Definition and Identification of Information Storage and Processing Capabilities as Possible Markers for Turing-universality in Cellular Automata
DDefinition and Identification ofInformation Storage and ProcessingCapabilities as Possible Markers forTuring-universality in Cellular Automata
Yanbo Zhang
Physical Department, University of Science and Technology of ChinaHefei, Anhui, P. R. China
To identify potential universal cellular automata, a method is devel-oped to measure information processing capacity of elementary cellularautomata. We consider two features of cellular automata: Ability tostore information, and ability to process information. We define localcollections of cells as particles of cellular automata and consider in-formation contained by particles. By using this method, informationchannels and channels’ intersections can be shown. By observing thesetwo features, potential universal cellular automata are classified intoa certain class, and all elementary cellular automata can be classifiedinto four groups, which correspond to S. Wolfram’s four classes: 1) Ho-mogeneous; 2) Regular; 3) Chaotic and 4) Complex. This result showsthat using abilities of store and processing information to characterizecomplex systems is effective and succinct. And it is found that theseabilities are capable of quantifying the complexity of systems.
1. Introduction
A universal system is a system that can execute any computer program.In other words, it is feasible for it to execute any algorithm [1]. It isfound that some systems with simple rules can be a universal system,such as rule 110 in elementary cellular automata [2, 1, 3]. Some tagsystems and cyclic tag systems are also proved to be universal, whichare also systems with simple rules [2, 3, 10]. Glider system, which isan idealized system to simulate particle process of real physics system,was also proved to be a universal system [2]. And particle machines inperiodic backgrounds was proved to be universal [7].The widespread existence of universal systems implies that someprocess with simple rules in the real world may be able to executesome algorithms or any algorithm. Because of the significant amountof algorithms, these systems’ behaviors can be changeful and complex,which was considered as a potential origin of complexity in [3, 8].
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For cellular automata can show the wide variety of complex phe-nomena in the real world, and cellular automata are also sufficientgenerality for a wide variety of physical, chemical, biological, and othersystems [11]. Identifying universal cellular automata will help peopleunderstand origins of cellular automata’s behaviors and find key dy-namics of computation.In this study, a method is developed to identify potential universalelementary cellular automata. Two abilities of a system are considered:1) Ability to store information and 2) Ability to process information.We found these two features can identify potential universal cellularautomata and quantify the complexity of systems.
Cellular Automata (CA for singular, CAs for plural) are ideal modelsfor physical systems in which space and time are discrete. And ele-mentary cellular automata (ECA for singular, ECAs for plural) is oneof the simplest kind of CAs.ECAs are dynamic systems defined by deterministic rules, workingon a 1-dimension list { c n } with n cells. Rules can be expressed byfunction F : c n ( t + 1) = F [ c n − ( t ) , c n ( t ) , c n +1 ( t )] , (1)where n ∈ Z .Therefore, c n ( t + 1) is the function of itself c n ( t ) and its two im-mediate neighbors: c n − ( t ) and c n +1 ( t ). Each c n ( t ) has two possiblestates, 0 or 1. So there should be a 2 = 8 length list R to definea rule, and there will be 2 = 256 different rules. When R is equalto { , , , , , , , } , by considering it as a binary code, it will equalto 30 in decimal base, which is the ECA rule 30.With a given initial list L , an ECA will apply the function F to allcells parallelly to update L t to L t +1 . i.e., L t F −→ L t +1 . (2)By doing this process repeatedly, a matrix M (rule) = ( L , L , . . . , L t )will be generated, which is the “space–time evolution”. Figure 1 showstwo space–time evolutions generated by ECA rule 30 and ECA rule 110,started with the same L .256 different ECAs can be classified. In this paper, we compareour work with Wolfram’s classification, which are class 1 ∼ Complex Systems , Volume (year) 1–1+
Figure 1.
