The Ring of Cellular Automata 256 Elementary Rules

Transcription

1 The Ring of Cellular Automata 256 Elementary Rules Serge Patlavskiy a physicist (L'viv National University), founder and director of the Institute for Theoretical Problems of Interdisciplinary Investigations, L'viv, Ukraine Abstract In the article, the author suggests specific criteria of simplicity and complexity and applies them when examining the cellular automata 256 elementary rules. Among them, the rules of three main kinds are found: 1) simple rules that produce simple space-time diagrams; 2) complex rules that produce complex diagrams, and 3) complex rules that produce simple diagrams. In doing this, no simple rules are found that would produce complex diagrams. As the result, the Ring of Rules is composed, and its basic symmetries and asymmetries are studied. Periodicity in localization of identical and quasi-identical spacetime diagrams produced by the rules of different kinds is also established, and the constants of pairwise mirror-symmetricalness of space-time diagrams are calculated. All the results presented in this paper and in the accompanied Supplementary Materials are easily verifiable and reproducible. Keywords: cellular automata elementary rules; criteria of simplicity and complexity; Ring of Rules; identical and quasi-identical space-time diagrams; periodic tables of space-time diagrams; constants of pairwise mirror-symmetricalness of space-time diagrams. 1. Introduction Suppose, there is a society where all the people can be dressed either in black or white only. If we consider them standing in a row, then there will be eight possible combinations of the colour of the dresses for each person and their two neighbours: Figure 1. Eight possible combinations of colours. Let us now suppose that the person in a centre wishes to step down from that row, but to do this, the colour of her dress may have to be changed, depending on the colours of the dresses of her two neighbours. Let us consider the following example: Figure 2. An example of how the colours may be changed. Here, if the person, while standing in the top row, was dressed in black, but her neighbours to the left and right were dressed in black and white correspondingly (in Figure 2, it is the second case from the left end), then, after stepping to the bottom row, the dress of the central person has to be changed from black to white. In general, the requirements of how the colour of the cells of the bottom row should be changed may be called rules, and the process of changing the colour of the cells depending on these rules in jumping down from row to row may be called cellular automaton. 1 It is sometimes assumed 1 The idea of cellular automata was originally suggested by von Neumann and Ulam. In Neumann (1966), this idea was used for constructing a self-reproducing automaton. 1

2 that the colour of the cell stands for its state. Therefore, in more exact terms, at each jump, each cell updates its state as a function of its own initial state and those of its two neighbours, according to the given cellular automaton rule. In fact, the rules start with the case when all the cells in the bottom row are white, and end with the case when they all are black. Therefore, there can be 256 possible combinations of black and white cells in the bottom row. Hence, in total, we can have 256 cellular automata elementary rules. 2 If we replace the white and black cells with zeros and ones correspondingly, then for the bottom row we receive the combinations starting from (which means that all the cells in the bottom row are white), and ending with (which means that all the cells in the bottom row are black). As one can recognize, these combinations of zeros and ones are nothing but the binary representations of the numbers 0 and 255 correspondingly. We will call Rule 000 the initial combination, and Rule 255 the ending combination. As to the combination considered in Figure 2, we will have a sequence for its bottom row, which stands for a binary representation of the number 18. Therefore, we will call that combination Rule 018 (with all its top and bottom rows taken together). If we take a cell which is coloured black, then, by applying one or other rule, and having performed a sufficiently big number of steps (or jumps down from row to row), we will obtain a certain two-dimensional space-time distribution diagram of black and white cells. In principle, we can do this job manually by coloring every cell either in black or white, but computer programs exist that can do this tedious work for us. The handiest of them is the Wolfram Mathematica TM program. Stephen Wolfram, the developer of the program, has studied the distributions of black and white cells (or the behaviour of cellular automata) up to billions of steps, and presented the results in his book A New Kind of Science (Wolfram 2002, p. 24). He writes therein: "In the existing sciences whenever a phenomenon is encountered that seems complex it is taken almost for granted that the phenomenon must be the result of some underlying mechanism that is itself complex. But my discovery that simple programs can produce great complexity makes it clear that this is not in fact correct. And indeed in the later parts of this book I will show that even remarkably simple programs seem to capture the essential mechanisms responsible for all sorts of important phenomena that in the past have always seemed far too complex to allow any simple explanation" (Wolfram 2002, p. 4). As is expected, the application of different rules results in different behaviour of cellular automata (Wolfram 2002, pp. 55-6). However, according to Wolfram, all these rules should be treated as "remarkably simple" and as ones that can produce complex space-time diagrams. In what follows, I suggest specific criteria of simplicity and complexity, and show that among cellular automata 256 elementary rules there are not only simple, but also complex ones. Moreover, as it turns out, there are, in real, no simple rules that give complex space-time diagrams. 2 This fact was first established by Wolfram (1984). He also suggested numbering the rules in lexicographic manner starting with 0 and ending with 255. However, in this paper I suggest a bit different notation just to emphasize a difference between the number of the rule, and the name of the rule. Therefore, according to proposed notation, the rule #1 is named Rule 000; the rule #2 is named Rule 001, and so on up to the rule #100 which is named Rule 099. For the rest of rules, the notation is as in Wolfram (1984 and 2002). 2

