Power Spectral Analysis o Elementary Cellular Automata Shigeru Ninagawa Division o Inormation and Computer Science, Kanazawa Institute o Technology, 7- Ohgigaoka, Nonoichi, Ishikawa 92-850, Japan Spectral analysis o elementary cellular automata is perormed. A power spectrum is calculated rom the evolution o 88 independent rules starting rom random initial conigurations. As a result, it is ound that rule 0 exhibits / noise during the longest time steps. Rule 0 has proved to be capable o supporting universal computation. These results suggest that there is a relationship between computational universality and / noise in cellular automata.. Introduction Cellular automata (CAs) are spatially and temporally discrete dynamical systems with large degrees o reedom. Since spectral analysis is one useul method or investigating the behavior o dynamical systems [], it is reasonable to apply it to studying the behavior o CAs. In this paper we deal with elementary CAs (ECAs), namely one-dimensional and twostate, three-neighbor CAs. ECAs have been investigated in detail or their simplicity ([2], appendix in [3]). Although the ECA rule space is not large, there is a wide variety o rules which exhibit regular, chaotic, or complex behavior. Spectral analysis was used or the investigation o the spatial structures produced by ECAs in connection with regular languages in [4], whereas we ocus on the temporal behavior o ECAs. 2. Spectral analysis o elementary cellular automata Let x i (t) be the value o site i at time step t in an ECA. The value o each site is speciied as 0 or. The site value evolves by iterating the mapping x i (t ) F(x i (t), x i (t), x i (t)). () Here F is an arbitrary unction speciying the ECA rule. The ECA rule is determined by a binary sequence with length 2 3 8, F(,,), F(,,0),..., F(0,0,0). (2) Electronic mail address: ninagawa@inor.kanazawa-it.ac.jp. ; 2008 Complex Systems Publications, Inc.
400 S. Ninagawa Thereore the total number o possible distinct ECA rules is 2 8 256 and each rule is abbreviated by the decimal representation o the binary sequence (2) as used in [2]. A parameter Λ is deined as the density o ones in the binary sequence (2) [5]. Out o the 256 ECA rules, 88 o them remain independent (appendix in [6]). With a ew exceptions, we use as representative rules those with smaller Λ values and the smaller decimal rule numbers adopted in [6]. The discrete Fourier transorm o the time series o state x i (t) or t 0,,..., T isgivenby T ˆx i ( ) x T i (t)exp i 2Πt T t 0. (3) It is natural to deine the power spectrum o a CA as N S( ) ˆx N i ( ) 2, (4) i where N means array size and the summation is taken over all cells in the array. The power S( ) at requency intuitively means the strength o the periodic vibration with period T/ in the evolution o observation length T. Another deinition o power spectrum is given by calculating spatial density beore the Fourier transorm. It might be useul to investigate the behavior o CAs with an oscillating density in time, although we do not ocus on them in this research. Since we are interested in the dynamics that a CA rule brings, rather than the behavior rom a particular initial coniguration which is elaborately designed, throughout this paper we use random initial conigurations in which each site takes state 0 or randomly with independent equal probabilities. Generally, the evolution o one-dimensional CAs tends to depend more heavily on an initial coniguration than that o two-dimensional CAs. So we use some dierent initial conigurations to grasp the typical behavior o a CA. In addition, an array size larger than 400 seems required to avoid singular behavior. We adopt periodic boundary conditions where each end o the array is connected like a ring. Spectral analysis revealed that 88 independent ECAs can be classiied into several categories according to the shape o their power spectra. Theresultsareasollows. 2. Category : Extremely low power density The power spectra in this category are characterized by an extremely low power density at almost all requencies. Figure shows the spacetime patterns (let) and the power spectra (right) o rule 32 (top) and rule
