Operator representation and logistic extension of elementary cellular automata
M. Ibrahimi, A. Güçlü, N. Jahangirov, M. Yaman, O. Gülseren, S. Jahangirov
OOperator representation and logistic extension of elementary cellular automata
M. Ibrahimi a , A. G¨uc¸l¨u b,c , N. Jahangirov b , M. Yaman d , O. G¨ulseren b,e , S. Jahangirov b,f a Centre de Physique Th´eorique (CPT), Turing Center for Living Systems, Aix Marseille Universit´e, 13009 Marseille, France b UNAM-Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey c Department of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey d Department of Aeronautical Engineering, University of Turkish Aeronautical Association, 06790, Ankara, Turkey e Department of Physics, Bilkent University, Ankara 06800, Turkey f Interdisciplinary Graduate Program in Neuroscience, Bilkent University, 06800, Ankara, Turkey
Abstract
We redefine the transition function of elementary cellular automata (ECA) in terms of discrete operators. The operator repre-sentation provides a clear hint about the way systems behave both at the local and the global scale. We show that mirror andcomplementary symmetric rules are connected to each other via simple operator transformations. It is possible to decouple therepresentation into two pairs of operators which are used to construct a periodic table of ECA that maps all unique rules in such away that rules having similar behavior are clustered together. Finally, the operator representation is used to implement a generalizedlogistic extension to ECA. Here a single tuning parameter scales the pace with which operators iterate the rules. We show that,as this parameter is tuned, many rules of ECA undergo multiple phase transitions between periodic, locally chaotic, chaotic andcomplex (Class 4) behavior.Emergence, a semantic gap between behavior and interac-tions, is a hallmark of dynamical systems. Examples of emer-gent behavior include: exchanging deals between thousandsof agents / companies making up the whole stock market; inter-actions between alternately expressing genes generating juxta-posed biological forms; series of firings between peculiarly in-terconnected neurons breeding functional activities in the brain.Given the particularities of such systems (multiple levels ofhierarchies, events and processes operating at broadly di ff er-ent space / time scales), complexity science has adopted cellu-lar automata (CA) [1] as much simpler computational modelsto specifically target the semantic gap between individual andglobal degrees [2, 3, 4]. These agent based models operate infully discrete domains and are known to generate large scaletypes of behavior only through local interactions (rules) [5, 6].While aiming to acquire a generic understanding applicableto all dynamical systems, a main hypothesis has gathered sev-eral attempts to bridge the spatio-temporal patterns observedin CA (phenotype) with their rule space (genotype). The lead-ing apprehension is Wolfram’s classification which postulatesthat the asymptotic behavior of a dynamical system lies in oneof these four classes: homogeneous, periodic, aperiodic andcomplex behavior [2, 7]. He introduced elementary cellularautomata (ECA) as a paradigmatic simple set of rules whichcomprise all these types.Several studies have attempted to understand how distinctgroups of rules act to generate similar types of asymptotic be-havior, eventually making up the four classes. Langton’s methodof labelling a parameter out of a unit interval to a certain rule Email address: [email protected] (S. Jahangirov) is a simple, yet often e ffi cient approach for predicting the class[8]. Also, further approaches have introduced entropies, meanfield descriptions or network analyses that help to understandthe relation between patterns and their respective rules [9, 10,11, 12, 13]. However, these analyses are either too generic tohold for, or too specific to apply to larger families of CA. Con-sequently, the quest for generalizing a method to any dynamicallattice system remains challenging and this motivates the needto approach CA rules in the light of a di ff erent perspective.Using the ECA set as an example, we suggest a fundamen-tal approach that redefines the transition function based on asimple intuition gained by visual inspection of the system scaledynamics. This approach brings the microscopic informationcloser to the large scale dynamics and thus helps understandthe properties of a system based on its “first principles”.Having