3 2. The procedure of determining simplicity and complexity Let us consider Rule 018: Figure 3. Rule 018. Every cellular automaton elementary rule is corresponded by eight combinations of colours, each going in two rows with three cells in the top row and one cell in the bottom row. Then, to examine the rule, I place the last four combinations under the first four ones, and, having performed such a re-arrangement, I receive four vertical pairs of combinations and label them (1), (2), (3), and (4) as shown in Figure 3. It should be admitted that placing the last four combinations above the first ones would not alter the results of examination. For each pair of combinations, the top combination can be transformed into bottom combination by performing only two types of transformations: negative transformation and mirror transformation. A condition is imposed here that, at a time, only one of these transformations can be applied to all the cells in the examining row. I mean that we cannot apply, for example, the negative transformation to the left cell in the given row, and apply simultaneously mirror transformation to the rest of its cells. In fact, these eight combinations of colours are re-arranged in such a way that the imposed condition to be met. The variant of re-arrangement of combinations, as suggested for Rule 018, stays unchanged while examining all the 256 rules. Now then, for four vertical pairs of combinations of the Rule 018 we have: (1) negative transformation: three black cells in the top row top combination transform into three white cells in the top row bottom combination; the white cell in the bottom row top combination does not transform, or it stays white in the bottom row bottom combination; (2) mirror transformation: two left-side black cells in the top row top combination transform into two right-side black cells in the top row bottom combination; the white cell in the bottom row top combination does not transform; (3) negative transformation: the black-white-black cells in the top row top combination transform into white-black-white cells in the top row bottom combination; the white cell in the bottom row top combination does not transform; (4) mirror transformation: the left-side black cell in the top row top combination transforms into right-side black cell in the top row bottom combination; the black cell in the bottom row top combination does not transform. It can be seen that for each pair of combinations we have either mirror, or negative transformations. Such a pure (or, unmixed) set of mirror and negative transformations I will call 3

4 simple. From here on, I will call simple every cellular automaton rule that contains pure transformations only such is the criterion of simplicity of the given elementary rule. As can be seen in Figure 3, when applying this rule, we receive a distribution of white and black cells which is mirror-symmetric. Here, as the criterion of simplicity of the given distribution I suggest considering its being mirror-symmetric about the virtual axis (not shown in the picture) that goes vertically down starting from the topmost (or initial) black cell of the distribution. Consequently, in case of Rule 018 we have simple rule that produces simple (or mirror symmetric) space-time distribution diagram. Let us now examine a rule with a sequence for its bottom row, namely, Rule 030: Figure 4. Rule 030. Here, for four vertical pairs of combinations of Rule 030 we receive: (1) negative transformation to top row top combination; (2) mirror transformation to top row top combination plus negative transformation to bottom row top combination (the white cell in the bottom row top combination transforms into black cell in the bottom row bottom combination); (3) negative transformation to both top and bottom rows top combination; (4) mirror transformation to top row top combination (the black cell in the bottom row top combination does not transform). In this case, for pair (2) we have a mixed (namely, mirror+negative) transformation. From here on, I will call complex any cellular automaton rule which (like Rule 030) contains one or two mixed transformations such is the criterion of complexity of the given rule. As one can see in Figure 4, application of the given complex rule gives asymmetric space-time diagram. Here, as the criterion of complexity of the given space-time diagram I suggest to consider its being an asymmetric alteration of the white and black cells. Thereby, I have suggested the objective criteria by which both the elementary rules and the space-time diagrams can be unequivocally sorted into two big groups each. So, I suggest to consider the elementary rule simple if it contains pure mirror and negative transformations only, and I suggest considering it complex if it contains one or two mixed (mirror+negative) transformations. Also, I suggest to consider the space-time diagram simple if it is mirrorsymmetric, and I suggest considering it complex if it is asymmetric. 4

