Designing Complex Dynamics in Cellular Automata with Memory
DDesigning Complex Dynamics in Cellular Automata with Memory
Genaro J. Mart´ınez − Andrew Adamatzky , Ramon Alonso-Sanz , , July 3, 2013Paper published at
International Journal of Bifurcation and Chaos International Center of Unconventional Computing, University of the West of England, BS16 1QY Bristol,United Kingdom http://uncomp.uwe.ac.uk/ Laboratorio de Ciencias de la Computaci´on, Escuela Superior de C´omputo, Instituto Polit´ecnico Nacional,M´exico, D.F. http://uncomp.uwe.ac.uk/LCCOMP/ Foundation of Computer Science Laboratory, Hiroshima University, Higashi-Hiroshima, Japan Centre for Chaos and Complex Networks, City University of Hong Kong, Hong Kong, P. R. China School of Science, Hangzhou Dianzi University, Hangzhou, P. R. China Centre for Complex Systems, Instituto Polit´ecnico Nacional, M´exico, D.F. Centro de Ciencias de la Complejidad, Universidad Nacional Aut´onoma de M´exico, M´exico, D.F. http://c3.fisica.unam.mx/ Laboratoire de Recherche Scientifique, Paris, France http://labores.eu/ Institut des Syst`emes Complexes en Normandie, Normandie, France http://iscn.univ-lehavre.fr/ ETSI Agronomos Estad´ıstica, Universidad Polit´ecnica de Madrid, Madrid, Espa˜na
Abstract
Since their inception at
Macy conferences in later 1940s complex systems remain the most controversialtopic of inter-disciplinary sciences. The term ‘complex system’ is the most vague and liberally used scientificterm. Using elementary cellular automata (ECA), and exploiting the CA classification, we demonstrateelusiveness of ‘complexity’ by shifting space-time dynamics of the automata from simple to complex byenriching cells with memory . This way, we can transform any ECA class to another ECA class — withoutchanging skeleton of cell-state transition function — and vice versa by just selecting a right kind of memory.A systematic analysis display that memory helps ‘discover’ hidden information and behaviour on trivial —uniform, periodic, and non-trivial — chaotic, complex — dynamical systems. keywords: elementary cellular automata, classification, memory, computability, gliders, collisions, complexsystems. a r X i v : . [ n li n . C G ] J un ontents d -spectrum classification (2003) . . . . . . . . . . . . . . . . . . . 294.2.6 Communication complexity classification (2004) . . . . . . . . . . . . . . . . . . . . . . . . 294.2.7 Topological classification (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.8 Power spectral classification (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.9 Morphological diversity classification (2010) . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.10 Distributive and non-distributive lattices classification (2010) . . . . . . . . . . . . . . . . 344.2.11 Topological dynamics classification (2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.12 Expressivity analysis (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.13 Normalised compression classification (2013) . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.14 Surface dynamics classification (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.15 Spectral classification (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.16 Bijective and surjective classification (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.17 Creativity classification (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Universal relations in ECAM classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ist of Figures ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Dynamics in ECAM on an arbitrary one-dimensional array and hypothetical evolution rule ϕ andmemory function φ m : τ with τ = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 (a) Typical fractal and (b) chaotic global evolution of ECA rule 126. (a) initially all cells in ‘0’but one in state ‘1,’ (b) evolution from random initial configuration with 50% of ‘0’ and ‘1’ states.Evolution on a horizontal ring of 387 cells with time going down up to 240 time steps). . . . . . . 115 Mean field curve for ECA rule 126. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Periodic patterns calculated from some exemplar attractors. . . . . . . . . . . . . . . . . . . . . . 137 The whole set of non-equivalent basins in ECA rule 126 from l = 2 to l = 18. . . . . . . . . . . . 148 De Bruijn diagram for the ECA rule 126. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Patterns calculated with extended de Bruijn diagrams, in particular from cycles of order ( x, x -shift in 2-generations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510 Filtered space-time configuration in ECA rule 126. . . . . . . . . . . . . . . . . . . . . . . . . . . 1611 (a) φ R min :3 displays a typical evolution of ECAM rule 126 with minority memory τ = 3,(b) φ R par :2 displays an evolution but now evolving with parity memory, and (c) the mostinteresting evolution with ECAM rule φ R maj :4 , where we can see the emergence of complexpatterns as gliders and glider guns. In this case a filter is selected for a best view of complexpatterns and their interactions. Snapshots start with same random initial conditions on a ring of296 cells evolving in 1036 generations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812 Filtered space-time configuration of ECAM φ R maj :4 evolving with a ring of 843 cells, periodicboundaries, starting just from one non-quiescent cell and running for 1156 steps. . . . . . . . . . 1913 Continued evolution to 2312 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014 Continued evolution to 3468 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115 Continuee evolution to 4624 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216 Continued evolution to 5780 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317 “Strong” ECAM class is able to reach some other classes. Starting from a Wolfram’s class (rule)and selecting some kind of memory inside strong ECAM class, one can reach some other classwith such a rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218 “Moderate” ECAM class is able to reach some other classes. Starting from a Wolfram’s class(rule) and selecting some kind of memory inside moderate
ECAM class, one each some other classwith such a rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219 “Weak” ECAM class is not able to reach other classes. Starting from a Wolfram’s class (rule)and selecting some kind of memory inside weak
ECAM class, one cannot reach some classes withsuch a rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4320 Every ECAM class has rules with behaviour class I, II, III, or IV. If you take one ECA rule witha kind of memory hence you can change to another class. “All memories” diagram show that itis possible to reach any class from some ECA enriched with memory, thus some ECAM is ableto reach any class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4421 Typical random evolution of φ R maj :4 from an initial configuration where 37% of cells take state’1’. The automaton is a ring of 767 cells. Evolution is recorded for 372 generations. . . . . . . . . 4522 We implement basic logic functions as not and and gates via collisions of gliders and a delay element. Single or pair of particles represent bits 0’s or 1’s respectively. . . . . . . . . . . . . . . 4633 A nand gate based in majority and not gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4624 (a) binary values by gliders codification, (b) scheme of a nand gate from majority and not gates with glider reaction, and (c)circuit based on four nand gates like a modified 7400 chip butnow with 18 pins (for the majority gate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4725 nand gate implemented from majority and not gates in φ R maj :4 . Inputs (a) 000 and (b) 010. 4826 nand gate implemented from majority and not gates in φ R maj :4 . Inputs (a) 100 and (b) 110. 494 Introduction