Power Spectral Analysis o Elementary Cellular Automata 40 Rule 32 0.0 #32 S() 0 00 200 300 400 500 600 Rule 08 0.0 #08 S() 0 00 200 300 400 500 600 Figure. Space-time patterns (let) and power spectra (right) o rule 32 (top) and rule 08 (bottom). The space-time patterns consist o 00 cells or 00 time steps. The power spectra are calculated rom the evolution o 700 cells or 024 time steps. 08 (bottom). Space-time patterns show conigurations obtained at successive time steps on successive horizontal lines in which black squares represent sites with value, white squares value 0. The power spectrum is calculated rom the evolution o 700 cells or 024 time steps. Only hal o the components o the spectrum are shown since the other hal are redundant. The y-axis is plotted on a logarithmic scale. The power density in the power spectrum o rule 32 is extremely low at all requencies except 0. This is ascribed to the initial transient behavior which vanishes ater the irst ew time steps. Other rules with similar power spectra are rules 0, 4, 8, 2, 3, 26, 40, 44, 72, 76, 77, 78, 04, 28, 32, 36, 40, 60, 64, 68, 72, 200, and 232. These rules are considered to be in class I, or a part o class II, with evolutions that lead to stable structures according to Wolram s classiication scheme in [7]. The power spectrum o rule 08 has a peak at 52 which is caused by periodic structures with period 2. Rules like this are a part o class II with evolutions that lead to periodic structures, namely rules, 5, 9, 23, 28, 29, 33, 37, 50, 5, 33, 56, and 78.
402 S. Ninagawa 2.2 Category 2: Broad-band noise The power spectra in this category are characterized by broad-band noise. In general, broad-band noise in a power spectrum means that the original time series is nonperiodic. This category is divided into two subcategories. A typical example o the irst subcategory 2-A is shown in the top o Figure 2. The power spectrum o rule 6 has a higher power density at all requencies compared with those in category. The space-time pattern o rule 6 yields uniormly shiting structures which move one cell let every time step. So the evolution with array size N becomes periodic with period N under periodic boundary conditions. Thereore the power spectrum with observation length T N has a undamental peak component at T/N and several harmonic peak components, while the power spectrum with T N is similar to white noise, which means the time series is random. But its randomness is mainly included in an initial coniguration rather than generated by a rule. Subcategory 2-A consists o rules 2, 3, 6, 7, 9, 0,, 4, 5, 24, 25, 26, 27, 34, 35, 38, 4, 42, 43, 46, 56, 57, 58, 74, 30, 34, 38, 42, 52, 54, 62, 70, and 84. Rule 6 0.0 #6 S() 0 00 200 300 400 500 600 Rule 90 S() 0.0 #90 0 00 200 300 400 500 600 Figure 2. Space-time patterns (let) and power spectra (right) o rule 6 (top) and rule 90 (bottom).
Power Spectral Analysis o Elementary Cellular Automata 403 The power spectra in subcategory 2-B are characterized by white noise. The bottom o Figure 2 shows the space-time pattern (let) and the power spectrum (right) o rule 90. The power spectrum in subcategory 2-B has an almost equal power density at all requencies except or rules 8 and 46, which have a peak with period 2. This result implies that evolution in subcategory 2-B is virtually orderless, although it is causally determined by a rule. Randomness in the evolution o subcategory 2-B owes much to the rule itsel rather than initial conigurations. Subcategory 2-B consists o rules 8, 22, 30, 45, 60, 90, 05, 06, 29, 46, 50, and 6 which coincide with class III [6]. By using an initial coniguration containing a single site with value, the dierence between subcategories 2-A and 2-B becomes clear. Subcategory 2-A resembles category in the shape o its power spectrum with the exceptions o rules 26 and 54, which exhibit broad-band noise. The power spectrum in subcategory 2-B, however, remains broad-band noise with the exception o rule 06, having an extremely low power density. Rules 26, 5, and 06 are considered as intermediates between these two subcategories. 2.3 Category 3: Power law spectrum The power spectrum in this category is characterized by power law at low requencies. Figure 3 shows the space-time patterns (let) and the power spectra (right) o rules 54 (top), 62 (middle), and 0 (bottom). The x and y axes in the power spectra are plotted on a logarithmic scale and the broken line represents the least square itting o the power spectrum rom to 0 by ln S( ) Α Βln,withΒ 0.5 or rule 54, Β.2 or rule 62, and Β.5 or rule 0. Rules 54 and 0 are considered to be in class IV, which is supposed to be capable o supporting universal computation [7] while there is a guess that the evolution o rule 62 is too simple to be universal [8]. 2.4 Exceptional rules There are two exceptional rules which do not belong to any o the three categories. Rule 73 is called locally chaotic because the array is divided into some independent domains by stable walls and chaotic patterns are generated in each domain [6]. Each domain has an individual period which contributes the peak in the power spectrum. Figure 4 shows the space-time pattern (let) and the power spectrum (right) o rule 73. The power spectrum o rule 204 has zero power density at all requencies except or 0 because the evolution o rule 204 retains the initial coniguration.
