈ R_d for d, e ∈ S. There

Second representation: Bipartite graph with welch sets

With the matrices G_L, G_R and M_φ, another bipartite graph $Q = L \times R$ is defined where $L$ is the family of left Welch sets and $R$ is the family of right Welch sets. For every s ∈ S, each set $L_{s} \in L$ and every $R_{s} \in R$ . There is a directed edge (L_s, R_s) labeled by s. The directed edges from $R$ into $L$ are implicitly defined by Property 3. That is, $(R_{a}, L_{b}) = R_{a} \cap L_{b}$ for each $R_{a} \in R$ and every $L_{b} \in L$ .

Fig. 9 presents de digraphs Q associated with the reversible automata previously used in Fig. 4. Bold lines describe the

Final remarks

This paper has presented an algorithm based on the properties of Welch indices in order to calculate a random reversible cellular automaton. This algorithm is able to calculate reversible automata with tens of states.

The properties of Welch indices are employed as well to specify two types of bipartite graphs, which represent the dynamics of reversible automata.

The first graph takes as nodes the Welch elements, and the edges among them are utilized to represent the evolution in both directions

Acknowledgment

This work has been supported by CONACYT project No. CB-2014-237323 and by IPN Collaboration Network “ Grupo de Sistemas Complejos del IPN ”.

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^☆: Dedicated to the memory of Harold V. McIntosh (1929–2015), a leader in the development of cellular automata theory, whose teaching and knowledge were essential to this work.

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Welch sets for random generation and representation of reversible one-dimensional cellular automata☆

Abstract

Introduction

Section snippets

Basic concepts of Welch sets

Properties of Welch sets

Algorithm to generate random Welch sets

First representation: bipartite graph with Welch elements

Second representation: Bipartite graph with welch sets

Final remarks

Acknowledgment

Opt. Lasers Eng.

Theor. Comput. Sci.

Discrete Math.

Commun. Nonlin. Sci. Numer. Simul.

Opt. Laser Technol.

Theor. Comput. Sci.

Inf. Sci.

Physica D

Commun. Nonlin. Sci. Numer. Simul.

Inf. Sci.

Reversible cellular automata with memory of delay type

Complexity