A complexity theory emerged from studies of computable problems in computer science and mathematicalfoundations of computation, when a need came to compare performance and resource-efficiency of algorithms.Typically time complexity (number of steps) and space complexity (memory of a single processor and numberof processors) are expressed in terms of a Turing machine or an equivalent mathematical device. Each specifickind of a Turing machine represents a certain class of complexity [Minsky, 1967], [Arbib, 1969], [Hopcroft &Ullman, 1987]. When related to complex systems meaning of the word ‘complexity’ is different and heavilydepends on its context. Complexity of a system is almost never quantified but often related to unpredictability.Theory of cellular automata (CA) refers to complexity its entire life [von Neumann, 1966], [Adamatzky &Bull, 2009], [Boccara, 2004], [Chopard & Droz, 1998], [Hoekstra et al., 2010], [Kauffman, 1993], [Margenstern,2007], [McIntosh, 2009], [Mainzer & Chua, 2012], [Mitchell, 2002], [Morita, 1998], [Margolus et al., 1986],[Poundstone, 1985], [Park et al., 1986], [Toffoli & Margolus, 1987], [Schiff, 2008], [Sipper, 1997], [Wolfram,1986], [Mart´ınez et al., 2013a], [Mart´ınez et al., 2013b]. Due to transparency of cellular automata structurestheir complexity can be measured and analysed [Wolfram, 1984a], [Culik II & Yu, 1988].An elementary cellular automaton (ECA) is a one-dimensional array of finite automata, each automatontakes two states and updates its state in discrete times according to its own state and states of its two closestneighbours, all cells update their state synchronously. Thus in 1980s Wolfram subdivided ECA onto fourcomplexity classes [Wolfram, 1984a]: • class I. CA evolving uniformly. • class II. CA evolving periodically. • class III. CA evolving chaotically. • class IV. Include all previous cases, known as the class complex .Also these classes can be defined in terms of CA evolution as follows: • If the evolution is dominated by a unique state of alphabet from any random initial condition, it belongsto class I . • If the evolution is dominated by blocks of cells which are periodically repeated from any random initialcondition, hence it belongs to class II . • If the evolution is dominated by sets of cells without some defined pattern for a long time from any randominitial condition, hence it belongs to class III . • If the evolution is dominated by non-trivial structures emerging and travelling along of the evolution spacewhere also uniform, periodic, or chaotic regions can coexist with these structures, it belongs to class IV .This class is named frequently as: complex behaviour , complex dynamics , or simply complex .Figure 1 illustrates the Wolfram’s classes by a selected ECA rule (following the Wolfram’s notation forECA [Wolfram, 1983]), and all evolutions begin with the same random initial condition. Fig. 1a shows ECArule 8 converging quickly to a homogeneous state, the class I. Figure 1b displays blocks of cells which evolveperiodically exhibiting a right shift, this is an interesting reversible ECA rule 15, the class II. Figure 1c displaysa typical chaotic evolution with ECA rule 126, where no regular patterns detected or no limit point can beidentified, the class III. Finally, Fig. 1d displays the so-called complex class or class IV with ECA rule 54.There we can see non-trivial patterns emerging in the evolution space, and such patterns conserve their formand travel along of the evolution space. The patterns collide with each other and annihilate or fuse, or undergosoliton-like transformations or produce new structures. These patterns are referred to as gliders in the CAliterature (glider is a concept widely accepted and popularised by Conway from its famous 2D CA Game of a) (b)(c) (d) Figure 1: Examples of space-time evolution of ECA rules: (a) class I, ECA rule 8, (b) class II, ECA rule15, (c) class III, ECA rule 126, (d) class IV, ECA rule 54 (a periodic background is filtered). All automataillustrated start their development at the same random initial condition with a density of 50% of states 0, light(light blue) dots, and states 1, dark (dark blue) dots. Each automaton is a horizontal ring of 385 cells evolvedfor 400 time steps. 6 ife [Gardner, 1970]). In space-time configurations developed by functions from class IV we can see regionswith periodic configurations, fragments of chaos, and well-defined non-trivial patterns. Frequently in complexrules the background is dominated by a stable state, such as happens in Conway’s Game of Life. In this case,particularly the complex ECA rule 54 and 110 can evolve with a periodic background (called ether) where thesegliders emerge and live. Gliders in GoL and other CAs as the 2D Brian’s Brain CA [Toffoli & Margolus, 1987]caught the attention of Langton and thus contributed to development of Artificial Life field [Langton, 1984],[Langton, 1986].Since the publication of the paper “Universality and complexity in cellular automata” in 1984 [Wolfram,1984a] there have been a number of disputes about validity of the classification. Wolfram selected certain ECArules to illustrate each class. Although, he commented textually that: k = 2 , r = 1 cellular automata aretoo simple to support universal computation [Wolfram, 1984a] (page 31). Nevertheless, in his book “CellularAutomata and Complexity” [Wolfram, 1994] ECA rule 110 was awarded its own appendix (Table 15, Structuresin Rule 110, pages 575–577). It contains specimens of evolution including a list of thirteen gliders compiled byLind, and also presents the conjecture that the rule could be universal. Wolfram wrote: One may speculate thatthe behaviour of rule 110 is sophisticated enough to support universal computation . Finally, in [Cook, 2004],[Wolfram, 2002] it was proved that ECA rule 110 is computationally universal because it simulates a novel cyclictag system with package of gliders and collisions on millions of cells. The paper written by Culick II and Yu titled “Undecidability of CA Classification Schemes” [Culik II &Yu, 1988], [Sutner, 1989] discussed the properties of Wolfram ECA classes and stated that it is undecidable towhich class a given cellular automaton belongs (page 177).Further attempts of ECA classification have been made in [Gutowitz et al., 1987], [Li & Packard, 1990],[Aizawa & Nishikawa, 1986], Ada94, [Sutner, 2009]. Gutowitz developed a statistical analysis in “Local struc-ture theory for cellular automata” [Gutowitz et al., 1987]. An extended classification of ECA classes with meanfield theory was proposed by McIntosh in “Wolfram’s Class IV and a Good Life” [McIntosh, 1990]. An inter-esting schematic diagram conceptualising classes in CA was made by Li and Packard in “The Structure of theElementary Cellular Automata Rule Space” [Li & Packard, 1990]. Patterns recognition and classification waspresented in “Toward the classification of the patterns generated by one-dimensional cellular automata” [Aizawa& Nishikawa, 1986]. An extended analysis of CA was presented in “Identification of Cellular Automata” byAdamatzky in [Adamatzky, 1994] relating to the problem of given a sequence of configurations of an unknownCA hence how to reconstruct the cell-state transition rule. Sutner has been discussed this classification and alsothe principle of equivalence computation in “Classification of Cellular Automata” [Sutner, 2009], with emphasisin class IV or computable CA. A fruitful approach with additive 2D CA was suggested by Eppstein [Eppstein,1999]. In this classification, class IV (called complex) is of particular interest because such rules present non-trivialbehaviour with a rich diversity of patterns (gliders) emerging and non-trivial interactions between them, glidersare referred as well as mobile self-localizations, particles, or fragments of waves. This feature was relevant toimplementation of a register machine in GoL [Berlekamp et al., 1982] to determine its universality. Thus Rendellhas developed an elaborated Turing machines in GoL with thousands of thousands of cells [Rendell, 2011a],[Rendell, 2011b]. Although across of the history, these bridges connection between complexity of a CA (or anyother dynamical system) and their universality is not always obvious [Adamatzky, 2002], [Mills, 2008].Other recommendable reference sources to mention include Mitchell’s “Complexity: A Guided Tour” [Mitchell,2009], Wolfram’s “A New Kind of Science” [Wolfram, 2002], Bar-Yam’s “Dynamics of Complex Systems” [Bar-Yam, 1997], and “The Universe as Automaton: From Simplicity and Symmetry to Complexity” [Mainzer &Chua, 2012] by Mainzer and Chua. Large snapshots of this large machine working in ECA rule 110 are available in http://uncomp.uwe.ac.uk/genaro/rule110/ctsRule110.html . You can see such discussion from Tim Tyler’s CA FAQ in http://cafaq.com/classify/index.php . One-dimensional cellular automata
CA are discrete dynamical systems, with a finite alphabet that evolve on a regular lattice in parallel. In thepaper we deal with one-dimensional cellular automata.
