Random Expansion Method for the Generation of Complex Cellular Automata
Juan Carlos Seck-Tuoh-Mora, Norberto Hernandez-Romero, Joselito Medina-Marin, Genaro J. Martinez, Irving Barragan-Vite
RRandom Expansion Method for the Generation ofComplex Cellular Automata
Juan Carlos Seck-Tuoh-Mora*, Norberto Hernandez-Romero,Joselito Medina-Marin, Genaro J. Martinez,Irving Barragan-ViteAAI-ICBI-UAEH. Carr Pachuca-Tulancingo Km 4.5.Pachuca 42184 Hidalgo. MexicoUnconventional Computing Centre, University ofthe West of England, BS16 1QY Bristol, United KingdomEscuela Superior de Computo, Instituto PolitecnicoNacional, MexicoSeptember 2020
Abstract
The emergence of complex behaviors in cellular automata is an areathat has been widely developed in recent years with the intention to gener-ate and analyze automata that produce space-moving patterns or glidersthat interact in a periodic background. Frequently, this type of automatahas been found through either an exhaustive search or a meticulous con-struction of the evolution rule. In this study, the specification of cellularautomata with complex behaviors was obtained by utilizing randomlygenerated specimens. In particular, it proposed that a cellular automatonof n states should be specified at random and then extended to anotherautomaton with a higher number of states so that the original automa-ton operates as a periodic background where the additional states serveto define the gliders. Moreover, this study presented an explanation ofthis method. Furthermore, the random way of defining complex cellu-lar automata was studied by using mean-field approximations for variousstates and local entropy measures. This specification was refined with agenetic algorithm to obtain specimens with a higher degree of complex-ity. With this methodology, it was possible to generate complex automatawith hundreds of states, demonstrating that randomly defined local inter-actions with multiple states can construct complexity. Keywords: Cellular automata, complexity, mean-field theory, local informa-tion, entropySubmitted to: Inf Sci 1 a r X i v : . [ n li n . C G ] S e p Introduction
Given that the emergence of complexity occurs through simple local interactionsbetween the automaton cells, complexity has been one of the most investigatedtopics in the study of cellular automata [1]. Researchers have intended to studyautomata that are capable of creating mobile structures (known as gliders) thatinteract in a periodic background. These structures do not disintegrate or endup dominating the space of evolutions but rather maintain a balance with theperiodic background while interacting with each other [2]. This is the typeof cellular automata that has been treated in this study. In particular, theone-dimensional case has been analyzed here.The study of gliders in cellular automata has been a popular field of researchsince an enormous amount of work began on the study of Conway’s Game-of-Lifecellular automaton. Some excellent studies on this automaton were the booksedited by Griffeath and Moore [3] and Adamatzky [4]. For the one-dimensionalcase, a boom was caused by Wolfram’s study [5] [6]. The implications of thegeneration of gliders in cellular automata peaked with the study conducted byCook, who, through the interaction of gliders, demonstrated that the cellularautomaton Rule 110 is universal [7].Multiple ways of generating complex cellular automata have been proposedand developed. We classify these methods in four main categories. However,these categories are not exclusive, and a study can simultaneously fall intomore than one of these at once. Given the impossibility of referring to all theworks related to the generation of complex cellular automata, we presentedonly a subset of the most relevant and recent studies in this field of research.Consequently, many other significant works have been omitted here.The first category is represented by the studies that used a more theoreticalapproach regarding the application of mathematical concepts. In this category,we include studies such as [8], which presented a cellular automaton whose evo-lution rule was an approximation of two-dimensional Kolmogorov complexity.The automaton was shown to be capable of simulating binary logic circuits. Inthe same study, a similar cellular automaton with increasing complexity pro-duced gliders that could be used as information carriers. Besides, a glider gunand logic gates were constructed as well. A probabilistic analysis that enabledus to predict the global behavior of configuration patterns in elementary cellularautomata was presented in [9], for which both the states were quiescent, fromasynchronism to full synchronism. With this, five automata exhibited complexphase transitions. The phase transitions in asynchronous cellular automatawere investigated in [10] by using the local structure theory in order to estimatedifferent types of second-order phase transitions. In [11] and [12], a mathemat-ical definition of gliders for one-dimensional cellular automata was presented toprovide a symbolic dynamics characterization demonstrating that glider