Static and Dynamical Equilibrium Properties to Categorise GeneralisedGame-of-Life Related Cellular Automata

K.A. HawickComputer Science, Institute for Information and Mathematical Sciences,

Massey University, North Shore 102-904, Auckland, New Zealandemail: k.a.hawick@massey.ac.nz

Tel: +64 9 414 0800 Fax: +64 9 441 8181

April 2012

ABSTRACTConways’s Game-of-Life (GoL) is one of a family of spa-tial cellular automata rule sets that employ the Mooreneighbourhood on a rectilinear lattice of cells. GoL gen-eralises to set of other automata using different neighbour-numbers that lead to birth and survival. These models ex-hibit different static properties and also different long termdynamical equilibrium points. We define some quantifi-able metrics and explore how some well known GoL vari-ants can be categorised into groupings, categorised by theirstatic properties and dynamical equilibria. We also exploreeffects due to differing initial crowding levels of the liveseed cell population, and how this affects the categories.

KEY WORDSstatistical mechanics; Game of Life; generalised rules; cel-lular automata; complex system.

1 IntroductionCellular Automata (CA) models [1, 13, 28] have longplayed an important role in exploring and understandingof the fundamentals of complex systems [17, 33, 34].

Studies of many CA systems are reported in the litera-ture and for many complex systems applications includ-ing: damage spreading [21]; network analysis [38]; foodchain systems [16]; lattice gases [37]; and fluid flow [7].Two fundamental and classic CAs that provide a basis formuch other work are the Wolfram’s one dimensional au-tomaton [36], and Conway’s Game of Life [14]. The Wol-fram system is especially useful because of it concisenessand the fact that the whole automaton rule state space iseasily described as a family of bit valued cell-wise truth ta-bles and can be systematically explored. Two dimensionalautomata can be also be formulated using the Wolfram ap-proach [26]. Conway’s Game of Life (GoL) however, also

Figure 1: Coral and Diamoeba Automata.

has a concise specification, but it is formulated in terms ofcounts of live neighbouring cells rather than specific pat-terns.

There is a space of similarly formulated automaton rules[24] in the same family as GoL [22]. Like the Wolframautomata not all the rules in the GoL family are “interest-ing” or can be classified as complex. The Conway precisespecification turns out to be special in the rule set spaceof the family, but there are some other related automatain the family that do have interesting properties and whichhave been given names. Figure 1 shows snapshots fromtwo such GoL family members - known as “Coral” and“diamoeba.”

There are a number of other variants of cellular automataincluding: the game of death [12], with an extra zombiestate added; ratchet automata [15]; and other variants ofGoL such as hyperbolic life [27]. GoL and its variants havealso also been studied: on other non-square geometries [5];on Penrose tilings [25]; on hexagonal lattices [4]; and ongeneralised networks [9]. Work has also been reported onthree dimensional GoL variants [2, 3].

This present article reports work exploring some of thetransient and long term dynamical properties of the GoLfamily of automata in two dimensions and on a square, pe-riodic lattice. In particular a preliminary attempt is made

to see how this GoL family can be categorised accordingto some quantitative metrics that can be obtained from nu-merical experiments. Attempts to automatically classifyautomata by their behaviours have been reported [19], al-though here we discuss a set of manually carried out ex-periments.

Much work has been done on studying the coherent struc-tures that occur in GoL and its variants [11]. It is possi-ble to implant specific patterns such as gliders, glider gunsand so forth to obtain specific sequences of GoL automataconfigurations. However, in this present paper we investi-gate GoL automata systems that have been randomly ini-tialised with an initial fraction of live cells. Providing wesimulate a large enough sample of large enough automatasystems, many different individual patterns can occur bychance, will interact and the system will eventually arrivea static or dynamical equilibrium. This approach supportsstudy of the statistical mechanical properties [31] of theautomata systems and the identification of which automatahave stable points [6] or not, as well as a framework tostudy universality properties [32] in cellular automata the-ory [35].

Although stochastic thermal or random effects [18, 20]have been introduced into cellular automata to study suchproperties, in this present work we restrict ourselves to en-tirely deterministic automata. We avoid asynchronous au-tomaton [8, 23, 29, 30] effects by using a two phase syn-chronous update implementation of the automata reportedhere. The only randomness or ambiguity introduced is inthe initial random pattern of live cells seeded.

The article is structured as follows: In Section 2 we de-scribe the notation for the GoL family of automata and thestatistical metrics that can be applied to study them numer-ically. In Section 3 we present some results for some ofthe automata in the family. We discuss the implications inSection 4 and summarise some conclusions and areas forfurther work in Section 5.

