Computational Universality and 1/f Noise in Elementary Cellular Automata
aa r X i v : . [ n li n . C G ] J u l Computational Universality and 1 /f Noise inElementary Cellular Automata
Shigeru NinagawaAugust 13, 2018
Abstract
It is speculated that there is a relationship between 1 /f noise and com-putational universality in cellular automata. We use genetic algorithms tosearch for one-dimensional and two-state, five-neighbor cellular automatawhich have 1 /f -type spectrum. A power spectrum is calculated fromthe evolution starting from a random initial configuration. The fitnessis estimated from the power spectrum in consideration of the similarityto 1 /f -type spectrum. The result shows that the rule with the highestaverage fitness has a propagating structure like other computationally uni-versal cellular automata, although computational universality of the rulehas not been proved yet. Cellular automata (CAs) is a d -dimensional array with a finite automaton re-siding at each site. Each automaton called cell takes the states of neighbouringcells as input and makes the transition of its state according to a set of tran-sition rules. CAs are also considered to be spatially and temporally discretedynamical systems with large degrees of freedom. It was proved that elemen-tary CA (ECA), namely one-dimensional and two-state, three-neighbor CA rule110 is computationally universal [1]. The ability to perform any algorithms iscalled computational universality. In addition, the evolution of rule 110 startingfrom a random initial configuration exhibits 1 /f noise [2]. Another example ofcomputationally universal CA is the Game of Life (LIFE) [3]. LIFE is a two-dimensional and two-state, nine-neighbor outer totalistic CA. It is supposedthat LIFE is capable of supporting universal computation, while the evolutionstarting from a random initial configuration has 1 /f -type spectrum [4]. Theseresults suggest that there is a relationship between computational universalityand 1 /f noise in CAs.However the range of frequencies in the power spectra of ECA rule 110 thatfits in power law is not broad. This is caused by the periodic background thatis a periodic pattern typically observed in the evolution of rule 110. Since theperiodic background does not play an essential role for performing computation,1e can guess that the essential feature of the evolution of rule 110 as a computingprocess is not lost by the removal of the periodic background and that 1 /f -typespectrum becomes clear by removing the periodic background.In this paper we study the influence of the removal the periodic backgroundon the shape of power spectrum of several ECA rules. In section 2 we explainthe method of spectral analysis of elementary cellular automata. In the follow-ing three sections we deals with ECA rule 110 as well as rule 54 and rule 62which exhibit power law in power spectrum. In the final section we discuss therelationship between computational universality and 1 /f noise in CAs. Let s x ( t ) ∈ { , } be the value of site x at time step t in an elementary CA. Thesite value evolves by iteration of the mapping, s x ( t + 1) = F ( s x − ( t ) , s x ( t ) , s x +1 ( t )) . (1)Here F is an arbitrary function specifying the elementary CA rule. The elemen-tary CA rule is determined by a binary sequence with length 2 = 8, F (1 , , , F (1 , , , · · · , F (0 , , . (2)Therefore the total number of possible distinct elementary CA rules is 2 =256 and each rule is abbreviated by the decimal representation of the binarysequence as used in [5]. Out of the 256 elementary CA rules 88 of them remainindependent (appendix of [7]).The discrete Fourier transformation of a time series of states s x ( t ) of site x for t = 0 , , · · · , T − s x ( f ) = 1 T T − X t =0 s x ( t )exp( − i πtfT ) f = 0 , , · · · , T − . (3)The power per site is defined as S ( f ) = N − X x =0 | ˆ s x ( f ) | , (4)where N denotes the total number of sites and the summation is taken in allsites. The period of the component at a frequency f in a power spectrum is givenby T /f . Throughout this paper we employ periodic boundary conditions wherethe sites are connected in a circle and each array is started from a random initialconfiguration where each site takes state 0 or state 1 randomly with independentequal probability.The spectral analysis on the evolution of 88 independent ECAs starting fromrandom initial configuration revealed that the power spectra of most of the rulesis white noise or Lorenzian type and that rule 54 , rule 62 and rule 110 has the2igure 1: Space-time pattern (left) and the filtered one (right) of rule110 startingfrom a random initial configuration with 200 cells for 200 time steps.power spectra of power law and especially rule 110 exhibits 1 /f noise duringthe longest time steps [2].we focus on the ECA rules which have power spectrum of power law, namelyrule 54, rule 62 and rule 110. The rule function of rule 110 is given by the following:1110 1101 1011 1000 0111 0101 0011 0000 . The upper line represents the 8 possible states of neighborhood and the lowerline specifies the state of the center cell at the next time step.Figure 1 (left) shows a typical example of the space-time pattern of rule 110starting from a random initial configuration of 200 cells for 200 time steps. Inspace-time pattern, configurations obtained at successive time steps in the evo-lution are shown on successive horizontal lines in which black squares representsites with value 1, white squares sites with value 0. We can observe periodicbackground of small white triangles with period seven in the space-time pattern.Figure 2 (left) is the power spectrum of rule 110 calculated from the evolutionstarting from a random initial configuration of 4000 cells for 4096 time steps.There are peaks at f = 585 (period:7) and its harmonics in the spectrum. Theexponent β of power spectrum is estimated by the least-squares fitting of thepower spectrum S ( f ) by ln( S ( f )) = α + β ln( f ). The residual sum of squares σ is given by σ = 1 f r f r X f =1 (ln( S ( f )) − α − β ln( f )) , (5)3 S ( f ) f 0.0001 0.001 0.01 0.1 1 10 100 1 10 100 1000 10000 S ( f ) f Figure 2: Left: Power spectrum of of rule 110. The broken line representsthe least-squares fitting of the power spectrum in the range of f = 1 ∼ β = − .