404 S. Ninagawa Rule 54 #54 ^(-0.5) 0.0 S() 0 00 000 Rule 62 #62 ^(-.2) 0.0 S() 0 00 000 Rule 0 0.0 ^(-.5) S() #0 0 00 000 Figure 3. Space-time patterns (let) and the power spectra (right) o rules 54 (top), 62 (middle), and 0 (bottom). The x and y axes in the power spectra are plotted on a logarithmic scale. The broken line in the power spectra represents the least square itting o the power spectrum rom to 0 by ln S( ) Α Βln, Β 0.5 or rule 54, Β.2 or rule 62, and Β.5 or rule 0.
Power Spectral Analysis o Elementary Cellular Automata 405 Figure 4. Space-time pattern (let) and power spectrum (right) o rule 73. 3. / noise in elementary cellular automata A luctuation with a power spectrum S( ) that is inversely proportional to requency is called / noise [9]. / noise can be observed in a wide variety o phenomena such as the voltage o vacuum tubes, the rate o traic low, and the loudness o music. However, its origin is not well understood. The most controversial problem in / noise is whether / noise lasts orever or not. Generally speaking, i there is inite correlation time Τ in a luctuation, the power spectrum with an observation length o T > Τ has an almost equal power density at requencies smaller than /Τ. Likewise i there is inite correlation time Τ in the evolution o a CA, the power spectrum with an observation length o T > Τ has an almost equal power density at requencies smaller than T/Τ. So the power spectrum becomes close to a lat line at low requencies as the observation length T becomes longer than the correlation time Τ. Moreover we can guess that the correlation time Τ depends on array size N because the average transient time steps T ave o rule 0, which is supposed to be relevant to Τ, increases algebraically with array size N, T ave N Α,withΑ.08 [0]. Thereore we investigate the value o the exponent Β o the power spectra S( ) Β or various observation lengths and array sizes to compare the behaviors o rules 0, 54, and 62. Since one-dimensional CAs in general have larger variations in the value o Β with initial conigurations than two-dimensional CAs, we calculate the value o Β averaged over some dierent random initial conigurations. Figure 5 shows the average Β o the exponent estimated by the least square itting o the power spectrum rom to 0 by ln S( ) Α Βln( ) over 400 random initial conigurations as a unction o array size N and observation lengtht or rules 0 and 54 (top), and rules 0 and 62 (bottom). The solid line represents Β o rule 0 and the broken line rules 54 (top) and 62 (bottom). Β o rules 0, 54,
406 S. Ninagawa <β> #0 #54 0-0.2-0.4-0.6-0.8 - -.2 -.4 0 200 400 N 600 800 6000 5000 4000 3000 T 2000 000 200 000 <β> #0 #62 0-0.2-0.4-0.6-0.8 - -.2 -.4 0 200 400 N 600 800 6000 5000 4000 3000 T 2000 000 200 000 Figure 5. Average Β o the exponent estimated by the least square itting o the power spectrum rom to 0 by ln S( ) Α Βln( ) or 400 random initial conigurations as a unction o array size N and observation length T or rules 0 and 54 (top), and or rules 0 and 62 (bottom). The solid line represents Β o rule 0 and the broken line rules 54 (top) and 62 (bottom). Β o rules 0, 54, and 62 at N 200, T 8000 equal.2, 0.4, and 0.6 respectively. and62atn 200, T 6000 equal.2, 0.4, and 0.6 respectively. On the whole, Β o rules 0, 54, and 62 increases as the array size N decreases or the observation length T increases. The reason Β o rule 0 at N 300 is close to zero is that singular behavior tends to occur requently in one-dimensional CAs with small array sizes. When the array size is larger than 300, Β o rule 0 is smaller than those o rules 54 and 62. This result means that the evolution o rule 0 exhibits / noise during the longest observation length o the ECAs.