2 Automata Notation & MetricsThe game of Live and its family of variant automata areimplemented using the Moore neighbourhood on a d = 2dimensional array of cells, where the number of (Moore)Neighbouring sites NM = 8, for d = 2 as shown in Fig-ure 2. We define the (square) lattice Length as L and hencethe number of sites N , typically N = Ld. We definethe number of live sites NL, and so the metric fractionfl = NL/N and similarly the number of dead sites ND,and fraction fD = ND/N .

The GoL community has developed various gener-alised notations for a GoL-like automata [1, 24]. We

8

1 2 3

4 5

6 7

Figure 2: Moore neighbourhood of 8 sites in 2-D.

elaborate these and use the more verbose form ofBn1,n2, .../Sm1,m2, .... This notation specifies therules that area applied to each site exactly once at eachtime step. The notation gives a comma-separated-list ofthe allowable numbers ni of live neighbours which giverise to Birth of a live site from a dead site before the slash,and likewise another comma-separated list of the allowablenumber mi of live neighbours that are necessary for a livesite to survive.

Algorithm 1 Synchronous automaton update cells algo-rithm.

for all i, j in (L,L) dogather Moore Neighbour-hoodMi,j

apply rule b[i][j]← {s[i][j],Mi,j}end forfor all i, j in (L,L) do

copy b[i][j]→ s[i][j]end for

Algorithm 2 CA Experimental proceduredeclare s[L][L]; b[L][L]for r = 0 to runs do

initialise s, st nL = pN live, rest deadfor t = 0 to te do

automaton update cellsend forfor t = te to tm do

automaton update cellsmake measurements

end forend fornormalise averages

The rules thus support death both from “overcrowding”and also from “lack of parents.” The classic Conway gameof life is thus specified by the notation B3/S2,3. We usethis notation to specify explicitly the 26 GoL variants weexperiment with below. Although GoL automata are well

studied, for completeness and to avoid ambiguity, we givethe algorithm for implementing these e automata.

Algorithm 2 shows the experimental procedure we deploy,based upon the automata update rule procedure specified inAlgorithm alg:synch-update. We use a two-phase updatealgorithm and so each rule is applied to change the site attime t + 1 based on its state and that of its neighbouringsites at time t.

The following metrics are used to study the automata ex-perimentally:

• the 1-step time correlation function C1 =1N

∑i,j s(i, j, t).s(i, j, t− 1)

• the fraction of neighbours that are the same as the sitethey surround. This is essentially a count of the like-like bonds, normalised by divining by (N.NM )

• likewise, the fraction of neighbours that are differentfrom the site they surround

• the number of monomers is the number of live sitesthat are completely surrounded by dead sites.

• the number of dimers is the number of connected-pairs of live sites that are each otherwise surroundedby dead sites.

The number of arbitrary sized cluster components of livesites is expensive to compute and we do not use it in thisstudy.

3 Experimental ResultsThe GoL family of automata were investigated numeri-cally. The reported experiments are based on averages over100 independently randomly initialised runs of a periodic2562 cell system, equilibrated for 512 time steps and thenrun with measured metrics recorded and averaged over asubsequent 512 steps.

Table 1 shows 26 members of the GoL family of automatasimulated for the specified times t on a 2562 periodic lat-tice, with random initial fractions of live sites of the speci-fied fractions p.

Figure 3 shows some of the metrics described in Section 2applied to the family of automata. The plots are of the1-step correlation function; the fraction of live and Va-cant(=dead) cells and the fraction of neighbour links thathave the same (live-live or dead-dead) cell on them. Theplots are sorted according to rank and the labelled pointsindicate the automata model. Note that the plot lines natu-rally group onto plateaux with sharp cliffs separating them.Automata of similar properties thus group together.

Figure 3: 1-Step Correlation and static fractional metrics.

Figure 4: Monomer and Dimer counts.

Figure 4 shows the number of monomers and dimers oflive cells for the different GoL automata with a startinginitialisation fraction of p = 0.5. The curves are againranked and points labelled by automata models. As can beseen most models do not lead to large numbers of dimers ormonomers, but a small number do exhibit large numbers.The grouping is split by a relatively sharp shoulder in thecurves.

Figure 5 shows a scatter-plot of the 1-step correlation met-ric plotted against the fraction of live sites. There are clearand obvious groupings with some annotations drawn (inred, online version) indicating which automata behave sim-ilarly.