A CA is a tuple (cid:104) Σ , ϕ, µ, c (cid:105) where d is a dimensional lattice and each cell x i , i ∈ N , takes a state from a finitealphabet Σ such that x ∈ Σ. A sequence s ∈ Σ n of n cell-states represents a string or a global configuration c on Σ. We write a set of finite configurations as Σ n . Cells update their states by an evolution rule ϕ : Σ µ → Σ,such that µ = 2 r + 1 represents a cell neighbourhood that consists of a central cell and a number of r -neighboursconnected locally. If k = | Σ | hence there are k r +1 neighbourhoods and k k r +1 evolution rules.Figure 2: Dynamics in ECA on an arbitrary one-dimensional array transformed for a specific evolution rule ϕ .An evolution diagram for a CA is represented by a sequence of configurations { c i } generated by the globalmapping Φ : Σ n → Σ n , where a global relation is given as Φ( c t ) → c t +1 . Thus c is the initial configuration.Cell states of a configuration c t are updated simultaneously by the local rule, as follows: ϕ ( x ti − r , . . . , x ti , . . . , x ti + r ) → x t +1 i (1)where i indicates cell position and r is the radius of neighbourhood in µ . Thus, the elementary CA representsa system of order ( k = 2 , r = 1) (in Wolfram’s notation [Wolfram, 1983]), the well-known ECA .To represent a specific ECA evolution rule we will write the evolution rule in a decimal notation, e.g. ϕ R represents the evolution rule 54. Thus Fig. 2 illustrates how an evolution dynamics works for ECA. Conventional CA are memoryless: new state of a cell depends on the neighbourhood configuration solely at thepreceding time step of ϕ . CA with memory are an extension of CA in such a way that every cell x i is allowed toremember its states during some fixed period of its evolution. CA with memory have been proposed originally byAlonso-Sanz in [Alonso-Sanz & Martin, 2003], [Alonso-Sanz, 2006], [Alonso-Sanz, 2009a], [Alonso-Sanz, 2009b],[Alonso-Sanz, 2011]. 8ence we implement a memory function φ , as follows: s ( t ) i = φ ( x t − τ +1 i , . . . , x t − i , x ti ) (2)where 1 ≤ τ ≤ t determines the degree of memory . Thus, τ = 1 means no memory (or conventional evolution),whereas τ = t means unlimited trailing memory. Each cell trait s i ∈ Σ is a state function of the series of states ofthe cell i with memory backward up to a specific value τ . In the memory implementations run here, commencesto act as soon as t reaches the τ time-step. Initially, i.e., t < τ , the automaton evolves in the conventional way.Later, to proceed in the dynamics, the original rule is applied on the cell states s as: ϕ ( . . . , s ( t ) i − , s ( t ) i , s ( t ) i +1 , . . . ) → x t +1 i (3)to get an evolution with memory. Thus in CA with memory, while the mapping ϕ remains unaltered, historicmemory of all past iterations is retained by featuring each cell as a summary of its past states from φ . We cansay that cells canalise memory to the map ϕ [Alonso-Sanz, 2009a].Figure 3: Dynamics in ECAM on an arbitrary one-dimensional array and hypothetical evolution rule ϕ andmemory function φ m : τ with τ = 3.Let us consider the memory function φ in a form of majority memory , φ maj → s i , x ti is to be adoptedas s ( t ) i , which implies no memory effect. These τ is even, in which case theeffect of memory may appear as somehow weaker , or simply different , compared to the effect of the odd τ − τ + 1 close lengths of memory. Thus, φ maj function represents the classic majority function. For three values[Minsky, 1967], then we have that: φ maj ( a, b, c ) : ( a ∧ b ) ∨ ( b ∧ c ) ∨ ( c ∧ a ) (4)Any map of previous states may act as memory (not only majority). Thus, minority, parity, alpha, . . . , orany CA rule acting as memory, weighted memory, . . . , etc. (for full details please see [Alonso-Sanz, 2009a],[Alonso-Sanz, 2011]).Evolution rules representation for ECAM in this paper is given in [Mart´ınez et al., 2010a], [Mart´ınez et al.,2010b], [Mart´ınez et al., 2011], [Mart´ınez et al., 2012a], [Mart´ınez et al., 2012b], as follows: φ CARm : τ (5)where CAR is the decimal notation of a particular ECA rule and m is the kind of memory used with a specificvalue of τ . This way, for example, the majority memory ( maj ) incorporated in ECA rule 30 employing fivesteps of a cell’s history ( τ = 5) is denoted simply as: φ R maj :5 . The memory is functional as the CA itself, seeschematic explanation in Fig. 3. In this section, we consider a particular case to illustrate the effect of memory, deriving in complex dynamicsfrom a chaotic rule [Mart´ınez et al., 2010b]. Here we deal with a chaotic ECA (class III), the evolution rule126. This is a special chaotic rule because such evolution yield sets of regular languages [Wolfram, 1984b],[McIntosh, 2009]. We can deduce from previous analysis that ECA rule 126 could contain another kind ofinteresting information. Selecting a kind of memory we will see that particularly ECAM φ R maj :4 displays alarge number of glider guns emerging from random initial conditions, and emergence of a number of non-trivialpatterns colliding constantly [Mart´ınez et al., 2010b]. The local-state transition function ϕ corresponding to ECA rule 126 is represented as follows: ϕ R = (cid:26) , , , , , , . ECA rule 126 has a chaotic global behaviour typical from Class III in Wolfram’s classification [Wolfram,1994] (Fig. 1). In ϕ R we can easily recognize an initial high probability of alive cells, i.e. cells in state ‘1’;with a 75% to appear in the next time and, complement of only 25% to get state 0. It will be always a newalive cell iff ϕ R has one or two alive cells such that the equilibrium reached when there is an overpopulationcondition. Figure 4 shows these cases in typical evolutions of ECA rule 126, both evolving from a single cell instate ‘1’ (Fig. 4a) and from a random initial configuration (Fig. 4b) where a high density of 1’s is evidently inthe evolution.While looking on chaotic space-time configuration in Fig. 4 we understand the difficulty for analysing therule’s behaviour and selecting any coherent activity among periodic structures without special tools.10 a) (b) Figure 4: (a) Typical fractal and (b) chaotic global evolution of ECA rule 126. (a) initially all cells in ‘0’ butone in state ‘1,’ (b) evolution from random initial configuration with 50% of ‘0’ and ‘1’ states. Evolution on ahorizontal ring of 387 cells with time going down up to 240 time steps).