Power Spectral Analysis o Elementary Cellular Automata 407 4. Origin o / noise in rule 0 In this section we compare two examples o the evolution o rule 0 rom distinct random initial conigurations to study the origin o / noise in the evolution o rule 0. Figure 6 shows a set o space-time patterns o the evolution o rule 0 rom a random initial coniguration o 200 cells or 000 time steps. The irst 200 time steps are shown on the let and the last 200 time steps are on the right. The evolution at irst exhibits transient behavior but becomes periodic at t 306 with period 750. The space-time pattern on the right shows that several gliders are moving monotonously within a periodic background. Figure 7 shows another set o space-time patterns rom dierent random initial conigurations. The evolution keeps transient behavior at t 999 although it eventually becomes periodic at t 246 with period 400. The space-time pattern on the right shows that several gliders interact complexly within a periodic background. Figure 8 shows the power spectra calculated rom the space-time patterns shown in Figures 6 and 7. The let one is calculated rom the evolution shown in Figure 6 and the right one rom Figure 7. The broken line represents the least square itting o the power spectrum rom to 0 by ln S( ) Α Βln, with Β 0.5 (let) and Β.5 (right). The let one is too lat to be considered as type /. Comparing these two examples o the evolution suggests that the dierence in the exponent o power spectra at low requencies depends on the duration o transient behavior. t 0 99 t 800 999 Figure 6. Space-time patterns o rule 0 rom a random initial coniguration. Each picture is 200 cells wide. The irst 200 steps o evolution are shown on the let and the last 200 steps on the right. The evolution becomes periodic at t 306 with period 750. Several gliders are moving monotonously in the last 200 time step evolution.
408 S. Ninagawa t 0 99 t 800 999 Figure 7. Space-time patterns o rule 0 rom a dierent random initial coniguration than the one used in Figure 6. Several gliders interact complexly in the last 200 steps o evolution. The evolution eventually becomes periodic at t 246 with period 400. Figure 8. Power spectra o rule 0 rom two random initial conigurations. The let one is calculated rom the evolution in Figure 6 and the right one rom Figure 7. The broken line represents the least square itting o the spectrum rom to 0 by ln S( ) Α Βln,withΒ 0.5 (let) and Β.5 (right). Intermittency in chaotic dynamical systems is one o the main mechanisms o / noise []. A system with a particular parameter value exhibits periodic behavior which is disrupted occasionally and irregularly by a burst. This burst persists or a inite duration, it stops and a new periodic behavior starts. Intermittent chaos occurs when the transition rom periodic to chaotic behavior takes place as the parameter is varied. During the transient behavior o rule 0 there are alternating periodic phases in which gliders are shiting in a periodic background without collisions and a burst that is caused by colliding gliders. It seems likely that the recurrence o the periodic phase and the burst in the transient behavior generates intermittency and causes / noise in rule 0.