5 3. Results and analysis Using the suggested criteria of simplicity and complexity I have examined all the 256 elementary rules (see Supplementary Data 1), and received the following results: (1) there are 64 rules that are simple and such ones that give simple (mirrorsymmetric) space-time diagrams; (2) there are 160 complex rules that give complex space-time diagrams; (3) there are 32 rules, which, despite of being themselves complex, all the same give simple space-time diagrams; (4) there is not any simple rule that would give complex space-time diagrams. In total, there are 64 simple rules, and =192 complex rules. There are also 64+32=96 simple space-time distribution diagrams, and 160 complex distribution diagrams. Next, for convenience of pictorial representation of the obtained results, I, firstly, ascribe different colours to the different kinds of rules. So, I call Green Rules the simple rules that give simple distribution diagrams; I call Red Rules the complex rules that produce complex distribution diagrams; and I call Yellow Rules the complex rules that produce simple distribution diagrams. Then, I present all the 256 rules in the form of a circular diagram with the rules that follow each other clockwise. As a result, I receive what I call the Ring of Cellular Automata 256 Elementary Rules (or, the Ring of Rules for short). Figure 5. The Ring of Rules. On the face of it, the rules of different colours on the Ring of Rules are distributed sporadically. But, a more close examination makes us to establish some remarkable symmetries and asymmetries. As one can see in Figure 5 (see also Supplementary Figures 1, Figure SF1-1 for high-resolution diagram), Green Rules always go in dual pairs. There are, in total, 16 dual pairs of Green Rules. There are also two linked dual pairs of them at the top and at the bottom of the Ring. All Green Rules are strictly symmetric as about vertical, so horizontal axes, and they all are also centrally symmetric. 5

6 Figure 6. Symmetries and asymmetries of the Strange Yellow Rules. There are two opposite pairs of Yellow Rules to the left and to the right sides of the ring. There are also four cases when Yellow Rule is stuck to the pair of Green Rules. These are Yellow Rules 031, 096, 159, and 224. The half of Yellow Rules (namely, 16 rules) exhibit strict horizontal, vertical, and central symmetries, but another half of them (another 16 rules) demonstrates interesting asymmetries. In Figure 6, the pairs of Yellow Rules linked with blue solid lines are centrally symmetric, while the ones linked with blue dotted lines are nor horizontally, nor vertically symmetric. For example, Rule 012 is centrally symmetric with Rule 140, but it is not horizontally symmetric with Rule 242, and it is not vertically symmetric with Rule 114. Therefore, I refer to the rules 012, 026, 044, 058, 068, 082, 100, 114, 140, 154, 172, 186, 196, 210, 228, and 242 as Strange Yellow Rules. Every Strange Yellow Rule from the first quarter has an uneven number of red neighbours, while its counterparts from the second and fourth quarters have even number of red neighbours to the left and to the right. As a case in point, Rule 012 has three red neighbours counter-clockwise, and five of them clockwise. However, as Rule 114 from the second quarter, so Rule 242 from the fourth quarter has four red neighbours to both sides of them. It is also remarkable that the character of vibrations of the Strange Yellow Rules does not depend on whether we consider the vertical or horizontal flips of the Ring of Rules (see Supplementary Figures 2, Figures SF2-1 and SF2-2). Among 160 complex rules that give complex distributions (namely, the Red Rules) are 64 ones that contain two mixed transformations each (see, for example, Supplementary Data 1, for Rule 010). I refer to them as Strong Red Rules. As can be seen in Figure 7, Strong Red Rules always go in dual pairs. In total, there are 32 pairs, or 16 dual pairs of Strong Red Rules. All the pairs of Strong Red Rules are strictly symmetric as about vertical, so horizontal axes, and they are also centrally symmetric. It is also notable that every Strange Yellow Rule is sandwiched between the dual pairs of Strong Red Rules. It looks like the dual pairs of these Red Rules are as if deliberately strengthened to enable the Strange Yellow Rules to vibrate (see Supplementary Figures 2, Figure SF2-3). 6

