On complex dynamics from reversible cellular automata
Juan Carlos Seck-Tuoh-Mora, Genaro J. Martinez, Norberto Hernandez-Romero, Joselito Medina-Marin, Irving Barragan-Vite
OOn complex dynamics from reversible cellularautomata
Juan Carlos Seck-Tuoh-Mora*, Genaro J. Martinez,Norberto Hernandez-Romero, Joselito Medina-Marin,Irving Barragan-ViteAAI-ICBI-UAEH. Carr Pachuca-Tulancingo Km 4.5.Pachuca 42184 Hidalgo. MexicoUnconventional Computing Centre, University of theWest of England, BS16 1QY Bristol, United KingdomEscuela Superior de Computo, Instituto PolitecnicoNacional, MexicoSeptember 2020
Abstract
Complexity has been a recurrent research topic in cellular automatabecause they represent systems where complex behaviors emerge fromsimple local interactions. A significant amount of previous research hasbeen conducted proposing instances of complex cellular automata; how-ever, most of the proposed methods are based on a careful search or ameticulous construction of evolution rules.This paper presents the emergence of complex behaviors based on re-versible cellular automata. In particular, this paper shows that reversiblecellular automata represent an adequate framework to obtain complexbehaviors adding only new random states.Experimental results show that complexity can be obtained from re-versible cellular automata appending a proportion of about two timesmore states at random than the original number of states in the reversibleautomaton. Thus, it is possible to obtain complex cellular automata withdozens of states. Complexity appears to be commonly obtained fromreversible cellular automata, and using other operations such as permu-tations of states or row and column permutations in the evolution rule.The relevance of this paper is to present that reversibility can be a usefulstructure to implement complex behaviors in cellular automata.
Keywords: Cellular automata, reversibility, complexitySubmitted to: Comm Nonlinear Sci Numer Simulat1 a r X i v : . [ n li n . C G ] S e p Introduction
Reversible cellular automata (RCAs) have received much attention in the lastyears due to their information conserving property, which offers a framework toanalyze interesting dynamics. An RCA is defined by two evolution rules whichare reversible each other yielding an invertible global dynamics. RCAs havemany possible uses to implement and analyze universal systems, expansivenessor conservation laws [24] [30] [33]. However, up to our knowledge, they havenot been applied to provide a systematic method to obtain complex cellularautomata.Rule 110 is a classic example of a complex cellular automaton; the evolutionrule can generate mobile particles (called self-localizations or gliders as well) ina periodic background. Thus, one way to obtain complexity is to define evolu-tion rules producing periodic backgrounds and the interaction of particles [12].Wolfram presented the best-known reference presenting a classification of com-plexity in elementary cellular automata in [45]. Since this seminal paper, otherclassifications were proposed; for instance, using the complexity of languages,equicontinuity and attractors [25]. Previous works have introduced instances ofcomplex cellular automata with different characteristics (dimension, number ofstates, neighborhood size, and topology). However, most of these methods arebased on a careful search or construction of evolution rules, and little attentionhas been paid to the systematic use of RCAs for this purpose.The present paper presents the application of RCAs for obtaining complexityin cellular automata. The desired behavior is achieved adding random states tothe original reversible evolution rule, other operations conserving reversibilitysuch as permutation of states and positions in the evolution rule are also in-vestigated. This process can generate extended cellular automata with periodicbackgrounds and interacting particles.
Much research in recent years has focused on the study of complexity in cellularautomata. Theoretical studies include the use of several techniques to detectand classify complexity in cellular automata. For instance, tools from sym-bolic dynamics such as shift equivalence [40], subshifts of finite type [10] andtopological entropy [49] [47] [19] [35]. Other techniques have also proposed, inparticular, decision algorithms [2], measuring their ability to store and processinformation by particles [50] and probability of words [4].Chaotic dynamical behaviors in the sense of topological entropy are consid-ered in [6] for invertible one-dimensional linear cellular automata with severalstates. This work implies that, depending on the selected measurement, re-versible cellular automata can generate interesting dynamical behaviors.Some of the current applications of complex cellular automata are devotedto simulate universal structures [18], emulate systems engineering with differ-ent levels of complexity [34] and model various cooperative strategies for the2eneration of new knowledge [22].A primary current focus of research is how to utilize different types of cel-lular automata to yield complexity; for instance, polynomial cellular automata[44], majority rules [17], Turing-universal cellular automata with prime andcomposite rules [38] and cellular automata with evolution rules defined withKolmogorov complexity [36]. A related work employed infinite Petri nets