Power Spectral Analysis o Elementary Cellular Automata 409 5. Conclusion In this research we perormed spectral analysis on the evolution o 88 independent elementary cellular automata (ECAs) starting rom random initial conigurations and concluded that rule 0 exhibits / noise during the longest time steps. Rule 0 has proved to be capable o supporting universal computation [8, 2]. These results suggest that there is a relationship between computational universality and / noise in CAs. But there remains the problem o how to decide array size and observation length to detect / noise in CAs. The average Β o the exponent o the power spectrum o rules 0, 54, and 62 in Figure 5 at N 00, T 6000 equal 0.4, 0.5, and 0.04 respectively. These power spectra are not regarded as type /. These examples make it clear that / noise in ECAs is observed only in a speciic range o array size N and observation length T. Since the guess is that / noise in CAs is caused by transient behavior, as mentioned in section 4, we have to estimate experimentally a proper range o array size and observation length in which the evolution remains transient to observe a type / power spectrum. Another example which possesses both the exhibition o / noise and the capability o supporting universal computation is the Game o Lie (Lie) which is a two-dimensional and two-state, nine-neighbor outer totalistic CA [3]. It is supposed that a universal computer can be constructed on the array o Lie by considering a glider as a pulse in a digital circuit. In addition, the evolution rom random initial conigurations has a type / spectrum [4]. Another important property o Lie is sel-organized criticality where the distribution D(T) otime steps required or the lattice to return to stability ollowing a random single-site perturbation is characterized by D(T) T.6 [5]. Since / noise and sel-organized criticality seem to be interrelated phenomena, rule 0 might have the property o sel-organized criticality as well. It must be noted that the power law in the power spectrum o rule 0 holds or one decade in the range o requencies ( 0) as shown in Figure 8 (right), whereas it holds or three decades ( 000) in Lie [4]. The periodic background which is observed in rule 0, but not in Lie, accounts or this dierence. The three peaks o the power spectrum o rule 0 at 43, 286, and 429 in Figure 8 (right) are the undamental, the second, and the third harmonic components caused by the periodic background with period seven respectively. The interaction between the periodic background and other structures broadens the peaks and contributes to the power in high requencies. The hypothesis o the edge o chaos has evoked considerable controversy [6]. This hypothesis says the ability to perorm universal computation in a system arises near a transition rom regular to chaotic
40 S. Ninagawa behavior such as class IV. So ar various statistical quantities, such as entropy and dierence pattern spreading rate, have been proposed to detect class IV quantitatively [7]. The exponent o the power spectrum at low requencies might be another useul index. Acknowledgments I would like to thank the reeree or instructive comments on the original manuscript. Reerences [] J. Crutchield, D. Farmer, N. Packard, R. Shaw, G. Jones and R. J. Donnelly, Power Spectral Analysis o a Dynamical System, Physics Letters, 76A (980) 4. [2] S. Wolram, Statistical Mechanics o Cellular Automata, Reviews o Modern Physics, 55 (983) 60 644. [3] S. Wolram (editor), Theory and Applications o Cellular Automata (World Scientiic, Singapore, 986). [4] W. Li, Power Spectra o Regular Languages and Cellular Automata, Complex Systems, (987) 07 30. [5] C. G. Langton, Studying Artiicial Lie with Cellular Automata, Physica D, 22 (986) 20 49. [6] W. Li and N. Packard, The Structure o the Elementary Cellular Automata Rule Space, Complex Systems, 4 (990) 28 297. [7] S. Wolram, Universality and Complexity in Cellular Automata, Physica D, 0 (984) 35. [8] S. Wolram, A New Kind o Science (Wolram Media, Champaign, IL, 2002). [9] M. S. Keshner, / Noise, Proceedings o the IEEE, 70 (982) 22 28. [0] W. Li and M. G. Nordahl, Transient Behavior o Cellular Automaton Rule 0, Physics Letters A, 66 (992) 335 339. [] Y. Pomeau and P. Manneville, Intermittent Transition to Turbulence in Dissipative Dynamical Systems, Communications in Mathematical Physics, 74 (980) 89 97. [2] M. Cook, Universality in Elementary Cellular Automata, Complex Systems, 5 (2004) 40. [3] E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways or Your Mathematical Plays, Volume 4 (A. K. Peters, Wellesley, second edition, 2004).
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