In this section we use a probabilistic analysis with mean field theory to uncover basic properties of ϕ R evolution space and its related chaotic behaviour. Such analysis we help us to explore the evolution space withspecific initial conditions, that might lead to discoveries of non-trivial behaviour.Mean field theory is a established technique for discovering statistical properties of CA without analysingevolution spaces of individual rules [McIntosh, 2009]. The method assumes that states in Σ are independentand do not correlate with each other in the local function ϕ R . Thus we can study probabilities of states in aneighbourhood in terms of the probability of a single state (the state in which the neighbourhood evolves), andprobability of the neighbourhood as a product of the probabilities of each cell in it. McIntosh in [McIntosh,1990] presents an explanation of Wolfram’s classes with a mixture of probability theory and de Bruijn diagrams,resulting in a classification based on mean field theory curve, as follows: • class I: monotonic, entirely on one side of diagonal; • class II: horizontal tangency, never reaches diagonal; • class IV: horizontal plus diagonal tangency, no crossing; • class III: no tangences, curve crosses diagonal.For one dimensional case, all neighbourhoods are considered as follows: p t +1 = k r +1 − (cid:88) j =0 ϕ j ( X ) p vt (1 − p t ) n − v (6)such that j is an index relating each neighbourhood and X are cells x i − r , . . . , x i , . . . , x i + r . Thus n is the numberof cells into every neighbourhood, v indicates how often state ‘1’ occurs in X , n − v shows how often state ‘0’11ccurs in the neighbourhood X , p t is the probability of cell being in state ‘1’ while q t is the probability of cellbeing in state ‘0’ i.e., q = 1 − p . The polynomial for ECA rule 126 is defined as follows: p t +1 = 3 p t q t . (7)Because ϕ R is classified as a chaotic rule, we expect no tangencies and its curve must cross the identity;recall that ϕ R has a 75% of probability to produce a state one. pq Figure 5: Mean field curve for ECA rule 126.Mean field curve (Fig. 5) confirms that probability of state ‘1’ in space-time configurations of ϕ R is 0.75for high densities related to big populations of 1’s. The curve demonstrates also that ϕ R is chaotic becausethe curve cross the identity with a first fixed point at the origin f = 0 and the absence of unstable fixed pointsinducing non stable regions in the evolution. Nevertheless, the stable fixed point is f = 0 . ϕ R . Of course a deeperanalysis is necessary for obtaining more features from a chaotic rule, so the next sections explain other techniquesto study in particular periodic structures. A basin (of attraction) field of a finite CA is the set of basins of attraction into which all possible states andtrajectories are driven by the local function ϕ . The topology of a single basin of attraction may be represented bya diagram, the state transition graph . Thus the set of graphs composing the field specifies the global behaviourof the system [Wuensche & Lesser, 1992].Generally a basin can also recognise CA with chaotic or complex behaviour following previous results onattractors [Wuensche & Lesser, 1992]. Thus we have Wolfram’s classes represented as a basin classifications,following the Wuensche’s characterisation: • class I: very short transients, mainly point attractors (but possibly also periodic attractors) very highin-degree, very high leaf density (very ordered dynamics);12 class II: very short transients, mainly short periodic attractors (but also point attractors), high in-degree,very high leaf density; • class IV: moderate transients, moderate-length periodic attractors, moderate in-degree, very moderateleaf density (possibly complex dynamics); • class III: very long transients, very long periodic attractors, low in-degree, low leaf density (chaotic dy-namics).The basins depicted in Fig. 7 show the whole set of non-equivalent basins in ECA rule 126 from l = 2 to l = 18 ( l means length of array) attractors, all they display not high densities from an attractor of mass oneand attractors of mass 14. This way, ECA rule 126 displays some non symmetric basins and some of themhave long transients that induce a relation with chaotic rules. (a) (b) (c) (d)
Figure 6: Periodic patterns calculated from some exemplar attractors.Particularly we can see specific cycles in Fig. 6 where the following structures could be found:(a) static configurations as still life patterns ( l = 8);(b) traveling configurations as gliders ( l = 15);(c) meshes ( l = 12);(d) or empty universes ( l = 14).The cycle diagrams expose only displacements to the left, and this empty universe evolving to the stablestate 0 is constructed all times on the first basin for each cycle, see Fig. 7.This way some cycles could induce a non trivial activity in rule 126, but the associated initial conditionsare not generally predominant. However some information could be derived from periodic patterns that have ahigh frequency inside this evolution space. This can be done by using filters. De Bruijn diagrams [McIntosh, 2009], [Voorhees, 1996] are proven to be an adequate tool for describing evolutionrules in one dimension CA, although originally they were used in shift-register theory (the treatment of sequences Basins and attractors were calculated with
Discrete Dynamical System
DDLab [Wuensche, 2011] available from l = 2 to l = 18.14here their elements overlap each other). Paths in a de Bruijn diagram may represent chains, configurations orclasses of configurations in the evolution space.For a one-dimensional CA of order ( k, r ), the de Bruijn diagram is defined as a directed graph with k r vertices and k r +1 edges. The vertices are labeled with elements of an alphabet of length 2 r . An edge isdirected from vertex i to vertex j , if and only if, the 2 r − i are the same that the 2 r − j forming a neighbourhood of 2 r + 1 states represented by i (cid:5) j . In this case, the edge connecting i to j is labeled with ϕ ( i (cid:5) j ) (the value of the neighbourhood defined by the local function) [Voorhees, 2008].