tosimulate the elementary cellular automaton Rule 110 [48].Given the relevance of complexity in cellular automata, some different tech-niques and approaches have been employed to study and characterize complex-ity. Hanson and Crutchfield propose domain filters to locate and classify parti-cles in elementary cellular automaton rule 54 [20]. The study of complex cellularautomata as a tilling problem has been applied for the analysis and extensionof topological properties in several dimensions [11]. The asymptotic behaviorof cellular automata in higher dimension according to a Bernoulli probabilitymeasure is studied in [9], showing that cellular automata have the same varietyand complexity of conventional Turing machines when self-organization emergesfrom random configurations. The limit behavior of cellular automata in an in-finite time evolution is computed and classified by an automatic method basedon finite automata and regular expressions in [8] and [39]. About techniques toidentify particles directly in a periodic background, the Z -parameter has alsobeen employed to find gliders in cellular automata [46]; other techniques alsoinclude adding memory to the evolution rule [27], the use of genetic algorithms[7] [42] and the application of de Bruijn diagrams [31] [28] [29].Entropy and density of states are common measurements used to detectcomplex dynamics in cellular automata, due to these are easily defined and lowtime-consuming to be calculated.Shannon entropy complemented with Kolmogorov complexity has been em-ployed to quantify the structural characteristics, in two-dimensional multi-statecellular automata [23] and periodic Coven cellular automata in [26]. The entropyof fractal type cellular automata is utilized in [16]. An extension of Lempel-Zivcomplexity measure to estimate the entropy density from random configura-tions has been addressed in [13]. The entropy of the stochastic Fukui-Ishibashitraffic model was resolved in [41] obtaining exact analytical solutions. Entropyvariations are measured in [3] to propose an information-based classification.Entropy is compared with the local structure theory to characterize densityfor elementary cellular automata in [14], illustrating the case of rule 26. Amean-field approximation for density has been investigated in [32] using ele-mentary cellular automata in nonoverlapping generations both with crowdedand dispersal neighborhoods. Mean-field approximation of density and MonteCarlo simulations were employed in probabilistic cellular automata to provethat they converge to a specific measure [37]. Second-order phase transition inasynchronous cellular automata has been analyzed in [15] using local structuretheory of density to predict a qualitative change from an active phase into a sta-tionary state with fluctuations. In [1] density in elementary cellular automatahas been employed to find particles and complex behaviors. In [43] density inelementary cellular automata has been used to define interpolation surfaces and3lassify periodic, chaotic and complex behaviors.These articles confirm that entropy and density are currently applied inrecent works for analyzing complexity. Therefore this paper uses these tools todetect complex cellular automata. A one-dimensional cellular automaton A is an array of cells whose dynamicsis locally defined by a finite set of states S and mapping (or evolution rule) ϕ : S m → S , where m is the neighborhood size of A . Thus, a cellular automatoncan be described as a tuple A = { S, m, ϕ } .A global state (or configuration) of A is specified by c : Z → S whichassigns to every cell a state of S . Every cell c i has associated a neighborhoodvector N i = { i, n . . . n m − } with m relative positions (including i ) specifyingthe neighbors of c i . The evolution rule is applied over every c N i to obtainthe new state of c i and a new configuration c (cid:48) . In this investigation, periodicboundary conditions are considered in the numerical simulations.Definition of the evolution rule can be extended for sequences of states largerthan the neighborhood size m . For an integer m (cid:48) > m and each sequence w ∈ S m (cid:48) , the evolution rule ϕ can be applied over every complete neighborhoodof w . Notice that there are m (cid:48) − m + 1 overlapping neighborhoods in w , then ϕ ( w ) = w (cid:48) ∈ S m (cid:48) − m +1 . Thus, ϕ : S m (cid:48) → S m (cid:48) − m +1 .This feature can be used to simulate the original cellular automaton withanother of neighborhood size 2 and a larger number of states. If we take se-quences of length m (cid:48) = 2 m −