Evolution of 4 typical rules from class 1 ∼
4. Rule 8 is in class 1,rule 4 is in class 2, rule 30 is in class 3, and rule 110 is in class 4.
2. Methodology
We consider two abilities of ECA rules: Ability to store information,and ability to process information. The ability to store informationwill make the system stable enough and do not have too much noise.Only when information can be stored, information can move stablyin a system, so that the whole system can be related. The ability toprocess information means interactions between information should befound in a system.We define a system can store information when its current localstates can be used to infer previous states at some location. It’s truethat some reversible systems can store all information at the wholesystem, but this information can hardly be used to infer the previousstates because many of them are computational irreducible. Thus, theparticle systems can cover the definition.We identify potential universal ECAs based on a theorem proposedin [7], which considers particle-like structures and their behavior insystems to identify Turing machines and UTM.A method was developed to extract particle patterns from ECAs to
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Input Output da t a and c on f i gu r a t i on s t Figure 2.
A typical particle machine. build “particle machines”, and to measure their computation ability bytaking into account their features. First, it is necessary to introduceparticles machines and define particles in ECAs.
A particles machine (PM), is a system in which particles can move,collide, annihilate and generate in a homogeneous medium. Figure 2shows a typical PM. Data and configurations are injected from left inthe form of particles, and by executing this system, particles will haveinteractions. Lines and dotted lines in this figure represent the pathsof particles. After time t , the system will generate an output. Theidentity of a particle includes position, phase, and velocity. Duringcollisions, particles can alter their identities, or be generated or anni-hilated. These changes of particles can be considered as a function ofparticles that participate in the collision, which is the collision func-tion. Some particles machines are proved to be Turing machines oruniversal Turing machine (UTM) in [7]. A PM is at least a Turingmachine when: 1) Identity of particles can change during collisions;2) Collision function is depending on identities of particles. For thefirst requirement, the identity of particles can change during collisions,also means new particles can be generated during collisions. And thesecond requirement means the result of a collision should depend ontypes of particles that participate in the collision. If no particles canbe generated or annihilated in collisions in a PM, then the PM is nota UTM. We define a local grid of cells in M (rule) as a particle in ECAs. Here weconsider one kind of particles: Their sequence may change periodicallyor not change through time. We call them “elementary particles”. Itwill be practical if we start with these simple kind of particles.Particles contain information, so that information can move in space, Complex Systems , Volume (year) 1–1+
Target particle : A. B.
Figure 3. A) . An illustration of how it takes a “target particle” P from amatrix generated by ECA rule 110. The rectangle with black frame is thetarget particle P , and gray rectangles mean there has a similar sequenceas P , which are linear particles L s. In this figure, the P and L s are ,and found 54 same sequences. B) . A figure of matrix M (110) . M (110) t,x = p when there is a L at { t, x } , or L t,x = 0. Dots at { t, x } means M (110) t,x = p . and have interaction with other information, which is a kind of compu-tation [9]. All identities of particles: Location, velocity, and sequence,can be computed by collisions. And all of these identities can be pre-served if there are no collisions.To extract particles’ identities from ECAs’ space–time evolutions, acertain sequence should be chosen for the research. We need to choosea sequence as a particle to study, which is the “target particle”. AsFigure 3.A shows, we choose target particle P at the center-bottom of aspace–time evolution and mark the same sequences as “linear particles” L s, which are the dots in Figure 3.B. P can be explained by the equation P = L t max ( p L , p R ), where L t isthe t –th row of the space–time evolution. p L and p R is the start-indexand the end-index for P .A particle P at ( t ; −→ x ), its location may be ( t (cid:48) ; −→ x (cid:48) ) at time t (cid:48) ( t (cid:48) < t ).We call the particle at ( t (cid:48) ; −→ x (cid:48) ) as P ’s father-particle P f . If let P be P ,the P f will be one of the L s.All L s in P ’s light cone are possible to be the father-particle of P (i.e. P f ), we assume that there is one and only one L is the P f , andeach L has probability p to be the P f . So when there are n L s, theprobability (i.e. p ) for a L i to be a father-particle is: p ( p, n ) = p (1 − p ) n − . (3)All the L i are drawn on a matrix M (rule) , such as Figure 3.B. M (rule) t,x = p when there is a L at { t, x } , or L t,x = 0. We call M (rule) “probability matrix”. The positions with black points will add a num- Complex Systems , Volume (year) 1–1+
Complex Systems average average matrix
Figure 4.