inter-action implies chaos in the sense of Li-Yorke. Moreover, ultradiscretization wasapplied in [13] in order to transform the multidimensional Allen-Cahn equationinto a cellular automaton, obtaining traveling wave solutions that were similarto those in the continuous systems. In [14], logical and algebraic properties2ere used to construct conditionally matching evolution rules in one and twodimensions to design replicating loops and square calculation tasks. In [15], theminimal Boolean form of binary cellular automata was applied to characterizetheir complexity. New tools to study self-organization in cellular automata withgliders were introduced in [16]. Initial configurations were defined according toan ergodic measure, and the limit measure described the asymptotic behav-iors of gliders. Mean-field theory, de Bruijn and subset diagrams, and memorywere used in [17] in order to display nontrivial constructions and quasi-chaoticbehavior in elementary cellular automaton Rule 22.Another category comprises of studies that proposed evolution rules inspiredby the physical, chemical, and biological phenomena. For instance, unconven-tional computer systems inspired by collision-based computing in cellular au-tomata that work through the interaction of gliders in a periodic backgroundwas investigated in [18]. Moreover, the ultradiscretization of reaction-diffusionpartial differential equations was developed in [19] in order to obtain evolutionrules with dynamical properties such as bistability, pulse annihilation, soliton-like preservation, and periodic pulse generation. A three-state hexagonal cellularautomaton was studied in [20], which was a discrete model of a reaction-diffusionsystem with inhibitor and activator reagents. The cellular automaton exhibitedgliders used to implement basic computational operations, and, thus, collision-based logical universality was demonstrated. Moreover, Actin cellular automatawere presented in [21], which were based on the behavior of a globular proteinthat forms long filaments for intracellular signaling. The model consisted of twobinary-state semi-totalistic automaton arrays that support gliders. Some prop-erties were predicted using Shannon entropy. In [22], actin cellular automatawere enriched with memory. This additional feature slowed the propagation ofgliders down, decreasing entropy and transforming some gliders to stationaryoscillators and stationary oscillations to still patterns. Furthermore, in [23],the simplicity of cellular automata was combined with a light-sensitive ver-sion of Belousov-Zhabotinsky reaction. The resulting model could simulate aone-bit full adder digital component. Three natural processes (spatial compe-tition and the distinction between the stronger and weaker agents as well asinertia) were studied in [24] in order to propose a cellular automaton modelin which spatial patterns resembled phase transitions, jumps, and discontinu-ous transitions. Additionally, the morphological complexity of the Gray-Scottreaction-diffusion system was studied in [25] through the use of Shannon en-tropy, Simpson diversity, Lempel-Ziv complexity, and expressivity to exhibitthat the Gray-Scott systems support wave-fragments and gliders. In [26], a cel-lular automaton with wave propagation characteristics on an excitable mediumwas presented. The automaton also used inhibitors to create a simplified basictransistor. A combination of two of those transistors could reproduce univer-sal logic gates. Besides, a lattice-gas cellular automaton was presented in [27]in order to mimic the self-organization process of swarm formation. Accord-ing to the parameters that define the automaton, the system may display agreat variety of patterns. Excitable cellular automata were studied in [28] inorder to simulate signal transmission in actin networks sampled from slime mold3hysarum polycephalum. Actin networks support directional transmission toimplement Boolean logical operations for designing actin circuitry. Actin cel-lular automata were investigated in [29], where two-state transition rules wereconsidered: a Game-of-Life-like automaton with two states and an excitableactin automaton with three states. Eight-argument Boolean functions were alsoused to implement the logical functions.A third category is represented by studies that analyzed the modificationsof the traditional cellular automaton model. For instance, elementary cellularautomata with memory capabilities were presented in [30], where several pat-terns showing glider interaction were obtained. In [31], the majority memorycan generate gliders in elementary cellular automata with chaotic behavior. In[32], the property of solitonic collisions of gliders was studied in elementary cel-lular automata. Moreover, memory was also employed in [33] and [34] to createcomplex behaviors and provide a classification of the dynamics in Rule 126 withmemory. Hybridization is another modification investigated in these studies.Specifically, a wide range of gliders was studied in [35] and [36], taking hybridcellular automata whose evolution was dependent on two or more