2, then the evolution rule can be applied as ϕ : S m (cid:48) → S m − ; this means that ϕ yields a mapping from two blocks of length m − K suchthat | K | = | S m − | and there is a bijection from K to S m − , then local dynamicsof ϕ can be simulated by an analogous evolution rule ϕ (cid:48) : K → K .This simulation shows that every one-dimensional cellular automaton A = { S, m, ϕ } can be simulated by another A (cid:48) = { K, , ϕ (cid:48) } , and we only have tostudy cellular automata of neighborhood size 2 to understand the other cases.In this case, the evolution rule ϕ can be represented by a matrix M whererows and columns are states in S and for every a, b ∈ S , M ( a, b ) = ϕ ( ab ). Rule 110 is composed of 2 states and neighborhood size 3; the evolution tableis defined by the binary representation of 110 as follows:4 0 0 00 0 1 10 1 0 10 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0Blocks of 4 cells evolve into blocks of 2 states; for instance, the block 0000evolves into 00, 0001 into 01 and so on. Every block of two states can be iden-tified with a number from 1 to 4. Hence Rule 110 can be represented withanother cellular automaton of 4 states and neighborhood size 2. The corre-sponding evolution rule is illustrated by a table M where the row and columnindexes are neighbors, and every entry is the evolution of every neighborhood.In the following, every evolution rule ϕ is represented by M ϕ . M = 1 2 3 41 1 2 4 42 3 4 4 33 1 2 4 44 3 4 2 1 (1)Evolution examples of this rule are presented in Fig. 1. Part (A) shows theevolution from a random initial configuration with 200 cells and 400 genera-tions. (B) starts from a unique, different state in the initial configuration. (C)shows the evolution from a random initial configuration with 400 cells and 800generations. (D) starts from a unique, different state in the initial configurationwith the same number of cells and generations. In all these examples, it is clearthe interaction of particles in a periodic background after a few generations.This paper proposes the following parameters to characterize complexity incellular automata. These criteria have been selected because they can be easilycalculated in computational experiments. These parameters are:1. Diversity degree of every state as row and column in M ϕ .2. Average of experimental density for every state measured in a number ofgenerations.3. Average of experimental entropy calculated in a number of generations.In M ϕ , the diversity degree of every row is measured with the number ofdifferent states and the repetitions of each state in the row. For every row i inthe evolution rule M ϕ , let U i ⊆ S be the set of states in row i and let F i be avector indicating the number of times that every state s ∈ U i appears in row i .This value is indicated by F i ( s ).The diversity of row i ( D i ) is calculated as:5 A) (B) (C) (D)
Figure 1: Evolutions of Rule 110 simulated with 4 states. D i = (cid:88) s ∈ U i (cid:18) | S | (cid:19) F i ( s ) (2)If a row i has states with high frequencies, these appear many times atrow i , and they have a pondered value close to 0. On the contrary, if row i has many states with few repetitions, they have low frequencies and ponderedvalues close to (1 / | S | ). Rows with D values close to 1 generate an extensiverange of states, and rows with D values close to 0 produce a low diversity ofstates. This parameter is applied analogously to the columns of M ϕ . Thus, wehave a measure for the diversity of states produced by every state (as a row orcolumn) of M ϕ . In complex cellular automata, we are going to look for evolutionrules with a mixture of states with high and low diversity degrees, which can beused to produce particles evolving in a periodic background.Another parameter used to characterize complexity is the average density ofstates measured during the evolution of a cellular automaton in various com-putational experiments. In particular, states defining a periodic backgroundhave a specific density and states delimiting particles have a different partic-ular density. This mixture of non-trivial densities is used to identify complexbehaviors.The last parameter employed to determine complexity is the informationentropy, measured for each state i ∈ S in every configuration c j during theevolution of a cellular automaton. Let | c j | be the number of cells in configuration6 j and let r ( i, c j ) be the number of cells with state i in c j . The entropy E c j ofconfiguration c j is calculated following Eq. 3. E c j = − (cid:88) i ∈ S (cid:18) r ( i, c j ) | c j | (cid:19) log (cid:18) r ( i, c j ) | c j | (cid:19) (3)This paper employs the average of information entropy evaluated in somecomputational experiments. Complex cellular automata have an asymptotic(not fixed or periodic) entropy behavior, indicating the average of bits neededto keep the information of every cell in each configuration.Figure 2 shows histograms to classify the diversity degrees of rows andcolumns in M (parts (A) and (B)), experimental densities and entropy ofRule 110. Histograms depict that rows have more diversity than columns, thisunbalance means that states as left neighbors may evolve in states different whenthey act as right neighbors. This feature allows the automaton to produce amixture of constructions generating a complex behavior. (A) (B)(C) (D) Figure 2: Properties of Rule 110 (diversity of row and column states, experi-mental densities and entropy).Experimental densities (part C) have been calculated taking the average of 10samples with 200 cells and 200 generations from random initial configurations.These densities are well differentiated in two groups, showing that state 4 is morecommon than the others due to the reproduction of a periodic background andthe propagation of particles.Finally, entropy (part D) has been calculated taking the average of everyconfiguration in 10 samples with 200 cells and 200 generations as well. In thiscase, entropy is around to 1 .
93, but it is not stationary. This value indicates7he interaction of particles in a periodic background, inducing a minimum butperceptible change in the dynamics of entropy in the automaton.