The plotted metrics are shown with error bars based on theestimated experimental standard deviations. These them-selves give some interesting insights into the behaviours

Rule Common t=128 t=256 t=128 t=256 t=128 t=256(B/S) Nameformat for Rule p=0.5 p=0.5 p=0.3 p=0.3 p=0.1 p=0.1

B1/S1 Gnarl

B1,3,5,7/S1,3,5,7 Replicator

B1,3,5,7/S0,2,4,6,8 Fredkin

B2/S Seeds

B2/S0 Live Free/Die

B2,3,4/S Serviettes

B3/S0,1,2,3,4,5,6,7,8 Life w/o Death

B3/S1,2,3,4 Mazectric

B3/S1,2,3,4,5 Maze

B3/S2,3 Conway

B3/S4,5,6,7,8 Coral

B3,4/S3,4 34 Life

B3,4,5/S4,5,6,7 Assimilation

B3,4,5/S5 Long Life

B3,5,6,7,8/S5,6,7,8 Diamoeba

B3,5,7/S1,3,5,8 Amoeba

B3,5,7/S2,3,8 Pseudo Life

B3,6/S1,2,5 2x2

B3,6/S2,3 HighLife

B3,6,8/S2,4,5 Move

B3,6,7,8/S2,3,56,7,8 Stains

B3,6,7,8/S3,4,6,7,8 Day & Night

B3,7,8/S2,3,5,6,7,8 Coagulations

B4,6,7,8/S2,3,4,5 Walled Cities

B4,6,7,8/S3,5,6,7,8 Vote4/5/Annl

B5,6,7,8/S4,5,6,7,8 Majority

Table 1: 2562 Automata for different rules at times 128 and 256, for p = 0.5, 0.3, 0.1.

1

2

3b

3a

Figure 5: Scatter-plot of fCorr vs fLive, with annotated groupings in red (colour, online).

of the different automata. Models like “coagulation” and“anneal” have the highest fluctuations - due to the slowlybut continually varying spatial pattern observed in thoseautomata over the time frame of the measurements.

4 DiscussionOne long term goal of this approach is to identify the dy-namical growth and equilibria properties of the differentautomata [10]. To this end it is useful to attempt a group-ing of classification based on temporal as well s spatial be-haviour. The experimental data gathered indicates that theGoL automata studied fall into one of the following cate-gories:

1. rapidly reach a single saturated state of almost alldead or live with at most minor fluctuations

2. steady growth in a complex structural pattern that then

subsequently reaches a saturated state

3. a dynamic equilibrium of structures that fluctuate andgives rise to a permanently changing pattern - theserefine into a) those with large scale structures and b)those that only exhibit short scale fluctuations

Table 2 was originally constructed somewhat subjectivelyafter examination of the time sequences of patterns ob-tained from the 26 automata. The method involved eye-balling the models as movies trying to describe some com-mon words from the observations. Somewhat amazingly,these however do appear to agree with the clusters foundupon examination of Figure 5. The category 3a - “dynamicequilibrium with long and multi-scale structures” clustertogether at the top left of the scatter plot.

The category 3b - “ dynamic equilibrium but with shortlength scale structures” cluster in a larger area at the lowerleft. The other two groups are at the top right area with

Category Game-of-Live Family Examples - Categorised by Subjective Observation1 Assimilation; Diamoeba; Live w/o Death; Replicator2 Coagulations; Coral; Day+Night; Majority; Maze; Mazectric; Move; Anneal3 a 2x2; Conway; Highlife; Pseudo Life3 b 34-Life; Amoeba; Fredkin; Gnarl; Live Free or Die; Long Life; Seeds; Serviettes; Stains; Walled Cities

Table 2: Categorisation of GoL Automata for p = 0.5

some differentiation between fast and slow equilibration atthe far right and middle right respectively.

The single exception to this grouping is the “replicator”automata. It exhibits the strange behaviour of remainingin fluctuation for a long time then suddenly saturating to asingle dead state, and this may be due to finite size effectsof the simulations.

In Figure 4 the monomers and dimer curves are less ob-vious discriminators by themselves. The category 3a au-tomata do however group closely on the mid or uppershoulder of the monomer and dimer curves, respectively.

Similarly, in Figure 3 the single metric averages do not pro-vide good discriminators individually, although there areobvious groupings of the 3a automata appearing together,and the category 1 also tend to be close together.

We might expect that a caveat of this analysis is that whenthe system is not sufficiently “thermodynamically largeenough” then:

• the pattern might otherwise continue to grow if it didnot reach the periodic boundaries

• the particular set of structural patterns that might leadto categories 3a or 3b cannot arise simply by chanceon a finite sized system.