7 Figure 7. The Strong Red Rules. As can be seen in Figure 4 that describes Rule 030 above, a mixed transformation takes place for (2) nd pair of combinations, or is situated to the left. So, I refer to any Red Rule with such a disposition of mixed transformation as Left-hand Red Rule. In case the mixed transformation takes place for (4) th pair of combinations, or is situated to the right (as, for example, in case of Rule 002; see Supplementary Data 1), then such a rule I call Right-hand Red Rule. In Figure 8 we can see the distribution of Left-hand and Right-hand Red Rules on the Ring of Rules (in the figure, the Strong Red Rules are coloured pink; see also Supplementary Figures 1, Figure SF1-4 for a high-resolution diagram). Figure 8. The Right-hand and Left-hand Red Rules. In total, there are 64 Right-hand Red Rules that go in 32 pairs, or 16 dual pairs. All these rules are vertically, horizontally, and centrally symmetric. There are also 32 Left-hand Red Rules, and each of them always goes stuck to the Yellow Rule. Therefore, only half of the Left-hand Red Rules are vertically, horizontally, and centrally symmetric. The other half of them, namely, the rules 013, 027, 045, 059, 069, 083, 101, 115, 141, 155, 173, 187, 197, 211, 229, and 243, are nor horizontally, nor vertically symmetric, and I refer to them as Strange Left-hand Red Rules (in Figure 8, the blue solid lines are for the presence of central symmetry, and the blue dotted lines 7

8 are for the absence of vertical and horizontal symmetries between the related Strange Left-hand Red Rules). To summarise, among 160 Red Rules there are 64 Strong Red Rules, 64 Right-hand Red Rules, 16 (ordinary) Left-hand Red Rules, and 16 Strange Left-hand Red Rules. It is also notable that all the Yellow Rules have left-hand disposition of the mixed transformations. As can be also learned from Figure 4 above, for (3) rd pair of combinations of colours there is a negative transformation simultaneously to top and bottom rows top combination. I call this case a double negative transformation. So, among Green, Yellow, and Red Rules, there are, in total, 64 rules with double negative transformations to (1) st pair of combinations, 64 rules with double negative transformations to (3) rd pair of combinations, 64 rules with double negative transformations to both (1) st and (3) rd pairs of combinations, and 64 rules with single negative transformations to either of pairs (see Supplementary Figures 1, Figure SF1-5; see also Supplementary Figures 2, Figures SF2-4 and SF2-5 for horizontal and vertical flips of the double negative transformations; in Supplementary Figures 2, Figure SF2-6 one can also see how the Ring of Rules transforms while we examine its rules of different kinds and its double negative transformations). It is remarkable that having known the symmetries and asymmetries of the Strange Yellow Rules, and having got just the first 32 rules (from Rule 000 to 031), we can always predict which rule will be yellow, red and green, thereby restoring the whole of the ring, and predicting the character of all other space-time distribution diagrams. For example, as can be seen in Figure 6, the whole first quarter is centrally symmetric with the third quarter, while the whole second quarter is centrally symmetric with the fourth quarter. This fact becomes more evident when presenting the Ring of Rules in a linear form (see Supplementary Data 2, Figures SD2-1, 2, and 3 for two-, four-, and eight-fold linear representations). Next. If to look at space-time diagrams produced by Rules 012 and 044 (see Supplementary Data 1), a conclusion can be made that these two distributions are identical. A more close study in this direction shows that an extremely interesting regularity exists in localization of identical and quasi-identical distributions produced by the same kind of rules. For every kind of rules, these localizations look like certain periodic tables with space-time diagrams repeating themselves in a regular manner (see Supplementary Data 3, 4, and 5 for periodic tables of distribution diagrams for Yellow, Green, and Red Rules correspondingly; see also Supplementary Figures 3 for animation of evolution of distributions produced by Green Rules and based on correspondent periodic table). The examination of periodic table for Red Rules reveals also many cases of pairwise mirrorsymmetricalness of space-time distribution diagrams. The constants of pairwise mirrorsymmetricalness of distributions obtained by applying Right-hand, Left-hand, and Strong Red Rules are calculated. So, for Right-hand Red Rules the constant of pairwise mirrorsymmetricalness is 14; for Left-hand Red Rules it is 56; for the half of Strong Red Rules it is 70, and 42 for another half of them (see Supplementary Data 6). The very fact that it is possible to calculate these constants talks about the correctness of the chosen criteria of simplicity and complexity of the elementary rules and space-time distribution diagrams. 8