Reversible cellular automata (RCAs) are a particular type of systems whereglobal information is conserved during the temporal dynamics. An RCA A hasan evolution rule ϕ such that there exists another inverse rule ϕ − (perhapswith a different neighborhood size) inducing an invertible global mapping. Itis clear that ϕ and ϕ − can be simulated simultaneously by evolution ruleswith neighborhood size 2 [5]. Therefore, this paper only analyzes RCAs with aneighborhood size of 2 in both rules. RCAs have been widely studied since theseminal article by Hedlund [21]. The local properties of an RCA A with bothinvertible rules with neighborhood size 2 can be resumed as: • Every sequence w ∈ S ∗ has | S | preimages. • For m > = 2, the preimages of every w ∈ S m have a set L w of initial stateswith | L w | = L , a common central part and a set R w of final states with | R w | = R , such that LR = | S | . • For each distinct w, w (cid:48) ∈ S m , it is fulfilled that | L w | = | L (cid:48) w | = L , | R w | = | R (cid:48) w | = R , | L w ∩ R w (cid:48) | = 1 and | R w ∩ L w (cid:48) | = 1.Values L and R are known as the Welch indices of A . These properties havebeen used in [5] to propose an algorithm to generate random RCAs, specifyingthe numbers of states and one of the Welch indices. This algorithm has beenused in the experimental results of this paper to create complexity from RCAs.The evolution rule in Eq. 1 can be obtained from a reversible cellular au-tomaton of two states and Welch index L = 2, as shown in Fig. 3-(A).Evolutions in Fig. 3 have 200 cells and 200 generations. The reversibleautomaton in (A) is just a right shift; more states can be added at random toobtain a new automaton with the dynamics in (B), where the shift behavioris conserved by states 2 and 3. Part (C) in the figure is obtained permutingstates 2 and 3; in this evolution, we can notice the production of particles in aperiodic background. Part (D) in the figure is produced by applying the samepermutation (23) of rows and columns. In this way, the last evolution showsthe complexity equivalent to Rule 110. This emergence of complex behavioris also illustrated with the histograms of state diversity, densities, and entropycalculated in numerical experiments with 10 samples. In our definition of complexity, we are looking for automata where there is aset of particles interacting in a periodic background. The idea of this work8
A) (B) (C) (D)
Figure 3: Transition from a reversible cellular automaton of two states to acomplex cellular automaton by adding random states and permutations. Statediversity, densities, and entropy are also depicted.9s to use RCAs as a framework to establish a periodic background and thenadd additional states at random to obtain complexity. The process consists ofgenerating an initial RCA with k states with the algorithm presented in [5].This algorithm can yield RCAs with dozens of states. After the RCA has beengenerated, an extended evolution rule is defined with n additional states. Thenew neighborhoods are outlined at random with uniform probability, using thewhole set of k + n states. This process is illustrated in Fig. 4, a RCA with k states is defined in part (A); after that, n extra states are added in part (B)and the new neighborhoods are specified at random taking the k + n states. M ' = m , · · · m ,k ... ... m k, · · · m k,k ) M ' = m , · · · m ,k a ,k +1 · · · a ,k + n ... ... ... ... m k, · · · m k,k a k,k +1 · · · a k,k + n a k +1 , · · · a k +1 ,k a k +1 ,k +1 · · · a k +1 ,k + n ... ... ... ... a k + n, · · · a k + n,k a k + n,k +1 · · · a k + n,k + n (A) (B) Figure 4: Extending an RCA to establish a complex cellular automata.Figure 5 presents an instance of this process; first a trivial RCA with 6 statesand Welch index L = 1 (a right shift) is defined (part (A)); we can notice that allstates have the same diversity measure classified in histograms below the tem-poral evolution sample of 200 cells and 200 generations. Experimental densitiesbelow histograms are conserved during the whole evolution, as we can expectfrom an RCA. Finally, the experimental entropy shows that the information isuniformly conserved during the dynamical behavior of the automaton.Random states were added to the evolution rule to obtain a cellular au-tomaton with 12 states (part(B)), temporal evolution presents the rise of someparticle-like structures, but the periodic background is still too dominant. His-tograms present more diversity in the evolution of every state, but experimentaldensities are still too close, and entropy is around 3 with small deviation asthe automaton evolves. When 18 states are considered (12 random additionalstates in part (C)), we have a mixture of particles interacting in a periodicbackground. Histograms show more state diversity; as rows, average diversityis predominant and as columns low and average diversity are similar. Densitiescan be well classified in three groups, indicating that some states are more fa-miliar to define the periodic background, and other states are used only in theparticle formation. Entropy has a more interesting behavior, centered around3 .
75 with small fluctuations close to 0 .
05 at both directions.Last part of Fig. 5 displays the case when 18 additional random stateswere considered. Histograms tend to average and high diversities, densities arealmost merged indicating that states are almost equiprobable, and entropy ismore unstable around 4 .
33, and with a deviation close to 0 .