The average matrix M (110) , generated with 10 probability matrices,with p equal to 0 .
01 for Equation (1). ber p . Each black point means there is a linear particle L of P at ( t, x ), ( t, x ) is the location of the black point. M t,x equals to p ( p, n ).The average M (rule) that generated with random initial lists: M (rule) = 1 N N (cid:88) i =1 M (rule)random , (4)will show some patterns that represent particles and particles’ behavior.We call M (rule) “average matrix”. Figure 4 shows how an averagematrix was generated.The meaning of an average matrix is, if a particle is found at thecenter-bottom of a space–time evolution, it may come from position ( t, x )with probability M t,x / (cid:80) t,x M t.x . So the pattern in an average ma-trix represents traces of particles. We calculate the average matrixwith N = 10 for each rule. By observing patterns of average matrices, the identity of particles canbe extracted. A typical average matrix is shown in Figure 4. If particlescan emerge, there will be some lines in the average matrix. Each linerepresents at least one particle, and their variations show interactionsbetween particles.The change of a line’s intensity with time represents interactionsbetween particles. Because if a particle is moving straight without anyinteractions, the lines’ intensity will not change through time. But ifthe particle can be generated by other particles, it will not be foundbefore it was created, so that the intensity will change through time,mostly, the intensity will get higher when t is getting higher. Complex Systems , Volume (year) 1–1+ - t D Rule 110 - t D Rule 54 - t D Rule 18 - t D Rule 76A. B.
Figure 5. A) . Four typical M . B) . The intensity change with time for fourtypical rules. The t -axis is time, and D -axis is the intensity of particle traces.It can be seen that the intensity D ( t ) may change with time for some rules.
3. Result
We get M for all rules, some typical M shown in Figure 5.We get numbers of particles’ traces for each ECA rules, which cor-respond to the number of particles. All traces are straight lines withvarious angles. For rules shown in Figure 5, rule 54 has 3 traces,rule 62 has 2 traces, rule 110 has more than 6 traces, and rule 18 hasa smooth trace. We use T (rule) to represent the count of traces, suchas T (54) = 3, which can be used as a parameter to classify ECAs.The intensity of traces may change through time. The result showsthat they have two kind behaviors: 1) Constant, 2) Variational (mostly,the intensity getting higher when t is getting higher). We use C (rule)to represent the existence of variation, such as C (54) = 1 (1 is varia-tional, 0 is constant ). These two behaviors can be used as a parameterto classify ECAs. In Figure 5, traces in rule 54, 62 and 110 are get-ting more obvious when time t gets higher. Figure 5 shows how theintensity of particle traces variation with time, where D ( t ) = max( L t ).Power law show in some rules, where D ( t ) ∼ ( t max − t ) − α , such asRule 146 and Rule 18, such power law also found by [12] (see Figure 10). To identify Turing machines and potential UTM, the two parameterswe mentioned above will be used to classify ECA rules into four classes.According to the theorem of particle machines [7], when T (rule) ≥ C (rule) = 1, then this ECA rule behave as a Turing machine andpotentially be a UTM. A particle machine that is a Turing machineshould have at least 2 particle traces so that it is possible to haveinteractions between particles. And traces’ intensity should change, Complex Systems , Volume (year) 1–1+
Complex Systems intensity variation (0 means constant, 1 means variational)number of paths mark Wolfram’s class1234
AB DC C process information: store information: Figure 6.
The classification of ECAs, divided by the number of paths and intensity variation . The number of paths associated with the ability to storeinformation, and intensity of variation associated with the ability to pro-cessing information. In each phase, rules will have similar behaviors. In thephase-A, all rules have both a high number of traces ( T ≥
2) and interac-tions that can generate particles, so that it is possible for these rules to havecomplex behaviors. The shape of a point represents its class in Wolfram’sclassification. Each point in this figure represents a rule, and their positionswere moved randomly ( ∼ .