evolutionrules. For example, Rules 168 and 133 produced complicated interactions thatwere classified by using a quantitative approach. In [37], a computationally uni-versal Brownian cellular automaton was described based on an asynchronouscellular automaton with local cell state transitions following a Poisson pointprocess, which demonstrates that random fluctuations can serve for efficientcomputation. The dynamics of randomized local neighborhood connections intwo-dimensional cellular automata were investigated in [38]. With the threestates, the automata held sustained rhythmic oscillations and glider dynamics.Using a totalistic cellular automaton, Game-of-Life-type rules were investigatedin [39], in which complexity emerged from the interactions between an activatingfactor and an inhibiting factor. Signed majority cellular automata were investi-gated in [40] to simulate different types of logic circuitry. It has been proven thatuniform asymmetric and non-uniform symmetric rules are universal. In [41], asimple triangular partitioned cellular automaton was proposed, where the nextstate of a cell was determined by the three adjacent parts of its neighbor cells.The evolution of some specimens with these features demonstrated the existenceof gliders and glider guns. Memory was used in [42] to simulate Fredkin gatesin a one-dimensional cellular automaton by the collision of gliders, solitons, andbinary interactions to obtain the final outputs. An algorithm was proposed in[43] to convert any Turing machine into a one-dimensional cellular automatonwith a 2-linear time dynamics. Moreover, three Turing machines were convertedinto three cellular automata: binary sum, Rule 110 and a universal reversibleTuring machine.A fourth category consisted of studies involved in systematic searches usingdifferent types of heuristic approaches in order to find complex cellular au-tomata. For instance, in [15], behavioral metrics were employed in a geneticalgorithm to select cellular automata similar to the Game of Life, while thespontaneous emergence of glider guns in cellular automata with two states intwo dimensions was developed in [44] and [45]. Those works applied an evolu-4ionary search for new glider guns, and an automatic process was provided toclassify glider guns that could implement collision-based universal cellular au-tomata. A surface and histogram-based classification for the periodic, chaoticand complex elementary cellular automata was presented in [46] by using thenearest-neighbor interpolation in order to analyze a diversity of surfaces.Previous studies demonstrated that the specification of complex cellular au-tomata has been proposed by a meticulous construction of evolution rules, mod-ification of the classical model or an exhaustive search of the suitable specimens.However, thus far, there has been no method to generate cellular automata withgliders that is general for any number of states and can consider the interactionof many states.This study presents an original method for the generation of cellular au-tomata in one dimension with hundreds of states by using randomly generatedautomata as its basis. These automata serve as support for generating a periodicbackground on which the random extensions on the said automata are capableof generating gliders. With this process, it is possible to obtain systematicallycomplex cellular automata with hundreds of states. The contribution of thisstudy lies in demonstrating that simple local interactions, that are specified atrandom and involve a large number of states, can achieve complexity (glidergeneration).
A cellular automaton A = { K, φ, v } consists of a finite set of states K , a vectorneighborhood of relative positions v = { v , v } , with v i ∈ Z , v ≤ v , and anevolution rule φ : K ( v − v +1) → K that maps blocks of states of size v − v + 1to individual states of K . The dynamics of A is defined by taking an initialconfiguration of m states c : Z m → K , where Z m is the set of integers from 0to m −
1. For 0 ≤ i ≤ m −
1, a neighborhood n i = c i + v . . . c i + v is generatedby taking periodic boundary conditions. This is how c i = φ ( n i ) is defined. Ingeneral, c ji = φ ( n j − i ) = φ ( c j − i + v . . . c j − i + v ).Thus, evolution rule φ induces a global mapping Φ : c j → c j +1 based onlocal interactions between the states of the cells of c j .In order to simplify the study, only cellular automata with neighborhoodsize 2 were considered, i.e., v = { v , v + 1 } , since all the other cases can besimulated to this representation by using more states, as explained below.Since the evolution rule maps a neighborhood of size v − v + 1 to anindividual state, the extension of this mapping to v − v neighborhoods willproduce a block of v − v states, where the extended vector of neighbor cells isspecified as v (cid:48) = { v , v − v − } . This vector contemplates 2( v − v ) states;therefore, this extended mapping can be represented as K v − v ) → K v − v .This means that a mapping of two blocks of size v − v evolves into a block ofthe same size.Thus, we can define a new set of states S such that | S | = K v − v ) , alongwith defining a new evolution rule ϕ : S → S such that ϕ simulates the same5ehavior as φ in the original automaton.This demonstrates that every cellular automaton in one dimension can besimulated by another A = { S, ϕ, v } with v = {− , } , which also entails anexponential increase in the number of states to perform the simulation.This way, we only consider cellular automata with a neighborhood size 2,since the other cases can be reduced to this. For this type of automaton, theevolution rule ϕ can be represented by a matrix M ϕ of order | S | × | S | suchthat for any two states s, y ∈ S , M ϕ ( s, y ) = ϕ ( sy ), i.e., each entry of M ϕ isdetermined by the evolution of the specified neighborhood by concatenating itsrespective row and column indices. Several studies have explored various ways of generating and measuring the pro-duction of gliders in cellular automata by analyzing the densities [47],mean-fieldtheory [17] and entropy [48], along with reviewing the emergence of structures inthe evolution space [49] [50]. As regards Shannon’s entropy, we can locally mea-sure entropy concerning the states of each cell, its history and its neighbors [51].These entropy measures can be used to numerically calculate the informationcontained in each cell and its relation to neighbor elements.These local entropy measures are easy to implement and useful for detectingthe existence of a periodic background and the generation of gliders. The firstlocal measure that we can use is the local active information storage (LAIS)to measure how much of the current and past information of a cell defines itsfuture state. For a state s ∈ S , LAIS is defined as follows:
LAIS ( s ) = log p ( s n +1 | s ( k ) n ) p ( s n +1 ) (1)In Eq. 1 , the conditional probability of having the state s at the time n + 1is reviewed, provided the block s ( k ) n of k past states from the time k − n + 1 tothe time n . This is divided by the probability of having the state s at the time n + 1.Thus, if a state is defined as the product of the periodic background, theconditional probability p ( s n +1 ) | s ( k ) n is expected to be high and, therefore, hasa positive LAIS value. On the other hand, if a well-defined background isnot present, the conditional probability is low, and its
LAIS value tends to benegative. Calculating the
LAIS value for each cell in a representative evolutionof a cellular automaton, and obtaining its average provides a numerical wayof identifying when an automaton is producing information locally from theperiodic background.Another measure of local information quantifies the effect that the infor-mation of a neighboring cell has concerning another one and its history. Thismeasure is known as the local transfer entropy (
LT E ). For cellular automata,this measure can be applied to both the right and the left of a given cell.6iven the state s ∈ S of a cell in a cellular automaton and the state of itsright neighbor y ∈ S , the right LT E of s can be defined as: LT E r ( s, y ) = log p ( s n +1 | s ( k ) n , y ) p ( s n +1 | s ( k ) n ) (2)The definition of the LT E on the left (
LT E l ( s, y )) is similar, although ittakes the state y ∈ S of the left neighbor.The entropy measures LT E r ( s, y ) and LT E l ( s, y ) provide us with an ideaof the local information that is transferred between the two neighboring cellsof a cellular automaton (either on the left or on the right) in relation to theprobability that the state s is generated by its history. Thus, for the informationcontained in a glider, p ( s n +1 | s ( k ) n , y ) > p ( s n +1 | s ( k ) n ), which produces positivevalues of LT E r or LT E l .As in the case of LAIS ( s ), the average values of LT E r and LT E l are cal-culated numerically through a representative evolution of a cellular automaton,providing us a measure to detect the information transferred between the cellsproduced by glider interactions. Rule 110 is the prototypical case of a complex cellular automaton. The followingtable presents the evolution rule, which involves 2 states and neighborhoods of3 cells. 000 001 010 011 100 101 110 1110 1 1 1 0 1 1 0Table 1: Representation of the evolution rule of Rule 110The simulation of Rule 110 with a 4-state cellular automaton produces thefollowing evolution rule:00 01 10 1100 00 01 11 1101 10 11 11 1010 00 01 11 1111 10 11 01 00 → q be the density of background states for this rule. The mean-fieldpolynomial for states 2 and 4 is: q (cid:48) = q + q (1 − q ) + (1 − q ) q (cid:48) = q + q − q + − q + q q (cid:48) = q − q + (4)where q (cid:48) is the resultant density, given an initial q . Figure 3 presents thegraph of the mean-field polynomial of the background states against the identityof the density so that the fixed points can be appreciated. Moreover, it showsthe experimentally measured density of these states in 30 evolution samples,each with 300 states and 300 evolutions.8igure 2: Evolution of the automaton of 4 states (left) applying a filter todifferentiate between the background and the gliders (right).The mean-field polynomial estimates a stable density at a value of 0 . .