4, this indicates that10
A) (B) (C) (D)
Figure 5: Obtaining complex behaviors from an RCA with 6 states and Welchindex L = 1.a chaotic behavior has been reached.Computational experiments indicate that it is more common to obtain com-plexity when additional random states are considered approximately a propor-tion of 1 : 2; that is, k states defining the RCA and around 2 k random statescompleting the evolution rule. This proportion yields a mixture between theemergence of states in the original RCA which act in the specification of a pe-riodic background, and additional states producing particles interacting in thisenvironment.A set of experiments has been done to observe the rate of complex behaviorsobtained when RCAs are expanded at random. Table 1 shows different casesof RCAs with distinct Welch indices. The first row shows the different cases( k, k + n ) where k is the number of states defining an RCA, and k + n is thetotal number of states in the final automaton. For every case, 100 differentevolution rules were generated and analyzed using diversity in the evolutionrule, the density of states and entropy, to obtain a percentage of complex rulesobtained with this process. These experiments have been defined with 400 cellsand 800 evolutions to facilitate the computational study.There are blanks in Table 1 because not every Welch index divides thenumber of states of each RCA. This table shows that complex behaviors are notstrange even when more states are considered, but complexity is more difficultto be obtained when higher values of Welch indices are considered.Figures 6, 7 and 8 present examples of some complex cellular automata11ize (6 ,
18) (7 ,
21) (16 ,
48) (18 ,
54) (19 , L = 1 22% 15% 21% 20% 20% L = 2 13% 17% 12% L = 3 8% L = 4 13%Table 1: Percentage of complex rules obtained from different RCAs.generated from RCAs. All cases have 800 cells and 1600 generations. Everyexample has two evolutions, one generated from a random configuration, andanother from a fixed state. Histograms classifying the diversity of states are alsodepicted, and the average of state density and entropy are displayed as well.Figure 6: Complex cellular automaton of 28 states from an RCA with 9 statesand Welch index L = 3. 12igure 6 shows a complex cellular automaton with 28 states generated froma RCA with 9 states and Welch index L = 3. There is a mixture in the diversityof states as rows or columns in the evolution rule. Densities describe 3 groups ofdifferent values and the entropy is asymptotically decreasing with fluctuations.Figure 7: Complex cellular automaton of 63 states from an RCA with 20 statesand Welch index L = 2.Figure 7 illustrates a complex cellular automaton with 63 states generatedfrom a RCA with 20 states and Welch index L = 2. Again, there is a combina-tion in the histograms depicting the diversity of states. Densities are groupedin 3 groups, and the entropy is asymptotically decreasing as well.13igure 8: Complex cellular automaton of 100 states from an RCA with 30 statesand Welch index L = 3.Figure 8 presents a complex cellular automaton with 100 states generatedfrom a RCA with 30 states and Welch index L = 3. Histograms have a mixtureof diversity, and now densities are grouped in 2 classes, entropy is asymptoticallydecreasing with fluctuations.In the previous experiments, a concrete value for entropy has not been char-acterized when complexity is generated. However, these experiments show thatentropy is close and always below log ( k + n ) to obtain complex behaviors. The algorithm used in this paper is characterized to produce reversible automatawith quiescent states; that is, for the reversible evolution rule ϕ generated bythe algorithm, the matrix M ϕ fulfills that every diagonal element M ϕ ( i, i ) = i .There are transformations over M ϕ which conserve reversibility; in particular,14he effect of permutation of states and permutation of row and columns areillustrated in this paper.For a RCA of k states with evolution rule ϕ represented by M ϕ , and apermutation π : { . . . k } → { . . . k } , a permutation of states over M ϕ defines anew RCA with rule ϕ (cid:48) such that M ϕ (cid:48) ( i, j ) = π ( M ϕ ( i, j )) for 1 ≤ i ≤ j ≤ k . Ina similar way, a permutation of rows and columns over M ϕ defines a new RCAwith rule ϕ (cid:48) such that M ϕ (cid:48) ( π ( i ) , π ( j )) = M ϕ ( i, j ).Figures 9 and 10 are instances of some complex cellular automata generatedfrom RCAs with left Welch index L = 1. These cases have 800 cells and 800generations. Figure 9 shows some cellular automata with 70 states generatedfrom a RCA with 20 states and Welch index L = 1. Part (A) is the original RCAwith quiescent states, part (B) shows a modification obtained by a permutationof states. Part (C) depicts another change using a permutation of rows andcolumns, and part (D) has been obtained applying both types of permutationsat the same time. The original automaton is almost dominated by chaoticbehavior; only some isolated parts are visible with a periodic structure derivedfrom the RCA. However, when a state permutation is applied over the reversiblepart, the dynamical behavior is closer to complexity. The same is observedin (C) and (D). Figure 10 illustrates some cellular automata with 135 statesgenerated from a RCA with 40 states and Welch index L = 1. Part (A) isbased on the original RCA, part (B) uses a permutation of states, part (C)applies a permutation of rows and columns, and part (D) is a mixture of bothpermutations.These examples demonstrate that complexity can be obtained even whendozens of states are considered, as long as some of the states determine a sta-ble dynamical structure; in the previous cases, reversibility brings an adequateframework to define particles in a periodic background. Meanwhile, the originalRCAs are close to chaos, permutations in states, rows, and columns provide astronger structure to the reversible part, provoking a more complex dynamicalbehavior. 15 A) (B)(C) (D)
Figure 9: Complex cellular automata of 70 states generated from an RCA with20 states, Welch index L = 1 and applying permutation of states and permuta-tions of rows and columns. 16 A) (B)(C) (D)