3) so that they can be seen clearly without toomany overlaps. which represents that new particles can be generated during collisions.So all rules can be classified into four classes: A). T ≥ C = 1;B). T < C = 1; C). T < C = 0; D). T ≥ C = 0.Figure 6 shows the final classification for all rules of elementarycellular automata. Each point represents a rule for an elementary cel-lular automaton. The x -axis is “number of traces”, and y -axis repre-sent the existence of information traces’ changes, where 0 means con-stant, 1 means variational. The shape of a point represents its class inWolfram’s classification. Complex Systems , Volume (year) 1–1+
In class A, rules have complex behaviors, and many particles withplentiful interactions can be found. The information here will be storedand processed. Then they can be considered as a Turing machine withenough complexity and computation ability, which was considered tohave connections with Turing universality [8, 14]; In class B, rules willgenerate some random patterns, particles have too many interactionswith the background, so that information traces are dissipated. Theinformation here cannot be stored; In class C, rules will generate con-tinuous or random structures without any complex behavior. Rules inthis class do not have particles or have particles but no interactions.In class D, rules will generate some structures that do not have enoughinteractions, which will not have any complex behavior either. Newparticles cannot be generated during collisions.Class C can be divided into two subclasses, as shown in Figure 6,separated by a dotted line. We use “Rule x ” to express the subclasses.C means the subclass of class C with T equal to 0. C means asubclass of class C with T equal to 1. In C , rules do not have anyparticles, the information here cannot be stored or processed. In C ,rules have particles but do not have interactions between particles. Theinformation here can only be stored but cannot be processed.When going through the dark curve in Figure 6 (anticlockwise), thefrequency of finding interactions is continually growing. And when thefrequency is higher than it in class A, it will generate too much noise,so particles and information will be scattered. When it is lower thanthe frequency in class A, the number of interactions is not enough to docomputation or universal computation, so the behavior is too simpleto get complex behaviors.Some typical rules in these 4 classes show in Table 1. All rules’classification are shown in Figure 7.The relation between this classification and Wolfram’s classificationwas also studied. According to Figure 8, Class C and D have a strongcorrelation to a certain Wolfram class, which is Class 2. While class A , B , and C contain some different Wolfram classes. Here the reducedentropy is used to measure the relation between the two classificationsbecause the Wolfram classification dose not have an order. The reducedentropy is defined as h = H/H max = H/ log n where H max is themaximum that entropy H could be.
4. Discussions
In this study, we consider two abilities as key dynamics for computa-tion:
1. Ability to store information;2. Ability to process information.
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ABC DC AB D C C information store abilityinformation process ability none low highhighlow high information store abilityhigh information process abilitylow information store abilityhigh information process abilityno information store abilitylow information process abilitylow information store abilitylow information process abilityhigh information store abilitylow information process ability Figure 7.
This is the final classification of all ECA rules with the methodintroduced in this paper. In this matrix, each kind of texture or color rep-resents a class defined by this paper (see the column at right side), and thenumbers over each square are the rule indexes. Each texture is associatedwith the ability of processing and storing information, which correspondingto the computation ability. The class A, which have both high informa-tion store and process ability, is considered having high computation ability.Rule 110, which is a UTM, is classified into this class.
The ability to store information means there should be particlesemerge in a system so that information can move in the system. Andin this way, the whole system can be connected and linked to be anentirety, which was considered as a common feature of complex sys-tems. Ability to process information means the system can computeinformation and execute algorithms.By using the coarse-grained method, robust patterns can be found,rules with different computation abilities are classified into a particularclass (class A, shown in Figure 6).All ECA rules can be classified into four classes, which correspondto Wolfram’s classification. All rules in class 1 and most rules in class 2(Wolfram’s classification), was found do not have interactions that cangenerate new particles. Most rules in class 3 are found do not haveenough particles to perform the universal computation. All rules inclass 4 are found classified into class A in this study. For rule 146,183, 18 and 22, which are classified into class 3 (chaotic) by Wolfram,are classified into class A in this study, which means these rules arecapable of doing complex computations. This result corresponds to
Complex Systems , Volume (year) 1–1+ ● ● ● ● ●■ ■ ■ ■ ■ � � � � � � � ���� ���� �������������� �� ���� ����� � � � �� � � � � � �� � � �� �� � � � � � ��� ����������� ������� ● ���� ■ ������� ������� Figure 8.