111 1 1 001
Figure 8: De Bruijn diagram for the ECA rule 126.The extended de Bruijn diagrams [McIntosh, 2009] are useful for calculating all periodic sequences by thecycles defined in the diagram. These ones also show the shift of a sequence for a certain number of generations .Thus we can get de Bruijn diagrams describing periodic sequences for ECA rule 126. (a) (b) (c)
Figure 9: Patterns calculated with extended de Bruijn diagrams, in particular from cycles of order ( x,
2) (thatmeans x -shift in 2-generations).The de Bruijn diagram associated to ECA rule 126 is depicted in Fig. 8. Figure 8 shows that there are twoneighbourhoods evolving into 0 and six neighbourhoods into 1. State 1 has higher frequency. This indicates a De Bruijn diagrams were calculated using NXLCAU21 designed by McIntosh; available in http://delta.cs.cinvestav.mx/~mcintosh/cellularautomata/SOFTWARE.html
Garden of Eden configurations [Amoroso & Cooper,1970] exist. These are configurations that cannot be constructed from other configurations, i.e., configurationswithout ancestors. In one dimension, the subset diagram can calculate quickly the Garden of Eden configurations,and the pair diagram can calculate configurations with multiple ancestors [McIntosh, 1990]. Classical analysisin graph theory has been applied to de Bruijn diagrams for studying topics such as reversibility [Nasu, 1978],[Mora et al., 2005]; in other sense, cycles in the diagram indicate periodic constructions in the evolution of theautomaton if the label of the cycle agrees with the sequence defined by its nodes, taking periodic boundaryconditions. Let us take the equivalent construction of a de Bruijn diagram in order to describe the evolution intwo steps of ECA rule 126 (having now nodes composed by sequences of four symbols); the cycles of this newdiagram are presented in Fig. 9.Cycles inside de Bruijn diagrams can be used for obtaining regular expressions representing a periodicpattern. Figure 9 displays three patterns calculated as: (a) shift − in 2 generations representing a pattern withdisplacement to the left, (b) shift 0 in 2 generations describing a static pattern traveling without displacement,and (c) shift +3 in 2 generations is exactly the symmetric pattern given in the first evolution.So, we can also see in Fig. 9 that it is possible to find patterns traveling in both directions, as gliders ormobile structures. But generally these constructions (strings) cannot live in combination with others structuresand therefore it is really hard to have this kind of objects with such characteristics. Although, ECA rule 126has at least one glider! This will be explained in the next section. Filters are essential tools for discovering hidden order in chaotic or complex rules. Filters were introduced in CAstudies by Wuensche who employed them to automatically classify cell-state transition functions, see [Wuensche,1999]. Also filters related to tiles were successfully applied and deduced in analysing space-time behaviour ofECA governed by rules 110 and 54 [Mart´ınez, 2006b], [Mart´ınez et al., 2006].Figure 10: Filtered space-time configuration in ECA rule 126.This way, we have found that ECA rule 126 has two types of two dimension tiles (which together work asfilters over ϕ R ): • the tile t = (cid:20) (cid:21) , and 16 the tile t = .Filter t works more significantly on configurations generated by ϕ R , the second one is not frequentlyfound although it is exploited when ECA rule 126 is altered with memory (as we will see in the next section).The application of the first filter is effective to discover gaps with little patterns traveling on triangles of ‘1’states in the evolution space. Although even in this case it may be unclear how a dynamics would be interpreted,a careful inspection on the evolution brings to light very small localisations (as still life), as shown in Fig. 10.This localisation emerging in ECA rule 126 and pinpointed by a filter is the periodic pattern calculated withthe basin (Fig. 6a), and with the de Bruijn diagram (Fig. 9b). The last one offers more information becausesuch cycles allow to classify the whole phases when this glider is coded in the initial condition. Circles in Fig. 10show some interesting regions that now are more clear with filters working. Some of them display very simplegliders (stationary), periodic meshes, and non-periodic structures emerging and existing inside chaotic patternsin several generations. CA with memory had open a new family of evolution rules with different and interesting dynamics [Alonso-Sanz,2009a], [Alonso-Sanz, 2011]. In this paper we explore three types of memory: minority , majority , and parity . Inthe latter case, s ( t ) i = x t − τ +1 i ⊕ . . . ⊕ x t − i ⊕ x ti .Figure 11 illustrates three different kinds of dynamics emerging in ECAM rule 126, for some values of τ . Exploring different values of τ , we found that large odd values of τ tend to define macrocells -like patterns[Wolfram, 1994], [McIntosh, 2009], while even values are responsible for a mixture of periodic and chaoticdynamics. Figure 11(a)i illustrates large periodic regions with few complex patterns traveling isolation developedby function φ R min :3 . Figure 11(b) shows the function φ R par :2 , its evolution is more interesting because wecan see the emergence of some complex patterns than also interact producing other types of complex structures,including mobile self-localizations or gliders. By exploring systematically distinct values of τ , we found that φ R maj :4 produces an impressive and non-trivial emergence of patterns traveling and colliding. Fig. 11(c)shows the most interesting evolution with well defined complex patterns, not just mobile self-localisations butalso the emergence of glider guns, they are complex patterns which travel on the evolution space emittingperiodically another kind of gliders.An interesting evolution is starting with a single non-quiescent cell. Particularly, φ R maj :4 displays agrowth complex behaviour. An example of this space-time configuration is given in Fig. 12 showing the first1152 steps, where in this case the automaton needed other 30,000 steps to reach a stationary configuration. Filteris convenient to eliminate the non relevant information about gliders. At the same figure we can see a numberof gliders, glider guns, still-life configurations, and a wide number of combinations of such patterns colliding andtraveling with different velocities and densities. Consequently, we can classify a number of periodic structures,objects, and interesting reactions. For full details about ECAM φ R maj :4 please see the paper [Mart´ınez etal., 2010b]. Another case was presented with the ECA rule 30 in [Mart´ınez et al., 2010a], and others ECA rulesin [Mart´ınez et al., 2012a].By selecting a majority memory function on the chaotic ECA rule 126 we can transform its dynamics tocomplex dynamics. Thus, for some CA rule with a memory m function φ and value τ we can derive a complexsystem from a chaotic system or vice versa, transform a chaotic system to complex.Further we explore systematically — on the 88 equivalent ECA rules — if memory functions are able tocover the Wolfram’s classes transforming each class. This way, we prove experimentally that each class may Evolutions of φ R maj : τ were calculated with OSXLCAU21 system available in http://uncomp.uwe.ac.uk/genaro/OSXCASystems.html a) (b) (c) Figure 11: (a) φ R min :3 displays a typical evolution of ECAM rule 126 with minority memory τ = 3, (b) φ R par :2 displays an evolution but now evolving with parity memory, and (c) the most interesting evolutionwith ECAM rule φ R maj :4 , where we can see the emergence of complex patterns as gliders and glider guns.In this case a filter is selected for a best view of complex patterns and their interactions. Snapshots start withsame random initial conditions on a ring of 296 cells evolving in 1036 generations.18igure 12: Filtered space-time configuration of ECAM φ R maj :4 evolving with a ring of 843 cells, periodicboundaries, starting just from one non-quiescent cell and running for 1156 steps.19igure 13: Continued evolution to 2312 steps.20igure 14: Continued evolution to 3468 steps.21igure 15: Continuee evolution to 4624 steps.22igure 16: Continued evolution to 5780 steps.23 ump to another class with a kind of memory. We will also show that by selecting a memory we can reach anyother class starting from any class. The full exploration is showed in the appendix ?? . There are varieties of CA classifications, including Wolfram’s classes [Wolfram, 1983], intra- and inter-classconnection probabilities [Li & Packard, 1990], λ -parameter [Langton, 1986], classification by patterns [Aizawa &Nishikawa, 1986], Z -parameter and attractors basin [Wuensche & Lesser, 1992], [Wuensche, 1999], local structureapproximation [Gutowitz et al., 1987], mean field and de Bruijn approximation [McIntosh, 1990], non-trivialcollective behaviours [Chat´e & Manneville, 1992], glider classification [Eppstein, 1999], equivalence computation[Sutner, 2009], morphology-bases classification [Adamatzky et al., 2006], nonlinear dynamics [Chua, 2006],[Chua, 2007], [Chua, 2009], [Chua, 2011], [Chua, 2012], [Mainzer & Chua, 2012], communication complexity[D¨urr et al., 2004], generative morphological diversity [Adamatzky & Mart´ınez, 2010], basis of lattice analysis[Gunji, 2010], genetic algorithms [Das et al., 1994], compression-based approach [Zenil, 2010], expressiveness(biodiversity) [Redeker et al., 2013], evolutionary computation [Wolz & de Oliveira, 2008].The present study with memory function opens a new and complementary properties on CA classes, pro-ducing a number of interesting properties. In this section, we propose a classification based in memory functions. This tables are published in [Mart´ınez].A ECAM is a ECA composed with a memory function, the new rule opens new and extended domain ofrules based in the ECA domain [Mart´ınez et al., 2010a].To derive a new rule from a basic ECA rule one should select an ECA rule and compose this rule with amemory function (in our analysis we have considered three basic functions : majority, minority, and parity).Therefore, the memory function will determine if the original ECA rule preserves the same class (respective toWolfram’s classes) or if it changes to another class.Following this simple principle, we know now that ECA rules composed with memory can be classified asfollows: strong , because the memory functions are unable to transform one class to another; moderate , because the memory function can transform the rule to another class and conserve the sameclass as well; weak , because the memory functions do most transformations and the rule changes to another differentclass quickly.Table 1 presents the ECA classification based in memory functions.