24, the residual sum of squares σ = 0 . β = − . σ = 0 .
1 1 1 1 11 0 0 0 11 0 0 1 11 0 1 1 1 1 1
Figure 3: Template used to remove the periodic background of rule 110.where f r is the number of data used for the calculation of σ and we set f r = 100.The broken line in the power spectrum represents the least-squares fitting of thepower spectrum S ( f ) in the range of f = 1 ∼
100 with the exponent of powerspectrum β = − .
24 and the residual sum of squares σ = 0 . s i g m a ^ betaoriginalfiltered Figure 4: Scattergram of the exponents β and the residual sum of squares σ of1000 power spectra of rule 110 calculated from the original space-time patterns(+) and from the filtered ones ( × ).left to the lower right searching for a section coincident with the template inFig. 3 and change the state zero into the state one in the matched section. Byremoving the periodic background from the original space-time pattern, we canget a filtered space-time pattern.Figure 1 (right) shows the filtered space-tine pattern obtained from the orig-inal one in Fig. 1 (left) by means of the algorithms mentioned above. Thecharacteristic behavior in the evolution of rule 110 , that is, the stationary andpropagating patterns and their interaction are readily apparent compared to theoriginal space-time pattern.Figure 2 (right) is the power spectrum calculated from the filtered space-time pattern obtained by removing the periodic background from the one whichwas employed to calculate the power spectrum shown in Fig. 2 (left). The peakat f = 585 (period:7) and its harmonics considerably contract and the spectrumseems to fit in power law in broader range of frequencies than the original one.The exponent of power spectrum estimated by the least squares method in therange of frequencies f = 1 ∼
100 is β = − .
85 and the residual sum of squaresis σ = 0 . β and the residual sum of square σ of power spectrum in the range of frequencies f = 1 ∼
100 for 1000 runsstarting from random initial configuration of 4000 cells for 4096 time steps.Figure 4 is the scattergram of the exponents β and the residual sum of squares σ of power spectra calculated from the original space-time patterns (+) andfrom the filtered ones ( × ). The pairs of parameters ( β , σ ) of the originalpower spectra and the filtered ones are evidently separated. The 95% confi-dence interval of population mean of β and σ are h β i = − . ± .
001 and h σ i = 0 . ± . h β i = − . ± .
002 and5igure 5: Space-time pattern (left) and the filtered one (right) of rule 54 startingfrom a random initial configuration with 200 cells for 200 time steps. h σ i = 0 . ± . Rule 54 and rule 62 are two other rules which exhibit power law in powerspectrum than rule 110 among 88 ECA rules. In this section we study theproperty of these rules.Figure 5 (left) shows the space-time pattern of rule 54 starting from a randominitial configuration with 200 cells for 200 time steps. The periodic backgroundof rule 54 has spatioal period 4 and temporal period 4. Figure 6 (left) is thepower spectrum of rule 54 calculated from the evolution starting from a randominitial configuration of 4000 cells for 4096 time steps. There are peaks at f =1024 (period:4) and its harmonics in the spectrum. The exponent of the powerspectrum estimated by the least squares method in the range of frequencies f = 1 ∼
100 is β = − .
909 and the residual sum of squares is σ = 0 . X i =0 s x + i ( t ) mod 2 → s x ( t ) . (6)Figure 5 (right) shows the filtered space-tine pattern obtained from the leftone. We can easily observe the stationary and propagating patterns and theirinteraction in the filterd space-time pattern in Fig. 5 (right).Figure 6 (right) shows the power spectrum calculated from the filtered space-time pattern obtained by removing the periodic background from the one whichwas employed to calculate the power spectrum shown in Fig. 6 (left). The peakat f = 1024 and its harmonics become blunt compared to the original one. Theexponent of power spectrum estimated by the least squares method in the rangeof frequencies f = 1 ∼
100 is β = − .