This figure shows the relation between Wolfram’s classificationand the classification in this paper. The orange line with dot markers is theaverage of W i , which is Wolfram’s classes of the rules in class i of this paper.The average numbers only make sense when all rule in a certain class (ofthis paper) have a same Wolfram class because the Wolfram class do nothave an order. The blue line with square markers is the reduced entropy ofWolfram classes of rules in certain class in this paper, which can measure thecorrelation between these two kinds of classification. The reduced entropy isdefined as h = H/H max = H/ log n , where H is the entropy of W i , and n isthe length of W i . the research [12]. Particles and interactions are found in rule 146,and it is shown that the intensity of traces in the average matrix iscorresponding to [12]. The differences of the classifications between thispaper’s and Wolfram’s come from the different criterions. For example,in Wolfram class 2, some rules shows particle interactions and othersnot, which were classified into different classes in this paper.Since the problems of storing and processing information can befound in various fields, such as chemical systems [13] and hydrodynam-ics [15, 16], and this method is not based on ECAs’ specific features, soit is potentially to be applied to other systems, such as birds flock [17],traffic flow [18], chaotic behaviors [15, 16], and complex networks [19].This method can also be used to quantify the complexity of systems,for UTM was considered having the highest complexity by [3], whichwill make people have a deeper understanding of complex behaviors. Acknowledgements
The author is grateful for suggestions and assistance from Dr. Lingfei Wuin University of Chicago, Dr. Qianyuan Tang in Nanjing University,Dr. Kaiwen Tian in University of Pennsylvania and Dr. Hector Zenilin Karolinska Institutet.
Complex Systems , Volume (year) 1–1+ Complex Systems
References [1] Edwin Roger Banks, “Universality in Cellular Automata,”
IEEE Annual Symposium (IEEE, 1970) , pp. 194–215.doi:10.1109/SWAT.1970.27[2] Matthew Cook, “Universality in Elementary Cellular Automata,”
Com-plex Systems , (1), 2004 pp. 1–40.[3] Stephen Wolfram, A New Kind of Science , Champaign: Wolfram MediaInc, 2002.[4] Genaro J. Martinez, “A Note on Elementary Cellular Automata Clas-sification,”
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International Journal ofBifurcation and Chaos , (09), 2013 pp. 1350159.[6] Hector Zenil, “Compression-based Investigation of the Dynamical Prop-erties of Cellular Automata and Other Systems,” arXiv preprint , 2009arXiv:0910.4042.[7] Mariusz H. Jakubowski, Ken Steiglitz, and Richard K. Squier, “WhenCan Solitons Compute?,” Complex Systems , (1), 1996 pp. 1–22.[8] J¨urgen Riedel, Hector Zenil, “Cross-boundary Behavioural Repro-grammability Reveals Evidence of Pervasive Turing Universality,” arXivpreprint , 2017 arXiv:1510.01671[9] Adamatzky, Andrew, and J´erˆome Durand-Lose. “Collision-based Com-puting,” Handbook of Natural Computing , Berlin Heidelberg: Springer,2012 pp. 1949–1978. “[10] John Cocke and Marvin Minsky, “Universality of Tag Systemswith P = 2,” Journal of the ACM , (1), 1964 pp. 15–20,doi:10.1145/321203.321206[11] Stephen Wolfram, “Statistical Mechanics of Cellular Au-tomata,” Review of Modern Physics , (3), 1983 pp. 601–644,doi:10.1103/RevModPhys.55.601.[12] Paul-Jean Letourneau, “Particle Structures in Elementary Cellular Au-tomaton Rule 146,” Complex Systems , (2), 2010 pp. 143.[13] Marcelo O. Magnasco, “Chemical Kinetics is Turing Univer-sal,” Physical Review Letters , (6), 1997 pp. 1190–1193,doi:10.1103/PhysRevLett.78.1190.