Nevertheless the classification clustering pattern from ex-amining the 1-step time correlation function scatter-plottedagainst the fraction of live cells does seem to give a goodcombined discriminator and grouping that matches obser-vation.

The fraction of initial live cells can be varied and a variantof figure 5 interpreted. We do not include these due tolack of space, but the effect is that if p is too low many ofthe automata patterns do not have enough material to formand so they die out with no live cells and 100% correlationforever.

Likewise if p is too large then overcrowding also preventsthe complex patterns from forming. The structure shownin Figure 5 is largely stable and unchanged if the stabilitycondition 0.3 < p <= 0.5 is satisfied.

5 ConclusionsWe have shown how a family of Game-of-Life-like Cellu-lar Automata can be formulated in terms of a common andextensible notation and systematically studied. We haveshown that some simple metrics help make preliminary at-tempts to categorise this family of automata into groups.

We have postulated four categories – originally conceivedin terms of a mix of temporal and spatial structural ob-servations. These map quite closely to the groupings thatemerge from a scatter plot of the long term averaged valuesof the fraction of live cells and the 1-step time correlationfunction.

There are interesting features in all the GoL automata stud-ied - this is no doubt why the community (See http://www.conwaylife.com/wiki) has troubled to givethese models specific names. However, perhaps the “mostinteresting” ones are those that seem to exhibit the high-est degree of complexity and not-coincidentally have beengiven the names with the word “life” in their names. Thesecategory 3a automata stabilise to a dynamic equilibria of arelatively low fraction of live cells, with a high degree oftime correlation between their states.

There is scope for further work in examining time-correlations longer than a single time-step and also com-ponent sizes larger than dimers. It seems likely that somemore discriminating combinations of static and temporalmetric can further refine the behaviours observed in cate-gory 3b.

In this work we have limited the study to known named au-tomata in the GoL family. There is scope to conduct a moresystematic search using the metrics discussed here to lookfor other category 3a automata. These criteria might alsobe usefully applied to higher dimensional GoL type mod-els where it is considerably harder to visualise the patternsand behaviours.

References[1] Adamatzky, A. (ed.): Game of Life Cellular Automata. No.

ISBN 978-1-84996-216-2, Springer (2010)[2] Bays, C.: Candidates for the game of life in three dimen-

sions. Complex Systems 1, 373–400 (1987)[3] Bays, C.: A new candidate rule for the game of three-

dimensional life. Complex Sytems 6, 433–441 (1992)[4] Bays, C.: A note on the game of life in hexagonal and pen-

tagonal tesellations. Complex Systems 15, 245–252 (2005)[5] Bays, C.: Game of Life Cellular Automata, chap. The

Game of Life in Non-Square Environments, pp. 319–329.Springer (2010)

[6] Binder, P.M.: (anti-) stable points and the dynamics of ex-tended systems. Physics Letters A 187, 167–170 (1994)

[7] Boghosian, B.M., Rothman, D.H., W.Taylor: A CellularAutomata Simulation of Two-Phase Flow on the CM-2Connection Machine Computer (Mar 1988), private Com-munication

[8] Capcarrere, M.S.: Evolution of asynchronous cellular au-tomata. In: Parallel Problem Solving from Nature Proc.PPSN VII. vol. 2439 (2002)

[9] Darabos, C., Giacobini, M., Tomassini, M.: Scale-free au-tomata networks are not robust in a collective computationaltask. In: et al., E.Y.S. (ed.) Proc. Cellular Automata: 5th In-ternational Conference on Cellular Automata for Researchand Industry. vol. LNCS 4173. Springer (2006)

[10] Eppstein, D.: Game of Life Cellular Automata, chap.Growth and Decay in Life-Like Cellular Automata, pp. 71–98. Springer (2010)

[11] Evans, K.M.: Larger than life: threshold-range scaling oflife?s coherent structures. Physica D 183, 45–67 (2003)

[12] Evans, M.S.: Cellular Automata - Brian’s Brain(2002), http://www.msevans.com/automata/briansbrain.html, department of Neurology, South-ern Illinois University School of Medicine

[13] Ganguly, N., Sikdar, B.K., Deutsch, A., Canright, G.,Chaudhuri, P.P.: A survey on cellular automata. Tech. rep.,Dresden University of Technology (2003)

[14] Gardner, M.: Mathematical Games: The fantastic combina-tions of John Conway’s new solitaire game ”Life”. Scien-tific American 223, 120–123 (October 1970)