086 and the residual sum of squares is6 S ( f ) f 0.001 0.01 0.1 1 10 100 1 10 100 1000 10000 S ( f ) f Figure 6: Left: Power spectrum of rule 54. The broken line represents theleast-squares fitting of the power spectrum in the range of f = 1 ∼
100 withthe exponent β = − . σ = 0 . β = − . σ = 0 . -0.005 0 0.005 0.01 0.015 0.02 0.025-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 s i g m a ^ betaoriginalfiltered Figure 7: Scattergram of exponents β and the residual sum of squares σ of1000 power spectra of rule 54 estimated from the original space-time patterns(+) and from the ones obtained by eliminating periodic background ( × ). σ = 0 . β and the residual sum of squares σ of 1000 power spectra of rule 54 calculated from the original space-timepatterns (+) and from the filtered ones ( × ). The 95% confidence interval ofpopulation mean of β and σ are h β i = − . ± .
002 and h σ i = 0 . ± . h β i = − . ± .
001 and h σ i = 0 . ± . f = 1365 (period:3). The exponent of the power spectrum estimated by theleast squares method in the range of frequencies f = 1 ∼
100 is β = − .
315 andthe residual sum of squares is σ = 0 . X i =0 s x + i ( t ) mod 2 → s x ( t ) . (7)Figure 8 (right) shows the filtered space-tine pattern obtained from the leftone. We can easily observe the stationary and propagating patterns and theirinteraction in the filterd space-time pattern in Fig. 8 (right).Figure 10 (right) shows the power spectrum calculated from the filteredspace-time pattern obtained by removing the periodic background from the onewhich was employed to calculate the power spectrum shown in Fig. 10 (left).The exponent of power spectrum estimated by the least squares method in therange of frequencies f = 1 ∼
100 is β = − .
234 and the residual sum of squaresis σ = 0 . β and the residual sum of squares σ of 1000 power spectra of rule 62 calculated from the original space-timepatterns (+) and from the filtered ones ( × ). The 95% confidence interval ofpopulation mean of β and σ are h β i = − . ± .
002 and h σ i = 0 . ± . h β i = − . ± .
001 and h σ i = 0 . ± . S ( f ) f 0.0001 0.001 0.01 0.1 1 10 100 1 10 100 1000 10000 S ( f ) f Figure 9: Power spectrumFigure 10: Left: Power spectrum of rule 62. The broken line represents theleast-squares fitting of the power spectrum in the range of f = 1 ∼
100 withthe exponent β = − . σ = 0 . β = − . σ = 0 . s i g m a ^ beta "n4000T4096.fit""n4000filT4096.fit" 0 0.05 0.1 0.15 0.2-1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 s i g m a ^ beta Figure 11: Scattergram of exponents β and the residual sum of squares σ of1000 power spectra of rule 62 estimated from the original space-time patterns(left) and from the filtered ones (right).9 Discussion
Generally speaking, the process capable of supporting computation needs thethree kinds of functions on information, that is, transmission, storage, and oper-ation of information. The transmission of information is achieved by propagat-ing patterns and the storage is done by stationary patterns while the operationof information is carried on by the interaction between those patterns.It is believed that rule 62 in not capable of supporting universal computationbecause the way in which information is transmitted is highly constrained [8].As an additional experiment, we calculated the exponent β of power spec-trum under the same conditions except for the time steps, T = 1024 and 2048.The 95% confidence interval of population mean of β are h β i = − . ± . T = 1024 h β i = − . ± . T = 2048 in original power spectraand h β i = − . ± . T = 1024 h β i = − . ± .
001 for T = 2048 infiltered ones. In the original power spectra | β | becomes large as the time steps T gets long while β hardly varies with T in the filtered power spectra. References [1] Cook, M.: Universality in elementary cellular automata, Complex Systems (2004) 1–40[2] Ninagawa, S.: 1/f Noise in elementary cellular automaton rule 110, Com-plex Systems XXXX (XXXX) XX–XX[3] Berlekamp, E.R., Conway, J.H., and Guy, R.K.:
Winning Ways for YourMathematical Plays , Vol.2, Academic Press, New York (1982)[4] Ninagawa, S., Yoneda, M., and Hirose, S.: 1 /f fluctuation in the ”Gameof Life”, Physica D (1988) 49–52[5] Wolfram, S.: Statistical mechanics of cellular automata, Rev. Mod. Phys., (1983) 601–644[6] Wolfram, S. (editor): Theory and Applications of Cellular Automata , WorldScientific, Singapore (1986)[7] Li, W., and Packard, N.: The structure of the elementary cellular automatarule space, Complex Systems (1990) 281–297[8] Wolfram, S.: A New Kind of Science , Wolfram Media, Champaign (2002)[9] Pomeau, Y., and Manneville, P.: Intermittent transition to turbulence indissipative dynamical systems, Commun. Math. Phys. (1980) 189–197[10] Boccacra, N., Nasser, J., and Roger, M.: Particlelike structures and theirinteractions in spatiotemporal patterns generated by one-dimensional de-terministic cellular automaton rules, Physical Review A (1991) 866–8751011] Mart´ınez, G. J., Adamatzky, A., and McIntosh H. V.: Phenomenology ofglider collisions in cellular automaton rule Rule 54 and associated logicalgates, Chaos, Solitons and Fractals28