Proposition 1.
Dynamics of CA from a ‘strong class’ can be changed by any memory function.
Proposition 2.
Dynamics of CA from ‘moderate class’ can be changed by at least one or more memoryfunctions.
Proposition 3.
Dynamics of CA from ‘weak class’ cannot be affected by none kind of memories studied inpresent paper.Memory classification presents a number of interesting properties.24lassificationtype num. rules strong
39 2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30, 34,35, 41, 42, 45, 46, 54, 56, 57, 58, 62, 94, 106,108, 110, 122, 126, 128, 130, 138, 146, 152,154, 162, 170, 178, 184. moderate
34 1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37,38, 40, 43, 44, 72, 73, 74, 77, 78, 104, 132,134, 136, 140, 142, 156, 160, 164, 168, 172. weak
15 0, 12, 19, 23, 36, 50, 51, 60, 76, 90, 105, 150,200, 204, 232.Table 1: ECAM classification.We have ECA rules which composed with a particular kind of memory are able to reach another classincluding the original dynamic. The main feature is that, at least, this rule with memory is able to reach everydifferent class. Rules with this property are called universal ECAM (5 rules). universal ECAM:
22, 54, 130, 146, 152.Particularly, all these UECAM are classified as strong in ECAM’s classification. strong:
22, 54, 130, 146, 152. moderate: - weak: -On the other hand, we have ECA that when composed with memory are able to yield a complex ECAM butwith elements of the original ECA rule. They are called complex ECAM (44 rules). Several of these complexrules are illustrated in Appendix A. complex ECAM:
6, 9, 10, 11, 13, 15, 22, 24, 25, 26, 27, 30, 33, 35, 38, 40,41, 42, 44, 46, 54, 57, 58, 62, 72, 77, 78, 106, 108, 110, 122,126, 130, 132, 138, 142, 146, 152, 156, 162, 170, 172, 178,184.and they can be particularised in terms of ECAM’s classification, as follows: strong:
9, 10, 11, 15, 22, 24, 25, 26, 30, 35, 41, 42, 46, 54, 57, 58,62, 106, 108, 110, 122, 126, 130, 138, 146, 152, 162, 170,178, 184. moderate:
6, 13, 27, 33, 38, 40, 44, 72, 77, 78, 132, 142, 156, 172. weak: -It is remarkable that none of the rules classified in weak class is able to reach complex behaviour. These setof rules are strongly robust to any perturbation in terms of ECAM’s classification.
In this instance, we will compare several ECA classifications reported in CA literature all along the CA-historyversus memory classification. 25 .2.1 Wolfram’s classification (1984)
Wolfram’s classification in “Universality and complexity in cellular automata”, establishes four classes: { uniform (class I), periodic (class II), chaotic (class III), complex (class IV) } See details in [Wolfram, 1994], [Martin et al., 1984], [Wolfram, 2002]. class I:
0, 8, 32, 40, 128, 136, 160, 168. strong: moderate:
8, 32, 40, 136, 160, 168. weak: class II:
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24, 25,26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 42, 43, 44, 46, 50, 51,56, 57, 58, 62, 72, 73, 74, 76, 77, 78, 94, 104, 108, 130, 132,134, 138, 140, 142, 152, 154, 156, 162, 164, 170, 172, 178,184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 26, 34, 35, 42, 46, 56, 57, 58, 62,94, 108, 130, 138, 152, 154, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 13, 14, 27, 28, 29, 33, 37, 38, 43, 44, 72, 73,74, 77, 78, 104, 132, 134, 140, 142, 156, 164, 172. weak:
12, 19, 23, 36, 50, 51, 76, 200, 204, 232. class III:
18, 22, 30, 45, 60, 90, 105, 122, 126, 146, 150. strong:
18, 22, 30, 45, 122, 126, 146. moderate: - weak:
60, 90, 105, 150. class IV:
41, 54, 106, 110. strong:
41, 54, 106, 110. moderate: - weak: - Li and Packard’s classification in “The Structure of the Elementary Cellular Automata Rule Space”, establishesfive ECA classes: { null, fixed point, periodic, locally chaotic, chaotic } .For details please see [Li & Packard, 1990]. null:
0, 8, 32, 40, 128, 136, 160, 168. strong: moderate:
8, 32, 40, 136, 160, 168. weak:
0. 26 xed point:
2, 4, 10, 12, 13, 24, 34, 36, 42, 44, 46, 56, 57, 58, 72, 76, 77,78, 104, 130, 132, 138, 140, 152, 162, 164, 170, 172, 184,200, 204, 232. strong:
2, 10, 24, 34, 42, 46, 56, 57, 58, 130, 138, 152, 162, 170,184. moderate:
4, 13, 44, 72, 77, 78, 104, 132, 140, 164, 172. weak:
12, 36, 76, 200, 204, 232. periodic:
1, 3, 5, 6, 7, 9, 11, 14, 15, 19, 23, 25, 27, 28, 29, 33, 35, 37,38, 41, 43, 50, 51, 74, 94, 108, 131(62), 134, 142, 156, 178. strong:
7, 9, 11, 15, 25, 35, 41, 62, 94, 108, 178. moderate:
1, 3, 5, 6, 14, 27, 28, 29, 33, 37, 38, 43, 74, 134, 142, 156. weak:
19, 23, 50, 51. locally chaotic:
26, 73, 154. strong:
26, 154. moderate: weak: - chaotic:
18, 22, 30, 45, 54, 60, 90, 105, 106, 132, 129(126), 137(110),146, 150, 161(122). strong:
18, 22, 30, 45, 54, 106, 122, 126, 110, 122, 146. moderate: - weak:
60, 90, 105, 150.