Figure 10: Complex cellular automata of 135 states generated from an RCAwith 40 states, Welch index L = 1 and applying permutation of states andpermutations of rows and columns. 17he previous examples are constructed with RCAs having a unitary Welchindex. It is interesting to investigate if the same dynamical behavior can be ob-tained with RCAs with nonunitary Welch indices. Figures 11, 12 and 13 presentinstances of complex cellular automata generated from RCAs with nonunitaryWelch indices. (A) (B)(C) (D) Figure 11: Complex cellular automata of 40 states generated from an RCAwith 12 states, Welch index L = 2 and applying permutation of states andpermutations of rows and columns. 18 A) (B)(C) (D)
Figure 12: Complex cellular automata of 40 states generated from an RCAwith 12 states, Welch index L = 3 and applying permutation of states andpermutations of rows and columns. 19 A) (B)(C) (D)
Figure 13: Complex cellular automata of 51 states generated from an RCAwith 15 states, Welch index L = 3 and applying permutation of states andpermutations of rows and columns.The previous figures depict cellular automata with 40 and 51 states producedfrom an RCA with 12 and 15 states and Welch indices L = 2 and L = 3. Part(A) is the original RCA with quiescent states and dynamical behavior closerto chaos, part (B) shows a modification obtained by a permutation of states.Part (C) presents a permutation of rows and columns, and part (D) has beenobtained with both types of permutations. Again, when some permutation isapplied over the reversible part, the dynamical behavior is closer to complexity,with particles interacting in a periodic background. Experimental results demonstrate that a proportion close to (3 ,
1) (number ofstates, number of reversible ones) is adequate to obtain complex behaviors from20 +n=18, k=6, L=2 k+n=23, k=7, L=1 k+n=24, k=8, L=2 k+n=31, k=9, L=3 k+n=33, k=10, L=2k+n=36, k=11, L=1 k+n=40, k=12, L=3 k+n=45, k=13, L=1 k+n=47, k=14, L=2 k+n=48, k=15, L=3k+n=52, k=16, L=4 k+n=55, k=17, L=1 k+n=59, k=18, L=3 k+n=62, k=19, L=1 k+n=63, k=20, L=2k+n=67, k=21, L=3 k+n=70, k=22, L=2 k+n=73, k=23, L=1 k+n=77, k=24, L=4 k+n=80, k=25, L=5k+n=83, k=26, L=2 k+n=90, k=27, L=1 k+n=94, k=28, L=4 k+n=100, k=29, L=1 k+n=105, k=30, L=1
Figure 14: Examples of complex cellular automata from 18 to 105 states ob-tained from RCAs with a mixture of Welch indices.RCAs, characterized by a diversity of states, entropy, and density of states.Meanwhile, the number of states in the original RCA is not determinant toobtain complexity and Welch indices have more influence to produce complexbehaviors. Nevertheless, if not general, it is not strange as well to find complex-ity expanding RCAs at random as it can be observed in Table 1. Permutationsof the reversible part (states or rows and columns) show an essential effect toreduce chaos and yield complex behaviors when more states are considered.Figure 14 presents a collage of different complex cellular automata from 18to 105 states obtained with RCAs defined from 6 up to 30 states and differentvalues of Welch indices. 21
Conclusions
This paper has demonstrated that RCAs define an adequate framework to pro-duce complex cellular automata when they are expanded at random. The pro-cedures presented in this paper can produce complexity with dozens of states.Meanwhile, the number of states is not a constraint to obtain complex behav-iors, Welch indices are more determinant to restrict the rise of complexity. Ourexperiments show a proportion close to k reversible states for 3 k total stateswhere complexity is more probable to be obtained.Permutations of the reversible part have proved their utility to transit fromchaos to complex behavior, generating a periodic background with interactingparticles even when a large number of states are considered. It is interesting toobserve how a large number of states reach self-organization defining particlesand background.Further work may imply the generation of complex cellular automata in amore intelligent or heuristic way than only expanding at random RCAs; for ex-ample, employing fuzzy or evolutionary algorithms. Besides, to prove that thesecellular automata are computationally universal is also an open problem to betreated. Another direction can be the application of more complex measuresand metrics to construct complex automata from RCAs. It implies more compu-tational time or resources, but the smart construction of complex automata canbe useful to reduce the number of specimens to review. An analytical methoddefining how to create complex automata from a given RCA would be desirable,even for a small number of states.Besides the application of intelligent or heuristic techniques, another kindof automata can be used to support complex behaviors; for instance, surjectiveautomata, the use of partitions from the full shift system as used in symbolicdynamics. Acknowledgment
This work has been supported by CONACYT project No. CB- 2017-2018-A1-S-43008 and by IPN Collaboration Network “ Grupo de Sistemas Complejos delIPN ”.
References [1] Andrew Adamatzky, Genaro Ju´arez Mart´ınez, and Juan Carlos Seck TuohMora. Phenomenology of reaction–diffusion binary-state cellular automata.
International Journal of Bifurcation and Chaos , 16(10):2985–3005, 2006.[2] Sebastien Autran and Enrico Formenti. More Decision Algorithms forGlobal Properties of 1D Cellular Automata.
Journal of Cellular Automata ,13(1-2):1–14, 2018. 223] Enrico Borriello and Sara Imari Walker. An Information-Based Classifica-tion of Elementary Cellular Automata.
Complexity , 2017.[4] L. Boyer, M. Delacourt, V. Poupet, M. Sablik, and G. Theyssier. mu-Limitsets of cellular automata from a computational complexity perspective.
Journal of Computer and System Sciences , 81(8):1623–1647, DEC 2015.[5] Juan Carlos Seck-Tuoh-Mora, Joselito Medina-Marin, NorbertoHernandez-Romero, Genaro J. Martinez, and Irving Barragan-Vite.Welch sets for random generation and representation of reversible one-dimensional cellular automata.
Information Sciences , 382:81–95, MAR2017.[6] Chih-Hung Chang and Hasan Akin. Some Ergodic Properties of One-Dimensional Invertible Cellular Automata.