61. The polynomialhas a discrepancy because it is approaching two states at the same time; there-fore, its prediction falls below the real value. However, the critical point is thatthe fixed value it predicts is stable, which can be observed in the graph and thepolynomial iteration.The polynomial suggests that the density of background states is preservedwithout being too high or low (i.e., around 0 . M ϕ = 1 2 3 41 4 1 3 12 4 4 2 43 2 3 1 34 3 2 3 39igure 3: Density estimated by the mean-field polynomial, fixed-point behaviorand close values in 10 iterations of the polynomial, and experimental density ofbackground states.Fig. 4 presents the examples of its regular and filter evolution by selectingthe states 3 and 4 as the background and taking 300 cells and 300 evolutions aswell as the density of this filter with the average of 30 samples.Figure 4: Regular evolution with filter and density of background states in a 4-state automaton.In this case, the background states tend to retain a density close to 0 .
56 asthe automaton evolves. However, the presence of structures interacting in theevolution space is not observed as in complex automata. We can use local en-tropy measures to characterize the information features in this automaton. Theanalysis of local entropies is carried out on the filtered evolution to facilitateour study. Thus, we only consider two types of states: background and addi-tional states, which make the computation of entropies easier. For the analysis,we take a sample filtered evolution of 10 ,
000 cells with random states and 600evolutions to approximate the values of
LAIS , LT E r and LT E l . Then, thosevalues are used to make the measurements in the blocks of states obtained inanother evolution and obtain an average value.In this example, we observe that the value of LAIS is 0 . LT E r =0 . LT l = 0 . AIS=0.3513 LTEr=0.0958 LTEl=0.0656
Figure 5: Local active information storage (LAIS) and local entropies
LT E r and LT E l in a 4-state automaton with filtered evolution.cells and 300 evolutions.Figure 6: Regular and filtered evolution of the 4-state automaton emulatingRule 110.We know that the background states present a density close to 0 . LAIS , LT E r and LT E l forthis evolution. First, 10 ,
000 cells with random states and 600 evolutions aretaken to approximate the local information values. Then, these measurementsare applied in the sample evolution with filter, calculating their averages withthe obtained blocks of the states.Figure 7 shows that the local information retained in the background states ishigher than the last example (
LAIS = 0 . .
6, an average
LAIS value close to 0 . AIS=0.6068 LTEr=0.08727 LTEl=0.05612
Figure 7: Local active information storage (LAIS) and local entropies
LT E r and LT E l in the filtered evolution of the 4-state cellular automaton emulatingRule 110. The purpose of this study is to demonstrate that it is possible to generatecomplex cellular automata with hundreds of states. For this, first, a cellularautomaton is randomly generated with a states. This automaton serves as a pe-riodic background. Then, the evolution rule of this first automaton is expandedwith b additional states so that we obtain a newly expanded automaton of a + b states. The neighborhoods resulting from the combination of the a initial stateswith the additional b states are filled at random with uniform probability aftertaking the whole of a + b states into account.Formally, the process can be defined as follows: To begin with, a cellularautomaton A = { S, ϕ, }} is randomly defined such that: M ϕ ( s, y ) = rand ( S ) ∀ s, y ∈ S (5)where rand ( S ) is a random state of S . Once A is defined, another automaton P = { S ∪ P, ϕ (cid:48) , } is defined based on it, where P is a new automaton with aset of states such that: M ϕ (cid:48) ( s, y ) = M ϕ ⇐⇒ s, y ∈ Srand ( S ∪ P ) ⇐⇒ s ∈ P ∨ y ∈ P (6)Notably, the evolution rule M ϕ (cid:48) is an extension of M ϕ that preserves theoriginal part for neighborhoods formed with the states of S but takes any ran-dom state in S ∪ P for the rest of the new neighborhoods. This definition of P allows the original automaton A to act as a periodic background because itis a closed subsystem in the evolution of P and for which some of their newneighborhoods produce background states randomly.Given the cellular automaton P = { S ∪ P, ϕ (cid:48) , } , with a = | S | and b = | P | ,there are ( a + b ) = a + 2 ab + b neighborhoods. We know that a neighbor-hoods produce only background states, while 2 ab + b neighborhoods randomly12enerate both background and additional states. Thus, due to uniform random-ness, the probability of generating any state in these 2 ab + b neighborhoodsis: 1 a + b . Hence, the probability of obtaining background states in the new neighbor-hoods is: aa + b Let q be the probability of having a background state in a given evolutionof the P automaton. Then, the probability of having a neighborhood formedby the background states is q , the probability of having neighborhoods formedby additional states is (1 − q ) = 1 − q + q and the probability of havingmixed neighborhoods is 2 q (1 − q ) = 2 q − q . The proportion of neighborhoodsformed with background states that evolve in background states is 1, and theproportion of the other types of neighborhoods that generate background statesis aa + b . In this way, the mean-field polynomial that approximates the density q (cid:48) of background states in the following evolution can be defined as: q (cid:48) = q + (cid:16) aa + b (cid:17) (cid:0) q − q (cid:1) + (cid:16) aa + b (cid:17) (cid:0) − q + q (cid:1) = (cid:16) − aa + b (cid:17) q + (cid:16) aa + b (cid:17) (7)Figure 8 shows the surface obtained by iterating the polynomial of the pre-vious equation for all the densities, with a = 4 and 0 ≤ b ≤
16. In the sectionon the right, we observe the surface crossed with the surface that representsthe identity of the densities. The intersection exemplifies the densities that arefixed points in Eq. 7.We are interested in obtaining the densities of the background states close to0 .