[14] Hector Zenil, J¨urgen Riedel, “Asymptotic Intrinsic Universality andReprogrammability by Behavioural Emulation,” arXiv preprint , 2016arXiv:1601.0033. Complex Systems , Volume (year) 1–1+ [15] St´ephane Perrard, Emmanuel Fort, and Yves Couder, “Wave-BasedTuring Machine: Time Reversal and Information Erasing,” PhysicalReview Letters , (9), 2016, doi:10.1103/PhysRevLett.117.094502.[16] Daniel M. Harris, Julien Moukhtar, Emmanuel Fort, Yves Couder,and John W. M. Bush, “Wavelike Statistics from Pilot-wave Dy-namics in a Circular Corral,” Physical Review E , (1), 2013,doi:10.1103/PhysRevE.88.011001.[17] Hanno Hildenbrandt, Cladio Carere, and Charlotte K. Hemelrijk, “Self-organized Aerial Displays of Thousands of Starlings: a Model,” Behav-ioral Ecology , (6), 2010 pp. 1349–1359, doi:10.1093/beheco/arq149.[18] Takashi Nagatani, “Density Waves in Traffic Flow,” Physical Review E , (4), 2000, doi:10.1103/PhysRevE.61.3564.[19] Dirk Brockmann and Dirk Helbing, “The Hidden Geometry of Com-plex, Network-driven Contagion Phenomena,” Science , (6164), 2013pp. 1337–1342, doi:10.1126/science.1245200. Appendix .1 Particles in ECAs
I define a local grid of cells in M as a particle in ECAs. Backgrounds arealso particles, which do not have any interactions with other particlesor themselves. According to the definition of particles in ECAs: P = L t max ( p L , p R ) . (.1)In this study, the size of a space–time evolution is (200 , P = L (100 − ,
100 + 2) . (.2)For the formula p ( p, n ) = p (1 − p ) n − . (.3)The number of p is a priori hypothesis, choosing a proper p will makeimages clear. Figure 9 shows that the formula with different p will notchange its whole behavior. Experiments show that choosing p = 0 . .2 Particles in Rule 146 Figure 10 show particles in the space-time for rule 146. These particlesare also introduced by [12].
Complex Systems , Volume (year) 1–1+ Complex Systems ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ �� �� �� �� �� �� ���������������������� � � ● � ��� ■ � ���� ◆ � ���� ▲ � ����� Figure 9.
Relation of p with different p and N . .3 Changes of lines’ intensity The change of a line’s intensity with time represents interactions be-tween particles. Because if a particle moves straight without any in-teractions, the lines’ intensity will remain unchanged through time.But if the particle can be generated by other particles, it will not befound before it was generated, so that the intensity of lines will changethrough time, mostly, the intensity will get higher when t is gettinghigher. To get particles’ changes of time, we define a function D ( t ) toget paths’ intensity: D ( t ) = max( L t ) . (.4)Figure 11 shows the procedure of extracting growth pattern of par-ticles and three examples for rule 149, rule 2 and rule 26. .3.1 The growth of particle traces’ intensity for rule 146 Particles were found in Rule 146 (shown in Figure 10), while alsofounded earlier in 2010 [12]. In that study, the intensity of particles inrule 146 has a power-law of the form n b ( t ) ∼ t − α , (.5)with α = 0 . ± . .4 Typical Rules for Four Classes Some space–time evolutions of typical rules in each class shown inTable 1.
Complex Systems , Volume (year) 1–1+
Particles are found in rule 146. +A. B.
C. D.
Figure 11.
Extracting growth pattern of particles. A) The growth of particles’intensity represents interactions of particles. B) An example of particle’sintensity, generated with rule 149, which has a growth pattern. C) Generatedwith rule 2, which do not has growth pattern. D) Generated with rule 26,with multiple particles and they all do not have growth pattern.
Complex Systems , Volume (year) 1–1+ Complex Systems t p Rule 146
Figure 12.
The points are the data for the growth of particles’ traces. Andthe line is the figure of function y = ( t max − t ) − . , which has the sameform as n b ( t ) ∼ t − α . It shows that the power–law also is shown in this kindof measurement, and it has a good fit when using the number of α from [12]. Table 1.
Typical Rules of Four Classes
ABCD