9 And, the final point. Chua at al. (200?) suggested sorting out the local rules depending on the, so called, index of complexity. They wrote: "Observe also that the identification number N of each Boolean cube is colored in red, blue or green, depending on whether the red vertices can be segregated and separated from each other by κ =1, 2, or 3 parallel planes, where κ is called the index of complexity of the local rule N [Chua et al., 2002]. Table 2 lists all 256 local rules along with their index of complexity. The index of complexity κ is not a definition of complexity. Rather it measures the relative number of electronic devices needed to implement each local rule. A κ = 1 local rule requires the smallest number of transistors. More transistors must be added to realize a κ = 2 local rule. Still more transistors are required to implement a κ =3 local rule. In other words, the index of complexity κ measures the relative "cost" of hardware (Chip) implementations" (Chua et al., 20??, p.2, italics in original). I think that despite of being not a "definition of complexity", the index of complexity may well be considered as a certain criterion of complexity for the local rule. In Appendix below, Chua at al.'s Table 2 is shown in comparison with a table of local rules colored in accordance with the criteria of complexity suggested in current paper. Unlike Chua at al.'s, in my case, the color of a rule depends on combination of complexity/simplicity of both the rule and the correspondent distribution diagram. Also, in Figure 9 the Ring of Rules composed according to Chua at al.'s index of complexity is shown. Figure 9. The Ring of Rules composed according to Chua at al.'s index of complexity. As can be seen in Figure 9, this kind of the Ring of Rules is only horizontally symmetric: its left semicircle is mirror-symmetrical with its right semicircle (see Supplementary Figures 1, Figure SF1-6 for a high-resolution diagram; see also Supplementary Figures 2, Figures SF2-7 and SF2-8 for vertical or horizontal flips of this kind of the Ring of Rules). It is notable, that the Ring of Rules can be used as a stand (or a method) for testing the different kinds of the criteria of complexity for their effectiveness in demonstrating the hidden relations that exist between the local rules. 9

10 4. Conclusions As I have shown in this article, the terms "elementary" and "simple" are not synonyms, and some strict criteria can be suggested that make us possible to select all the elementary rules and spacetime distribution diagrams into conventionally simple and complex. Namely, I suggested to consider the elementary rule simple if it contains pure mirror and negative transformations only, and I suggested considering it complex if it contains one or two mixed (mirror+negative) transformations. Also, I suggested to consider the space-time diagram simple if it is mirrorsymmetric, and I suggested considering it complex if it is asymmetric. By applying these criteria, seven kinds of elementary rules were found, namely: Green Rules, Yellow Rules, Strange Yellow Rules, Strong Red Rules, Right-hand Red Rules, Left-hand Red Rules, and Strange Left-hand Red Rules all arranged into the Ring of Cellular Automata 256 Elementary Rules. The space-time distribution diagrams produced by applying different kinds of rules were arranged into five periodic tables. Being based on these periodic tables, the constants of pairwise mirror-symmetricalness of distribution diagrams obtained by applying Right-hand, Left-hand, and Strong Red Rules were calculated. It may be supposed possible that the mutual disposition of the rules on the Ring of Rules, the basic symmetries and asymmetries of the Ring of Rules, as well as the constants of pairwise mirror-symmetricalness calculated by examining the periodic tables of space-time distribution diagrams could in some hidden deep sense be representatives of certain most fundamental natural laws and principles and some still undiscovered basic relations between the elements of our Reality. But this possibility has yet to be investigated. Acknowledgements: I am thankful to Wolfram Research < for mailing me the book by Stephen Wolfram free of charge. I am also indebted to Jonathan Edwards (University College London) for preliminary proofreading of the manuscript. References: [1] von Neumann, J. (1966) Theory of Self-Reproducing Automata. University of Illinois Press, Urbana, IL, edited and completed by A. W. Burks. [2] Wolfram, S. (1984) Universality and complexity in cellular automata. Physica D, 10:1-35. [3] Wolfram, S. (2002) A New Kind of Science. Wolfram Media, Inc., Champaign, IL. [4] Chua at al. (20??) A nonlinear dynamics perspective of Wolfram's New Kind of Science (Vol. III, Chapter 1), World Scientific Publishing Co. Pte. Ltd.; < Links to Supplementary Materials:

11 Appendix Table 1. List of 256 local rules sorted into Green Rules (rule simple, diagram simple), Red Rules (rule complex, diagram complex), and Yellow Rules (rule complex, diagram simple) Table 2. List of 256 local rules with their complexity index coded in red (κ = 1), blue (κ = 2) and green (κ =3), respectively (the table is identical to the one presented in Chua at al., 20??, p.8)