Journal of Cellular Automata ,11(2-3):247–261, 2016.[7] Rajarshi Das, Melanie Mitchell, and James P Crutchfield. A genetic algo-rithm discovers particle-based computation in cellular automata. In
Inter-national Conference on Parallel Problem Solving from Nature , pages 344–353. Springer, 1994.[8] Pedro P. B. De Oliveira, Eurico L. P. Ruivo, Wander L. Costa, Fabio T.Miki, and Victor V. Trafaniuc. Advances in the study of elementary cellularautomata regular language complexity.
Complexity , 21(6):267–279, JUL-AUG 2016.[9] Martin Delacourt and Benjamin Hellouin de Menibus. Characterisationof Limit Measures of Higher-Dimensional Cellular Automata.
Theory ofComputing Systems , 61(4, SI):1178–1213, NOV 2017. 32nd Symposium onTheoretical Aspects of Computer Science (STACS), Garching, Germany,Mar 04-07, 2015.[10] Alberto Dennunzio, Enrico Formenti, and Luca Manzoni. Limit Proper-ties of Doubly Quiescent m-Asynchronous Elementary Cellular Automata.
Journal of Cellular Automata , 9(5-6):341–355, 2014.[11] Alberto Dennunzio, Enrico Formenti, and Michael Weiss. Multidimensionalcellular automata: closing property, quasi-expansivity, and (un)decidabilityissues.
Theoretical Computer Science , 516:40–59, JAN 9 2014.[12] David Eppstein. Which life-like systems have gliders ?, 2002.[13] E. Estevez-Rams, R. Lora-Serrano, C. A. J. Nunes, and B. Aragon-Fernandez. Lempel-Ziv complexity analysis of one dimensional cellularautomata.
Chaos , 25(12), DEC 2015.[14] Henryk Fuks. Minimal entropy approximation for cellular automata.
Jour-nal of Statistical Mechanics-Theory and Experiment , Feb 2014.2315] Henryk Fuks and Nazim Fates. Local structure approximation as a predic-tor of second-order phase transitions in asynchronous cellular automata.
Natural Computing , 14(4, 1-2, SI):507–522, Dec 2015. 3rd InternationalWorkshop on Asynchronous Cellular Automata and Asynchronous DiscreteModels (ACA) held as a Satellite Workshop of the 11th International Con-ference on Cellular Automata for Research and Industry (ACRI), Krakow,POLAND, SEP, 2014.[16] N. Ganikhodjaev, M. Saburov, and Asimoni N. R. Mohamad. Periodic Cel-lular Automata of Period-2.
Malaysian Journal of Mathematical Sciences ,10(1, S):131–142, Feb 2016.[17] Vladimir Garcia-Morales. Diagrammatic approach to cellular automata andthe emergence of form with inner structure.
Communications in NonlinearScience and Numerical Simulation , 63:117–134, OCT 2018.[18] Eric Goles, Pedro Montealegre, Kevin Perrot, and Guillaume Theyssier.On the complexity of two-dimensional signed majority cellular automata.
Journal of Computer and System Sciences , 91:1–32, FEB 2018.[19] Junbiao Guan and Fangyue Chen. Complex Dynamics in Elementary Cel-lular Automaton Rule 26.
Journal of Cellular Automata , 10(1-2):137–147,2015.[20] JE Hanson and JP Crutchfield. Computational mechanics of cellular au-tomata: An example.
Physica D , 103(1-4):169–189, APR 15 1997. Work-shop on Lattice Dynamics, UNIV DENIS DIDEROT, PARIS, FRANCE,JUN 21-23, 1995.[21] G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamicalsystem.
Mathematical Systems Theory , 3(4):320–375, Dec 1969.[22] Georg Jaeger. Using Elementary Cellular Automata to Model Different Re-search Strategies and the Generation of New Knowledge.
Complex Systems ,27(2):145–157, 2018.[23] Mohammad Ali Javaheri Javid, Tim Blackwell, Robert Zimmer, and Mo-hammad Majid al Rifaie. Analysis of information gain and Kolmogorovcomplexity for structural evaluation of cellular automata configurations.
Connection Science , 28(2, SI):155–170, APR 2 2016.[24] Jarkko Kari. Reversible Cellular Automata: From Fundamental Classi-cal Results to Recent Developments.
New Generation Computing , 36(3,SI):145–172, JUL 2018.[25] P Kurka. Languages, equicontinuity and attractors in cellular automata.
Ergodic Theory and Dynamical Systems , 17(2):417–433, APR 1997.[26] Weibin Liu and Jihua Ma. Topological Entropy of Periodic Coven CellularAutomata.
Acta Mathematica Scientia , 36(2):579–592, Mar 2016.2427] Genaro J Martinez, Andrew Adamatzky, and Ramon Alonso-Sanz. Design-ing complex dynamics in cellular automata with memory.