6, remembering that the mean-field polynomial is underestimating the density.Therefore, we are searching for a behavior owing to which there is a balancebetween the background states and the additional states for the formation ofgliders in a periodic background.In this manner, we can extend the cellular automaton of 4 states to onebetween 10 and 14 states in order to obtain a density that allows the formationof gliders.To take an approximation of the number of additional states needed to obtaincomplex behaviors, let us assume b is proportional to a , i.e., b = αa . Then, Eq.7 can be expressed as: q (cid:48) = (cid:18) − aa + αa (cid:19) q + (cid:18) aa + αa (cid:19) (8)13igure 8: (A) Surface representing the density of the background states approx-imated by mean-field polynomial. (B) The same surface against the identity;the desired densities are represented by the white ellipse.Now, suppose we want to find the α ratio so that the mean-field equationpreserves the density of the background states. We now have the followingequation: q = (cid:18) − aa + αa (cid:19) q + (cid:18) aa + αa (cid:19) (9)By solving α , we establish Eq. 10: q = q − (cid:16) aa + αa (cid:17) q + (cid:16) aa + αa (cid:17) ( a + αa ) q = ( a + αa ) q − aq + aaq + αaq = αaq + aαq − αq = 1 − qα = − qq − q = − qq (1 − q ) = q (10)The proportion of additional states concerning the number of backgroundstates can be approximated by inversing the density of background states thatone wishes to have in the evolution of the automaton.This proportion indicates that we should look for the number of appropriateadditional stages in order to obtain the complex behaviors that use a randomcellular automaton as a periodic background.Let us take a cellular automaton A = { , ϕ, } defined randomly as:Let us take this automaton as the basis for extending it to an automatonwith a complex behavior. Eq. 10 tells us that if we want a density of q = 0 . α α = 1 / . P = { , ϕ (cid:48) , } .14igure 9: Random cellular automaton of 4 statesFigure 10: Randomly extended cellular automaton based on a 4-state automa-ton.Figure 10 shows that the extended evolution rule has 12 states comparedwith the original 4 (also indicated in the figure). Moreover, it depicts a sampleand a filtered evolution, demonstrating the background states (from 1 to 4) inone color and the additional states in another. Furthermore, the experimentaldensity of the background states is close to an average of 0 .
75, which is abovethe 0 . LAIS is lower compared to the 4-state cellular automatonthat emulates Rule 110 but with its
LT E r value is higher and LET l value issimilar. Let us consider another cellular automaton A = { , ϕ, } , which is also15 AIS=0.459 LTEr=0.1303 LTEl=0.0577
Figure 11: Local information and transfer entropies for the extended cellularautomaton of 12 statesrandomly defined as:Figure 12: Random cellular automata of 4 states.The previous automaton has a periodic behavior in a few evolutions. We takethis automaton as the basis for extending it to an automaton with a complexbehavior but add more states in it than in the previous one. Thus, we expandour rule to obtain a new cellular automaton P = { , ϕ (cid:48) , } .Figure 13 presents the extended evolution rule of 14 states along with asample and the filtered evolution. The value of the experimental density of thebackground states is close to an average of 0 . LAIS is lower compared to the 4-state cellularautomaton that simulates Rule 110. Moreover, although its
LT E r value issimilar, its LET l value is again a little lower. The density equation (Eq. 10)and the experiment indicates that we must look for an α ratio between 2 and 2 . a initial states. 16igure 13: Cellular automaton of 14 states extended randomly from a 4-stateautomaton. LAIS=0.391 LTEr=0.089 LTEl=0.0475
Figure 14: Local information and transfer entropies for the extended cellularautomaton of 14 states.