distilled the interactions into iterative operations,we employ these operations and the symmetries of the system torewrite the transition function in a more intuitive notation. Ourapproach provides a framework that links the similarities anddi ff erences observed at the phenotype level to an operator basedtranslation of the rule space. Furthermore, this framework en-ables us to implement a generalized logistic extension to ECA[14] where a single parameter scales the pace with which op-erators iterate the system. As a result, the binary state space ofECA is expanded into a Cantor set and in turn we get a chance toobserve transitions between classes [5]. In particular, we revealseveral complex (Class 4) instances that are not reachable inthe standard ECA. More interestingly, the behavioral di ff erencebetween some rules sharing similar genetic code is diminishedupon the logistic extension.ECA are time dependent one dimensional infinite strings Preprint submitted to Elsevier October 6, 2020 a r X i v : . [ n li n . C G ] O c t f sites S t = { s tn }| ∞ n = −∞ of a binary state space s i ∈ { , } . InECA a Rule defines how the value of a certain site is iterated s t + i = f S s ti based on its current value and the values of itsnearest neighbors, through a transition function f S ( s ti − , s ti , s ti + ).Given the binary state space, there are eight possible configu-rations of a three site neighborhood, resulting in 2 =
256 pos-sible mappings, i.e rules. Mapping of Rule 30 is shown as anexample in Fig. 1(a). The name “30” of this rule comes fromthe binary to decimal transformation of the string 00011110obtained from the particular mapping of the eight configura-tions listed in the order shown in Fig. 1(a). ECA possesses twoimportant symmetries: complementary and mirror. Here com-plementary means flipping the state in every site of the array.Hence, if Rule A and Rule B are complementary (mirror) sym-metric, then running Rule A with a certain initial string willgive the complementary (mirror) image of running Rule B withthe complementary (mirror) version of that string. When thesesymmetries are taken into account, the number of unique rulesreduces to 88 (and not 64 since mirror and / or complementarysymmetries of certain rules are equivalent to themselves).Simple observations on ECA runs reflect visual structuresof uniform, stable, oscillatory or irregular patterns. These struc-tures are prone to a mixture of three types of fundamental iter-ations [ s ti → s t + i ], namely: decay [0 | → → | → ff erent colors). Each group has cen-tral cells with values 0 and 1 that are mapped in four di ff erentways: [0 →
0, 1 → →
0, 1 → →
1, 1 → →
1, 1 → = ).The operator representation of Rule 30 becomes DSOG. Sym-metric counterparts of rules are easily constructed in operatorrepresentation. As shown in Fig. 1(b), to get the mirror symme-try of a rule, one needs to switch the operators in the group IIIand group IV. To get the complementary of a rule, one needs toreplace all D operations (if any) with G and vice versa. Sym-metries of the Rule 30 (DSOG) found by these transformationsare presented as an example in Fig. 1(c).Symmetric (I and II) and asymmetric (III and IV) sets ofoperators are decoupled from each other with respect to bothmirror and complementary transformations. Hence, it is in-structive to arrange ECA in a “periodic table” by placing possi-ble symmetric sets as abscissa and asymmetric sets as ordinate.However, using all 16 pairs of operations in both axes leadsto many repetitions of rules that are identical under mirror andcomplementary transformations. This can be avoided by real-izing that, for example, a symmetric set “DO” becomes “GO”under complementary transformation while remaining the same Figure 1: (a) Representation of the Rule 30 in terms of operators. (b) Trans-formations needed to switch between mirror and complementary symmetriesof a rule. (c) Switching between the Rule 30 and its symmetries using operatorrepresentation. under mirror transformation. Omitting one of these pairs erasesa whole column of repetitions. Continuing in this fashion onecan reach at a 10 ×