InternationalJournal of Bifurcation and Chaos , 23(10):1330035, 2013.[28] Genaro J Mart´ınez, Andrew Adamatzky, and Harold V Mcintosh. On therepresentation of gliders in rule 54 by de bruijn and cycle diagrams. In
Proceedings of the 8th international conference on Cellular Automata forReseach and Industry , pages 83–91. Springer-Verlag, 2008.[29] Genaro J Martinez, Harold V McIntosh, Juan CST Mora, and Sergio VCVergara. Determining a regular language by glider-based structures calledphases fi 1 in rule 110.
Journal of Cellular Automata , 3(3):231–270, 2008.[30] Genaro J. Martinez and Kenichi Morita. Conservative Computing in aOne-dimensional Cellular Automaton with Memory.
Journal of CellularAutomata , 13(4):325–346, 2018.[31] Genaro Ju´arez Mart´ınez, Harold V McIntosh, and Juan Carlos Seck TuohMora. Gliders in rule 110.
International Journal of Unconventional Com-puting , 2(1):1, 2006.[32] J. R. G. Mendonca. A probabilistic cellular automata model for the dy-namics of a population driven by logistic growth and weak Allee effect.
Journal of Physics A-Mathematical and Theoretical , 51(14), Apr 6 2018.[33] Kenichi Morita. Computation in reversible cellular automata.
InternationalJournal of General Systems , 41(6, SI):569–581, 2012.[34] Stefano Nichele, Mathias Berild Ose, Sebastian Risi, and Gunnar Tufte.CA-NEAT: Evolved Compositional Pattern Producing Networks for Cel-lular Automata Morphogenesis and Replication.
IEEE Transactions onCognitive and Developmental Systems , 10(3):687–700, SEP 2018.[35] Yangjun Pei, Qi Han, Chao Liu, Dedong Tang, and Junjian Huang. ChaoticBehaviors of Symbolic Dynamics about Rule 58 in Cellular Automata.
Mathematical Problems in Engineering , 2014.[36] Bar Y. Peled and Avishy Y. Carmi. Complexity Steering in Cellular Au-tomata.
Complex Systems , 27(2):159–175, 2018.[37] A. D. Ramos and A. Leite. Convergence Time and Phase Transition ina Non-monotonic Family of Probabilistic Cellular Automata.
Journal ofStatistical Physics , 168(3):573–594, Aug 2017.[38] Jurgen Riedel and Hector Zenil. Rule Primality, Minimal Generating Setsand Turing-Universality in the Causal Decomposition of Elementary Cel-lular Automata.
Journal of Cellular Automata , 13(5-6):479–497, 2018.[39] Eurico L. P. Ruivo and Pedro P. B. de Oliveira. Inferring the Limit Be-havior of Some Elementary Cellular Automata.
International Journal ofBifurcation and Chaos , 27(8), JUL 2017.2540] Eurico L. P. Ruivo, Pedro P. B. de Oliveira, Fabiola Lobos, and Eric Goles.Shift-equivalence of k-ary, one-dimensional cellular automata rules.
Com-munications in Nonlinear Science and Numerical Simulation , 63:280–291,OCT 2018.[41] Alejandro Salcido, Ernesto Hernandez-Zapata, and Susana Carreon-Sierra.Exact results of 1D traffic cellular automata: The low-density behavior ofthe Fukui-Ishibashi model.
Physica A-Statistical Mechanics and Its Appli-cations , 494:276–287, MAR 15 2018.[42] Emmanuel Sapin, Olivier Bailleux, and Jean-Jacques Chabrier. Researchof complex forms in cellular automata by evolutionary algorithms. In
Inter-national Conference on Artificial Evolution (Evolution Artificielle) , pages357–367. Springer, 2003.[43] Juan Carlos Seck-Tuoh-Mora, Joselito Medina-Marin, Genaro J. Martinez,and Norberto Hernandez-Romero. Emergence of density dynamics by sur-face interpolation in elementary cellular automata.
Communications inNonlinear Science and Numerical Simulation , 19(4):941–966, Apr 2014.[44] Bertrand Stone. Line complexity asymptotics of polynomial cellular au-tomata.
Ramanujan Journal , 47(2):383–416, NOV 2018.[45] S Wolfram. Cellular Automata as Models of Complexity.
Nature ,311(5985):419–424, 1984.[46] Andrew Wuensche. Classifying cellular automata automatically: Findinggliders, filtering, and relating space-time patterns, attractor basins, andthe z parameter.
Complexity , 4(3):47–66, 1999.[47] Junkang Xu, Erlin Li, Fangyue Chen, and Weifeng Jin. Chaotic propertiesof elementary cellular automata with majority memory.
Chaos Solitons &Fractals , 115:84–95, OCT 2018.[48] Dmitry A. Zaitsev. Simulating Cellular Automata by Infinite Petri Nets.
Journal of Cellular Automata , 13(1-2):121–144, 2018.[49] Kuize Zhang and Lijun Zhang. Generalized Reversibility of TopologicalDynamical Systems and Cellular Automata.
Journal of Cellular Automata ,10(5-6):425–434, 2015.[50] Yanbo Zhang. Definition and Identification of Information Storage and Pro-cessing Capabilities as Possible Markers for Turing Universality in CellularAutomata.