Given a random cellular automaton with a states, the previous results provideus with a set of parameters to generate complex cellular automata. We look forextended automata with 3 a to 3 . a states, with the LAIS value greater than0 . LT E r and LT E i greater than 0 . a states and their new neighborhoods are defined uni-formly and randomly among all the possible states. In each iteration, individualsare ranked after considering their local information and transfer entropy values(weighted equally). Based on this qualification, a refined population is selectedby using a tournament strategy. With this population improved, each rule iscrossed with another randomly selected rule, with a crossing probability of 0 . . Then, each modified rule is mutated only in the neighborhoods of the statesthat contain at least one additional element, with a probability of 0 .
1. If thismodified rule improves on the original one, it takes its place; otherwise, theoriginal rule remains.This simple genetic algorithm calculates the cellular automata with complexbehaviors with 2, 8, 16 and 32 initial states that are used as the backgroundand extend later to the 7, 28 , 56 and 112 states, respectively.
LAIS=0.574 LTEr=0.062 LTEl=0.0397
Figure 15: Randomly extended cellular automaton of 7 states from a cellularautomaton of 2 states.The cellular automaton in Figure 15 uses as a random automaton of a = 2states as a basis and then extends with a proportion of α = 2 .
5. Hence, wefinally have a + αa = 7 states . By filtering the evolution in the backgroundand additional states, the movement of gliders in a periodic background canbe appreciated. The experimental tests demonstrate that the density of thebackground states is close to 0 .
7, while the local entropy values are close tothose demonstrated by the emulation of Rule 110 with 4 states.18
AIS=0.479 LTEr=0.094 LTEl=0.035
Figure 16: Randomly extended cellular automaton of 28 states using a randomcellular automaton of 8 states as background.The cellular automaton in Figure 16 takes a random automaton with 8states as the background. In the initial evolution, it is possible to perceive theemergence of structures in a multicolored periodic background. The filter offers abetter perception of these structures by differentiating between the backgroundstates and the additional states. Its experimental density is close to 0 .
62, whilethe values of the local information and transfer entropies are close to thosecorresponding to Rule 110 simulated with 4 states.
LAIS=0.467 LTEr=0.088 LTEl=0.044
Figure 17: Randomly extended cellular automaton of 56 states from a cellularautomaton of 16 states 19s more states are employed, it is more complicated to perceive the forma-tion of gliders in a periodic background defined by multiple states. In the au-tomaton of Figure 17, we have a background specified for 16 states and extendedwith neighborhoods whose evolution was specified uniformly and randomly to56 states. Notably, the use of the filter makes the appearance of the glidersthat move in the periodic background clear. The experimental density is closeto 0 .
65, and the local information and transfer entropies values are again closeto those observed in the emulation of Rule 110.
LAIS=0.452 LTEr=0.086 LTEl=0.047
Figure 18: Randomly extended cellular automaton of 112 states from a cellularautomaton of 32 states.In Figure 18, we can notice that the emergence of complex patterns can bevery complicated just by inspecting the evolutions with multiple states. More-over, we can observe this phenomenon for the automaton specified from theone which formerly had 32 states (that were used as the background) and areto be later extended to 112 states. The rise of gliders is evident in a periodicbackground when filtering the evolution. The experimental density of the back-ground states is close to 0 .
6, and the local entropies are close to those observedby the 4-state automaton that emulates Rule 110.The following examples demonstrate various complex cellular automata ob-tained through the process described above (with the change from 100 to 400background states as the basis) to extended cellular automata from 340 to 1340states. The evolutions display 500 cells and 1000 evolutions from first a randomcondition and then from a single state that is different from others. Both the op-tions are filtered in the background states of one color and the additional statesof another in order to appreciate the appearance of the gliders in a periodicbackground. 20
A) a=100, b=340 (B) Filter(C) From a single di ff erent state (D) Filter (E) LAIS=0.4122 (F) LTEr=0.085(G) LTEl=0.035 Figure 19: Cellular automta of 340 states extended at random from a back-ground automata of 100 states. 21
A) a=200, b=680 (B) Filter(C) From a single di ff erent state (D) Filter (E) LAIS=0.4224(F) LTEr=0.084(G) LTEl=0.031 Figure 20: Cellular automta of 680 states extended at random from a back-ground automata of 200 states. 22