10 table that has all 88 unique Rules withonly 12 repetitions. While constructing this table, one needs todecide which repeating columns to erase and how to arrange therows and the columns that are left at the end. The table that wehave constructed, after evaluating numerous options based onmathematical and aesthetic criteria, is presented in Fig. 2. The12 repetitions that appear at the corners of the table are removedfor clarity. Note that, every adjacent row and column share atleast one common operator which means that every adjacentrule on the Table share at least three common operators.The Periodic Table presented in Fig. 2 o ff ers a systematic“bird’s eye” view of all 88 unique rules of ECA. Rules domi-nated by similar simple patterns (homogeneous, vertical lines,diagonal lines, horizontal stripes) tend to appear together. Therules that show rich behavior populate the “fertile crescent”along the diagonal where simple rules with contradicting pat-terns are expected to overlap. Among these rich rules, the onesthat have common features are also brought together. Rule pairs18, 146 and 122, 126 are striking examples of this. Despite thechaotic nature of these rules, starting a run with one of themand switching to the other rule results in the same pattern that isproduced without the switching. This is because, Rule 18 (122)and Rule 146 (126) share the same mapping, except for the con-figuration 111 (010) which is mapped to 0 in the former and 1in the latter. This 111 (010) configuration is “washed out” in afew steps and is never visited again. This e ff ect is also presentif one starts with the Rule 26 and continues with the Rule 154but not the other way around.The Periodic Table of ECA also resonates with the findingsof Li and Packard [9] in their classic study on the structure of2 igure 2: Periodic table of the elementary cellular automata (ECA). Rules corresponding to operators representations (in the order I, II, III, IV) and their mirror andcomplementary counterparts (if di ff erent) are presented below each box in increasing order. Each box presents a run starting with a random sequence of 100 binarydigits evolved for 100 time steps according to the Rule that is named by the smallest number. Periodic boundary condition is used. Chaotic, locally chaotic andcomplex rules are highlighted with red, blue and purple squares, respectively. Rules that acquire aperiodic behavior upon the logistic extension are highlighted withgreen squares. igure 3: Definition of the operation regions based on a configuration. L , C and R correspond to the values at the left, center and right cell of a configuration. the ECA rule space. They found two clusters of chaotic rules (inthis context it includes the complex rules 54 and 110). ChaoticA includes Rules 18, 22, 30, 54, 146, and 150 while Chaotic Bhas Rules 60, 90, 106, 110, 122, and 126. As seen in Fig 2, theyappear as clusters at the bottom left and top right of the “fertilecrescent”, respectively. The authors found Rule 45 to be sepa-rated from the clusters but in our Table we find it connected tothe cluster B. Furthermore, clusters A and B are connected overa bridge of locally chaotic Rule 26 in the Table. There are noother chaotic rules in the row and the column of the Rule 105,which was also found to be isolated by Li and Packard, but itis connected to the cluster B over a bridge of locally chaoticRule 73.The operator representation can further illuminate the stud-ies on the computational irreducibility of ECA. In particular, itis interesting to examine the rules that are detached from thecoarse-graining network investigated by Israeli and Goldenfeld[15]. They have shown that, Rule 105 can be course grained bythe Rule 150. In the operator representation, these rules appearas OOSS and SSOO, respectively. Furthermore, both DGDG(Rule 60) and DDGG (Rule 90) can be coarse-grained by them-selves. Finally, the authors were unable to coarse-grain fourunique rules: 30, 45, 106 and 154. In the operator represen-tation, they happen to be DSOG, OGDS, OSGD, and SDOG.These make up four unique rules that involve all four opera-tors while avoiding two complementary symmetric operators(D and G) in the same mirror symmetric set. In other words,the rules that were found to be irreducible are the ones that ap-pear the most asymmetric in the operator representation. Webelieve that these mere observations can guide further studiesin this subject.Recently, we have introduced the logistic extension of two outer-totalistic CA: Game of Life and Rule 90. This extensionis achieved via introduction of a parameter, λ , that tunes the dy-namics of CA. λ = λ is tuned below 1, the binary statespace extends into a Cantor set and the systems expand theircomplexity through series of deterministic transitions [14]. Inparticular, the Rule 90 which is aperiodic at λ = λ ∼ .