[15] Hastings, M.B., Reichhardt, C.J.O., Reichhardt, C.: Ratchetcellular automata. Phys. Rev. Lett. 90, 247004 (2003)

[16] Hawick, K.A., Scogings, C.J.: A minimal spatial cellu-lar automata for hierarchical predator-prey simulation offood chains. In: International Conference on ScientificComputing (CSC’10). pp. 75–80. WorldComp, Las Vegas,USA (July 2010), www.massey.ac.nz/˜kahawick/cstn/040

[17] Hernandez-Montoya, A.R., Coronel-Brizio, H., Rodriguez-Achach, M.: Game of Life Cellular Automata, chap.Macroscopic Spatial Complexity of teh Game of Life Cel-lular Automaton: A sSimple Data Analysis, pp. 437–450.Springer (2010)

[18] Kaneko, K., Akutsu, Y.: Phase Transitions in two-dimensional stochastic cellular automata. J.Phys.A. Letters19, 69–75 (1986)

[19] Kunkle, D.R.: Automatic Classification of One-Dimensional Cellular Automata. Master’s thesis, RochesterInstitute of Technology (2003)

[20] Lynch, J.F.: On the threshold of chaos in random booleancellular automata. Random Structures & Algorithms 6(2-3),239–260 (March-May 1995)

[21] Martin, B.: Damage spreading and mu-sensitivity on cel-lular automata. Ergod. Th. and Dynam. Sys. 27, 545–565(2007)

[22] Martinez, G., Adamatzky, A., Morita, K., Margenstern, M.:Game of Life Cellular Automata, chap. Computation withCompeting patterns in Life-Like Automaton, pp. 547–572.Springer (2010)

[23] Nehaniv, C.L.: Evolution in asynchronous cellular au-tomata. In: Proc ICAL 2003 - Eighth Int. Conf. on ArtificialLife. pp. 65–73. MIT Press (2003)

[24] de Oliveira, G.M.B., Siqueira, S.R.C.: Parameter charac-terization of two-dimensional cellular automata rule space.Physica D: Nonlinear Phenomena 217, 1–6 (2006)

[25] Owens, N., Stepney, S.: Investigations of game of life cellu-lar automata rules on penrose tilings: lifetime and ash statis-tics. Journal of Cellular Automata 5, 207–225 (2010)

[26] Packard, N., Wolfram, S.: Two-dimensional cellular au-tomata. J. Stat. Phys. 38, 901–946 (1985)

[27] Reiter, C.A.: The game of life on a hyperbolic domain.Comput. & Graphics 21(5), 673–683 (1997)

[28] Sarkar, P.: A brief history of cellular automata. ACM Com-puting Surveys 32, 80–107 (2000)

[29] Sipper, M., Tomassini, M., Capcarrere, M.S.: Evolvingasynchronous and scalable non-uniform cellular automata.In: In Proceedings of International Conference on ArtificialNeural Networks and Genetic Algorithms (ICANNGA97.pp. 67–71. Springer-Verlag (1997)

[30] Suzudo, T.: Spatial pattern formation in asynchronous cel-lular automata with mass conservation. Physica A: Stat.Mech. and Applications 343, 185–200 (November 2004)

[31] Wolfram, S.: Statistical Mechanics of Cellular Automata.Rev.Mod.Phys 55(3), 601–644 (1983)

[32] Wolfram, S.: Universality and complexity in cellular au-tomata. Physica D 10, 1–35 (1985)

[33] Wolfram, S.: Cellular Automata as models of complexity.Nature 311, 419–424 (Oct 1984)

[34] Wolfram, S.: Complex systems theory. Tech. rep., Institutefor Advanced Study, Princton, NJ 08540 (6-7 October 19841985), presented at Santa Fe Workshop on ”A response tothe challenge of emerging synthesis in science”

[35] Wolfram, S.: Twenty problems in the theory of cellular au-tomata. Physica Scripta T9, 170–183 (1985)

[36] Wolfram, S.: Theory and Applications of Cellular Au-tomata. World Scientific (1986)

[37] Wylie, B.J.: Application of Two-Dimensional Cellular Au-tomaton Lattice-Gas Models to the Simulation of Hydrody-namics. Ph.D. thesis, Physics Department, Edinburgh Uni-versity (1990)

[38] Yey, W.C., Lin, Y.C., Chung, Y.Y.: Performance analysis ofcellular automata monte carlo simulation for estimating net-work reliability. Expert Systems with Applications 36(5),3537–3544 (2009)