A) a=300, b=1020 (B) Filter(C) From a single di ff erent state (D) Filter (E) LAIS=0.398(F) LTEr=0.093(G) LTEl=0.033 Figure 21: Cellular automta of 1020 states extended at random from a back-ground automata of 300 states. 23
A) a=400, b=1340 (B) Filter(C) From a single di ff erent state (D) Filter (E) LAIS=0.413(F) LTEr=0.091(G) LTEl=0.029 Figure 22: Cellular automta of 1340 states extended at random from a back-ground automata of 400 states. 24able 2 provides the summary of the characteristics of the proportion ofadditional states, the density of background states and the local entropies ofthe complex cellular automata presented in this study (Sections 4 to 6).background states total states α density LAIS LT E r LT E l This study demonstrates how complexity in cellular automata can be randomlyobtained by using a smaller automaton as the background and extending it togenerate another larger automaton with multiple states in which gliders areformed, which interact in a periodic background. The method proposed inthis study offers the possibility of generating complex cellular automata withhundreds of states.While generating complex automata, the approximation of the density ofthe background states through mean-field polynomials as well as local informa-tion and transfer entropy measures are useful for specifying a simple geneticalgorithm to look for evolution rules with multiple states that produce complexbehaviors.Many aspects have been identified for future researchers to explore. Forinstance,they can use the complexity classifications that have been presentedin other works to characterize the complexity obtained through the proposedmethod.In this method, only one subautomaton is specified as a generator of a peri-odic background. Moreover, the investigation of the application of two or moresubautomata for this task has also been proposed in addition to using operationssuch as permutations and reflections of states on these subautomata in orderto investigate whether it is possible to generate different types of dynamics andif there is a variation in the proportion of additional states needed to obtaincomplex behaviors. 25he extension of random automata proposed in this work uses a uniformrandom distribution. Another task for future researchers would be to investigateother types of probabilistic distributions in order to investigate the type ofcomplexity that can be achieved through them.However, the random extension of an automaton is not the only way to ob-tain complex behaviors; other methods can also serve the same purpose. Forexample, the composition of several evolution rules may also lead to the genera-tion of multiple state automata. However, it is relevant to know in which casesthis operation may be able to produce complex behaviors.Different measures and tools can also be applied to the ones already proposedin this study to detect and measure the complexity and produce more elaborateconstructions such as certain types of gliders or glider guns, which are essentialfor the implementation of structures that are capable of performing computingtasks.
Acknowledgment
This study was supported by the National Council for Science and Technology(CONACYT) with the project numbers CB-2014-237323 and CB-2017-2018-A1-S-43008, along with IPN Collaboration Network “ Grupo de Sistemas Complejosdel IPN ”.
ReferencesReferences [1] M. Mitchell, Complexity: A guided tour, Oxford University Press, 2009.[2] Y. Bar-Yam, Dynamics of complex systems, CRC Press, 2019.[3] D. Griffeath, C. Moore, New constructions in cellular automata, OxfordUniversity Press on Demand, 2003.[4] A. Adamatzky, Game of life cellular automata, Vol. 1, Springer, 2010.[5] S. Wolfram, A new kind of science, Vol. 5, Wolfram media Champaign, IL,2002.[6] S. Wolfram, Cellular automata and complexity: collected papers, CRCPress, 2018.[7] M. Cook, Universality in elementary cellular automata, Complex systems15 (1) (2004) 1–40.[8] B. Y. Peled, A. Y. Carmi, Complexity Steering in Cellular Automata, Com-plex Systems 27 (2) (2018) 159–175. doi:{10.25088/ComplexSystems.27.2.159} . 269] N. Fates, D. Regnault, N. Schabanel, E. Thierry, Asynchronous behav-ior of double-quiescent elementary cellular automata, in: Correa, JR andHevia, A and Kiwi, M (Ed.), LATIN 2006: THEORETICAL INFORMAT-ICS, Vol. 3887 of LECTURE NOTES IN COMPUTER SCIENCE, CLEI;CMM; CONICYT; Int Federat Informat Proc, 2006, pp. 455–466, 7th LatinAmerican Symposium on Theoretical Informatics (LATIN 2006), Valdivia,CHILE, MAR 20-24, 2006. doi:{10.1007/11682462\_43} .[10] H. Fuks, N. Fates, Local structure approximation as a predictor of second-order phase transitions in asynchronous cellular automata, NATURALCOMPUTING 14 (4, 1-2, SI) (2015) 507–522, 3rd International Workshopon Asynchronous Cellular Automata and Asynchronous Discrete Models(ACA) held as a Satellite Workshop of the 11th International Conference onCellular Automata for Research and Industry (ACRI), Krakow, POLAND,SEP, 2014. doi:{10.1007/s11047-015-9521-6}doi:{10.1007/s11047-015-9521-6}