6. The operator representa-tion presented here enables us to go beyond the outer-totalisticrules and generalize the logistic extension to all ECA. We firstdefine four regions of operation for each group (I, II, III and IV)as shown in Fig. 3. The coordinates of a configuration [ L , C , R ](denoting left, center and right sites, respectively) defined as thesums x ≡ L + R + (mod 2) and y ≡ L + C + (mod 2) de-termine in which operation region it falls. As shown in Fig. 3,the eight possible binary configurations appear at the centers ofthe regions that correspond to their group definitions shown inFig 1(a). Hence, the configuration [ L , C , R ] determines the op-eration region which in turn determines the corresponding op-erator based on the rule at hand. Depending on the operator, thevalue of a site is updated according to the following formulae:Decay ⇒ s t + = (1 − λ ) s t Stability ⇒ s t + = s t Oscillation ⇒ s t + = (1 − λ ) s t + λ, if s t ≤ (1 − λ ) s t , if s t > Growth ⇒ s t + = (1 − λ ) s t + λ where s t and s t + are the values of the central site at thecurrent and the next time step, respectively. These equations,consistent with the operator notation, make up the new form ofthe transition function. Note that this generalization is consis-tent with the special case of the Rule 90 that we have reportedearlier [14].Significant changes in dynamics can occur when x or y passesover from one region to another. This happens when the sum L + C or L + R is equal to the critical thresholds 0.5 or 1.5. Thevalues that L , C , and R can take is dictated by the λ -dependentCantor set. Hence, one can expect these changes at the valuesof λ that mark the equality of binary sums to the critical thresh-olds. As λ is tuned below 1, the first time such a transitionoccurs is when 2 λ = .
5. After this point, some of the rulesstart behaving di ff erently than their original version.Rules that exhibit chaotic, locally chaotic or complex be-havior pass through multiple phase transitions while going be-tween these regimes. As seen in Fig. 4, chaotic Rule 18 be-comes complex at λ = .
73 mimicking (but not exactly copy-ing) the complex patterns seen in one of its neighbors, Rule 54.Another locally chaotic rule close by, Rule 82, also mimics theRule 54 behavior at λ = . L + C . This isclear in the distinct behavior of Rule 26 (the mirror symmetryof Rule 82) which has a mixture of chaotic and locally chaotic4 igure 4: 150 ×
150 cells snapshots at a later stage of a 1000 × λ . The color bar shown at the top mapsthe range between the minimum and maximum cell values for each snapshot.Both conventional and operator representation of the rules are given below eachpanel. behavior at λ = .
74. However, complementary rules behavein the same way under the logistic extension. For example,the behavior of complementary Rules 90 and 165 are the sameat λ = . λ = .
74. They also canbecome locally chaotic like Rule 110 at λ = .
74 or becomechaotic like Rule 124 (not shown in Fig. 4) which resembles itsneighboring chaotic Rule 60 at λ = .
72. Rules that are chaoticor locally chaotic can behave in a complex fashion as exempli-fied by Rule 86 (mirror symmetry of the Rule 30) at λ = . λ = .
68, respectively.Some of the Rules that are originally periodic can acquireaperiodic behavior. For example, Rule 38 becomes locally chaoticat λ ∼ .
69 and chaotic at λ ∼ .
61. Periodic rules can also be-come complex, for example Rule 37 and Rule 46 at λ = . ff erwhile generating inter-class transitions and disclosing inert be-haviors of periodic rules. We believe that logistic extension tothe operator-based representation may be useful to explore hid-den features in other complex systems, such as discrete latticemodels [16] and boolean genetic networks [17].S. J. acknowledges support from the Turkish Academy ofSciences - Outstanding Young Scientists Award Program (T ¨UBA-GEB˙IP). References [1] J. von Neumann, Theory of Self-Reproducing Automata, edited and com-pleted by A.W. Burks (University of Illinois Press, Urbana, IL, 1966).[2] S. Wolfram. Cellular automata as models of complexity, Nature 311(1984) 419.[3] A. Deutsch and S. Dormann,
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