A Note on Elementary Cellular Automata Classification

In his seminal book “Theory and Applications of Cellular Automata”, Wolfram had included an extended number of properties in one-dimensional CA, displayed in “Tables of Cellular Automata Properties” section [36]. Here we have reviewed and updated such a classification with the WolframAlpha engine.²²2WolframAlpha computational knowledge engine http://www.wolframalpha.com/.

Wolfram’s classification is shown in Table 2. This classification establishes that from random initial conditions a CA will reach in a long time, one of the following classes of global behaviour.

• Class I: evolve to uniform behaviour;

• Class II: evolve to periodic behaviour;

• Class III: evolve to chaotic behaviour;

• Class IV: evolve to complex behaviour.

The main interest of chaotic rules relate to developing cryptography, random number generators, and fields of attraction. However, the so called class IV or complex rules have captured most attention given their potential for computational universality, and their applications in artificial life by the simulations of particles, waves, mobile self localizations, or gliders.³³3Complex Cellular Automata Repository http://uncomp.uwe.ac.uk/genaro/Complex_CA_repository.html. Their capacity to contain intrinsically complex systems [38, 35, 16, 1, 31, 14]. This kind of discrepancy between chaotic rules, and complex rules capable of computational universality, are discussed in the CA literature (for details please see [25, 26]).

Wuensche and Lesser did a detailed analysis on basin of attraction fields and rule clusters of equivalences in ECA, published in their book “The Global Dynamics of Cellular Automata” written in 1992 [33].

In this book, Wuensche recognises three main transformations that relate equivalent ECA rules expressed in rule clusters. Such transformations are: reflection, negation, and complementation. Moreover, these equivalences can be explored quickly and automatically for other CA with free software Discrete Dynamics Lab⁴⁴4Discrete Dynamics Lab (DDLab) http://www.ddlab.org/. [41].

Rule clusters classification displays 88 equivalences (as is shown in Table 1) in ECA, and besides establishes basically three kind of symmetries: symmetric, semi-asymmetric, and full-asymmetric set of rules [33].

Also, these equivalences become refined in that some transformations (refection, negation, or complement) produce identity or another transformation, causing a rule cluster of 4 equivalent rules and their 4 equivalent compliments, to collapse. Where reflection produces the same rule, the rule is called symmetric (otherwise asymmetric) – but the reflection algorithm allows for semi-asymmetric and fully-asymmetric rules.

Memory classification on ECA is investigated in “Designing Complex Dynamics with Memory” [19]. A ECAM is a ECA composed with a memory function, the new rule open a new and extended domain of rules based in the ECA domain [17].

Basically, if you select an ECA rule and compose this rule with a memory function (in our analysis we have considered three basic functions : majority, minority, and parity). Hence, we will achieve a new rule with a base of a ECA rule. Of course, a number of features from the original ECA rule will be more evident on its ECAM generalisation. Therefore, the memory function will determine if the original ECA rule preserves the same class (respect to Wolfram’s classes) or if it changes to another class.

Following this simple principle, we know now that ECA rules composed with memory can be classified in three classes:

• strong, because the memory functions are unable to transform one class to another;

• moderate, because the memory function can transform the rule to another class and conserve the same class as well;

• weak, because the memory functions do most transformations and the rule changes to another different class quickly.

This way, the next table displays the ECA classification based in memory functions.

Memory classification presents a number of interesting properties.

We have ECA rules which composed with a particular kind of memory are able of reach another class including the original dynamic. The main feature is that, at least, this rule with memory is able to reach every different class. Rules with this property are called universal ECAM (5 rules).

Particularly, all these UECAM are classified strong in ECAM’s classification.

On the other hand, we have ECA that when composed with memory are able to yield a complex ECAM but with elements of the original ECA rule. They are called complex ECAM (44 rules).

It is remarkable that none of the rules classified in weak class is able to reach complex behaviour. These set of rules are strongly robust to any perturbation in terms of ECAM’s classification.

Wolfram’s classification in “Universality and complexity in cellular automata”, establishes four classes:

{uniform (class I), periodic (class II), chaotic (class III), complex (class IV)}

For details please see [37, 38].

Li and Packard’s classification in “The Structure of the Elementary Cellular Automata Rule Space”, establishes five ECA classes:

{null, fixed point, periodic, locally chaotic, chaotic}.

For details please see [15].

Wuensche’s equivalences in “The Global Dynamics of Cellular Automata”, establishes three ECA kinds of symmetries:

{symmetric, semi-asymmetric, full-asymmetric}.

For details please see [33].

Also, Wuensche establishes a set of “maximally chaotic” rules or known as “chain rules” (for details please see [40]).

Index complexity in “A Nonlinear Dynamics Perspective of Wolfram s New Kind of Science. Part I: Threshold of Complexity”, establishes three ECA classes:

{red ($k=1$), blue ($k=2$), green ($k=3$)}.

For details please see [8].

Density parameter with $d$-spectrum in “Experimental Study of Elementary Cellular Automata Dynamics Using the Density Parameter”, establishes three ECA classes:

{P, H, C}.

For details please see [11].

Communication complexity classification in “Cellular Automata and Communication Complexity”, establishes three ECA classes:

{bounded, linear, other}.

For details please see [10].

Additionally, bound class can be refined in other four subclasses.

Topological classification in “A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden”, establishes six ECA classes:

{period-1, period-2, period-3, Bernoulli $\sigma_{t}$-shift, complex Bernoulli-shift, hyper Bernoully-shift}.

For details please see [7].

Power spectral classification in “Power Spectral Analysis of Elementary Cellular Automata”, establishes four ECA classes:

{category 1: extremely low power density, category 2: broad-band noise, category 3: power law spectrum, exceptional rules}.

For details please see [27].

Morphological diversity classification in “On Generative Morphological Diversity of Elementary Cellular Automata”, establishes five ECA classes:

{chaotic, complex, periodic, two-cycle, fixed point, null}.

For details please see [4].

Distributive and non-distributive lattices classification in “Inducing Class 4 Behavior on the Basis of Lattice Analysis”, establishes four ECA classes:

{class 1, class 2, class 3, class 4}.

For details please see [13].

Topological classification in “A Full Computation-Relevant Topological Dynamics Classification of Elementary Cellular Automata”, establishes four ECA classes:

{equicontinuous, almost-equicontinuous, sensitive, sensitive positively expansive}.

For details please see [30, 6].

Also, this classification can be refined into three sub-classes: weakly periodic, surjective, and chaotic (in the sense of Denavey).

This is a classification by the evolution of a configuration consisting of an isolated one surrounded by zeros, that is a bit different from conventional ECA classifications previously displayed. In “Expressiveness of Elementary Cellular Automata”, we can see five ECA kinds of expressivity:

{0, periodic patterns, complex, Sierpinski patterns, finite growth}.

For details please see [28].

Normalised compression classification in “Asymptotic Behaviour and Ratios of Complexity in Cellular Automata Rule Spaces”, establishes two ECA classes:

{$C_{1,2}$, $C_{3,4}$}.

For details please see [42].

Expressivity classification in “Emergence of Surface Dynamics in Elementary Cellular Automata”, establishes three ECA classes:

{type A, type B, type C}.

For details please see [23].

Spectral classification in “A Spectral Portrait of the Elementary Cellular Automata Rule Space”, establishes four ECA classes:

{DE/SFC, DE/SFC SFC, EB, S}.

For details please see [29].

In this section, we have just bijective and surjective classification (personal communication, Harold V. McIntosh and Juan C. Seck Tuoh Mora):

{bijective, surjective}.

For details please see [21, 22].

Creativity classification in “On Creativity of Elementary Cellular Automata”, establishes four ECA classes:

{creative, schizophrenic, autistic savants, severely autistic}.

For details please see [5].