Wuensche in “The Global Dynamics of Cellular Automata”, establishes three ECA kinds of symmetries: { symmetric, semi-asymmetric, full-asymmetric } .For details please see [Wuensche & Lesser, 1992]. symmetric:
0, 1, 4, 5, 18, 19, 22, 23, 32, 33, 36, 37, 50, 51, 54, 72, 73,76, 77, 90, 94, 104, 105, 108, 122, 126, 128, 132, 146, 150,160, 164, 178, 200, 204, 232. strong:
18, 22, 54, 108, 122, 126, 128, 146, 178. moderate:
1, 4, 5, 32, 33, 72, 73, 77, 104, 132, 160, 164. weak:
0, 19, 23, 36, 50, 51, 76, 90, 105, 150, 200, 204. semi-asymmetric:
2, 3, 6, 7, 8, 9, 12, 13, 26, 27, 30, 34, 35, 38, 40, 41,44, 45, 58, 62, 74, 78, 106, 110, 130, 134, 136, 140,154, 162, 168, 172. strong:
2, 7, 9, 26, 30, 34, 35, 41, 45, 58, 62, 106, 110, 130,154, 162. moderate:
3, 6, 8, 13, 27, 38, 40, 44, 74, 78, 134, 136, 140, 168,172. weak:
12. 27 ull-asymmetric:
10, 11, 14, 15, 24, 25, 28, 29, 42, 43, 46, 57, 60, 138,142, 152, 156, 170, 184. strong:
10, 11, 15, 24, 25, 42, 46, 57, 138, 152, 170, 184. moderate:
14, 28, 29, 43, 142, 156. weak: chain rules:
30, 45, 106, 154. strong:
30, 45, 106, 154. moderate: - weak: - Index complexity in “A Nonlinear Dynamics Perspective of Wolframs New Kind of Science. Part I: Thresholdof Complexity”, establishes three ECA classes: { red ( k = 1 ), blue ( k = 2 ), green ( k = 3 ) } .For details please see [Chua et al., 2002]. red ( k = 1 ):
0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 19, 23, 32, 34,35, 42, 43, 50, 51, 76, 77, 128, 136, 138, 140, 142, 160, 162,168, 170, 178, 200, 204, 232. strong:
2, 7, 10, 11, 15, 34, 35, 42, 128, 138, 162, 170, 178. moderate:
1, 3, 4, 5, 8, 13, 14, 32, 43, 77, 136, 140, 142, 160, 168. weak:
0, 12, 19, 23, 50, 51, 76, 200, 204, 232. blue ( k = 2 ):
6, 9, 18, 22, 24, 25, 26, 28, 30, 33, 36, 37, 38, 40, 41, 44,45, 54, 56, 57, 60, 62, 72, 73, 74, 90, 94, 104, 106, 108, 110,122, 126, 130, 132, 134, 146, 152, 154, 156, 164. strong:
9, 18, 22, 24, 25, 26, 30, 41, 45, 54, 56, 57, 62, 94, 106, 108,110, 122, 126, 130, 146, 152, 154. moderate:
6, 28, 33, 37, 38, 40, 44, 72, 73, 74, 104, 132, 134, 156, 164. weak:
36, 60, 90. green ( k = 3 ):
27, 29, 46, 58, 78, 105, 150, 172, 184. strong:
46, 58, 184. moderate:
27, 29, 78, 172. weak: .2.5 Density parameter with d -spectrum classification (2003) Density parameter with d -spectrum in “Experimental Study of Elementary Cellular Automata Dynamics Usingthe Density Parameter”, establishes three ECA classes: { P, H, C } .For details please see [Fat`es, 2003]. P:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 50,51, 56, 57, 58, 62, 72, 74, 76, 77, 78, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 156, 160, 162, 164, 168,170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 56, 57, 58, 62, 108,128, 130, 138, 152, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 132, 134, 136, 140, 142, 156, 160, 164,168, 172. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232. H:
26, 41, 54, 73, 94, 110, 154. strong:
26, 41, 94, 110, 154. moderate: weak: - C:
18, 22, 30, 45, 60, 90, 105, 106, 122, 126, 146, 150. strong:
18, 22, 30, 45, 106, 122, 126, 146. moderate: - weak:
60, 90, 105, 150.
Communication complexity classification in “Cellular Automata and Communication Complexity” establishesthree ECA classes: { bounded, linear, other } .For details see [D¨urr et al., 2004]. bounded:
0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 19, 24, 27, 28, 29, 32,34, 36, 38, 42, 46, 51, 60, 71(29), 72, 76, 78, 90, 105, 108,128, 130, 136, 138, 140, 150, 154, 156, 160, 162(missing),170, 172, 200, 204. strong:
2, 7, 10, 24, 34, 42, 46, 108, 128, 130, 138, 154, 162, 170. moderate:
1, 3, 4, 5, 8, 13, 15, 27, 28, 29, 32, 38, 72, 78, 136, 140, 156,160, 172. weak:
0, 12, 19, 36, 51, 60, 76, 90, 105, 150, 200, 204.29 inear:
11, 14, 23, 33, 35, 43, 44, 50, 56, 58, 77, 132, 142, 152, 168,178, 184, 232. strong:
11, 35, 56, 58, 152, 178, 184. moderate:
14, 33, 43, 44, 77, 132, 142, 168. weak:
23, 50, 232. other:
6, 9, 18, 22, 25, 26, 30, 37, 40, 41, 45, 54, 57, 62, 73, 74, 94,104, 106, 110, 122, 126, 134, 146, 164. strong:
9, 18, 22, 25, 26, 30, 41, 45, 54, 57, 62, 94, 106, 110, 122,126, 146. moderate:
6, 37, 40, 73, 74, 104, 134, 164. weak: -Additionally, bound class can be refined in other four subclasses. bounded by additivity:
15, 51, 60, 90, 105, 108, 128, 136, 150,160, 170, 204. strong:
15, 51, 108, 128, 170. moderate: weak:
60, 90, 105, 150, 204. bounded by limited sensibility:
0, 1, 2, 3, 4, 5, 8, 10, 12,19, 24, 29, 34, 36, 38, 42,46, 72, 76, 78, 108, 138,200. strong:
2, 10, 24, 34, 42, 46, 108,138. moderate:
1, 3, 4, 5, 8, 29, 38, 72, 78. weak:
0, 12, 19, 36, 76, 200. bounded by half-limited sensibility:
7, 13, 28, 140, 172. strong: moderate:
13, 28, 140, 172. weak: - bounded for any other reason:
27, 32, 130, 156, 162. strong: moderate:
27, 32, 156. weak: - Topological classification in “A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII:Isles of Eden”, establishes six ECA classes: { period-1, period-2, period-3, Bernoulli σ t -shift, complex Bernoulli-shift, hyper Bernoully-shift } .30or details please see [Chua et al., 2007]. period-1:
0, 4, 8, 12, 13, 32, 36, 40, 44, 72, 76, 77, 78, 94, 104, 128,132, 136, 140, 160, 164, 168, 172, 200, 204, 232. strong:
94, 128. moderate:
4, 8, 13, 32, 40, 44, 72, 77, 78, 104, 132, 136, 140, 160, 164,168, 172. weak:
0, 12, 36, 76, 200, 204, 232. period-2:
1, 5, 19, 23, 28, 29, 33, 37, 50, 51, 108, 156, 178. strong: moderate:
1, 5, 28, 29, 33, 37, 156. weak:
19, 23, 50, 51. period-3: strong: moderate: - weak: - Bernoulli σ t -shift:
2, 3, 6 , 7, 9, 10, 11, 14, 15, 24, 25, 27, 34, 35,38, 42, 43, 46, 56, 57, 58, 74, 130, 134, 138,142, 152, 162, 170, 184. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56,57, 58, 130, 138, 152, 162, 170, 184. moderate:
3, 6, 14, 27, 38, 43, 74, 134, 142. weak: - complex Bernoulli-shift:
18, 22, 54, 73, 90, 105, 122, 126, 146, 150. strong:
18, 22, 122, 126, 146. moderate: weak:
90, 105, 150. hyper Bernoully-shift:
26, 30, 41, 45, 60, 106, 110, 154. strong:
26, 30, 41, 45, 110, 154. moderate: - weak: Power spectral classification in “Power Spectral Analysis of Elementary Cellular Automata”, establishes fourECA classes: { category 1: extremely low power density, category 2: broad-band noise, category 3: power law spectrum,exceptional rules } .For details please see [Ninagawa, 2008]. 31 ategory 1 extremely low power density:
0, 1, 4, 5, 8, 12, 13, 19,23, 26, 28, 29, 33, 37,40, 44, 50, 51, 72, 76, 77,78, 104, 128, 132, 133(94),136, 140, 156, 160, 164,168, 172, 178, 200, 232. strong:
26, 94, 128, 178. moderate:
1, 4, 5, 8, 13, 28, 29, 33,37, 40, 44, 72, 77, 78, 104,132, 136, 140, 156, 160,164, 168, 172. weak:
0, 12, 19, 23, 50, 51, 76,200, 232. category 2 broad-band noise:
2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24,25, 26, 27, 30, 34, 35, 38, 41, 42, 43,45, 46, 56, 57, 58, 60, 74, 90, 105, 106,129(126), 130, 134, 138, 142, 146, 150,152, 154, 161(122), 162, 170, 184. strong:
2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30,34, 35, 41, 42, 45, 46, 56, 57, 58, 106,122, 126, 130, 138, 146, 152, 154, 162,170, 184. moderate:
3, 6, 14, 27, 38, 43, 74, 134, 142, . weak:
60, 90, 105, 150. category 3 power law spectrum:
54, 62, 110. strong:
54, 62, 110. moderate: . weak: . exceptional rules:
73, 204. strong: - moderate: weak: Morphological diversity classification in “On Generative Morphological Diversity of Elementary Cellular Au-tomata”, establishes five ECA classes: { chaotic, complex, periodic, two-cycle, fixed point, null } .See details in [Adamatzky & Mart´ınez, 2010]. 32 haotic:
2, 10, 18, 22, 24, 26, 30, 34, 42, 45, 56, 60, 73, 74, 90, 94,105, 106, 126, 130, 138, 150, 152, 154, 161(122), 162, 170,184. strong:
2, 10, 18, 22, 24, 26, 30, 34, 42, 56, 94, 106, 122, 126, 130,138, 152, 154, 162, 170, 184. moderate:
73, 74. weak:
60, 90, 105, 150. complex:
54, 110. strong:
54, 110. moderate: - weak: - periodic:
18, 26, 60, 90, 94, 154. strong:
18, 26, 94, 154. moderate: - weak:
60, 90. two-cycle:
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24, 25,27, 28, 29, 33, 34, 35, 36, 37, 38, 42, 43, 44, 46, 50, 51, 56,58, 74, 76, 106, 108, 130, 132, 134, 138, 140, 142, 152, 156,162, 164, 170, 172, 178, 184, 204. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56, 58, 106, 108,130, 138, 152, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 13, 14, 27, 28, 29, 33, 37, 38, 43, 44, 74, 132,134, 140, 142, 156, 164, 172. weak:
12, 19, 23, 36, 50, 51, 76, 204. fixed point:
0, 2, 4, 8, 10, 11, 12, 13, 14, 24, 32, 34, 36, 40, 42, 43, 44,46, 50, 56, 57, 58, 72, 74, 76, 77, 78, 104, 106, 108, 128,130, 132, 136, 138, 140, 142, 152, 160, 162, 164, 168, 170,172, 178, 184, 200, 204, 232. strong:
2, 10, 11, 24, 34, 42, 46, 56, 57, 58, 106, 108, 128, 130, 138,152, 162, 170, 178, 184. moderate:
4, 8, 13, 14, 32, 40, 43, 44, 72, 74, 77, 78, 104, 132, 136,140, 142, 160, 164, 168, 172. weak:
0, 12, 36, 50, 76, 200, 204, 232. null:
0, 8, 32, 40, 72, 104, 128, 136, 160, 168, 200, 232. strong: moderate:
8, 32, 40, 72, 104, 136, 160, 168. weak:
0, 200, 232. 33 .2.10 Distributive and non-distributive lattices classification (2010)
Distributive and non-distributive lattices classification in “Inducing Class 4 Behavior on the Basis of LatticeAnalysis”, establishes four ECA classes: { class 1, class 2, class 3, class 4 } .See details in [Gunji, 2010]. class 1:
0, 32, 128, 160, 250(160), 254(128). strong: moderate:
32, 160. weak: class 2:
4, 36, 50, 72, 76, 94, 104, 108, 132, 164, 178, 200, 204,218(164), 232, 236(200). strong:
94, 108, 178. moderate:
4, 72, 104, 132, 164. weak:
36, 50, 76, 200, 204, 232. class 3:
18, 22, 54, 122, 126, 146, 150, 182(146). strong:
18, 22, 54, 122, 126, 146. moderate: - weak: class 4: strong: moderate: . weak: . Topological classification in “A Full Computation-Relevant Topological Dynamics Classification of ElementaryCellular Automata”, establishes four ECA classes: { equicontinuous, almost-equicontinuous, sensitive, sensitive positively expansive } .See details in [Sch¨ule & Stoop, 2012], [Cattaneo et al., 2000]. equicontinuous:
0, 1, 4, 5, 8, 12, 19, 29, 36, 51, 72, 76, 108, 200, 204. strong: moderate:
1, 4, 5, 8, 29, 72. weak:
0, 12, 19, 36, 51, 76, 200, 204.34 lmost-equicontinuous:
13, 23, 28, 32, 33, 40, 44, 50, 73, 77, 78,94, 104, 128, 132, 136, 140, 156, 160,164, 168, 172, 178, 232. strong:
94, 128, 178. moderate:
13, 28, 32, 40, 73, 77, 78, 104, 132, 136,140, 156, 160, 164, 168, 172. weak:
23, 50, 232. sensitive:
2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24, 25, 26, 27, 30, 34,35, 37, 38, 41, 42, 43, 45, 46, 54, 56, 57, 58, 60, 62, 74, 106,110, 122, 126, 130, 134, 138, 142, 146, 152, 154, 162, 170,184. strong:
2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30, 34, 35, 41, 42, 45,46, 54, 56, 57, 58, 62, 106, 110, 122, 126, 130, 138, 146, 152,154, 162, 170, 184. moderate:
3, 6, 14, 27, 37, 38, 43, 74, 134, 142. weak: sensitive positively expansive:
90, 105, 150. strong: - moderate: - weak:
90, 105, 150.Also, this classification can be refined into three sub-classes: weakly periodic, surjective, and chaotic (in thesense of Denavey). weakly periodic:
2, 3, 10, 15, 24, 34, 38, 42, 46, 138, 170. strong:
2, 10, 15, 24, 34, 42, 46, 138, 170. moderate:
3, 38. weak: - surjective:
15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170, 204. strong:
15, 30, 45, 154, 170. moderate: - weak:
51, 60, 90, 105, 150, 204. chaotic (in the sense of Denavey):
15, 30, 45, 60, 90, 105, 106, 150,154, 170. strong:
15, 30, 45, 106, 154, 170. moderate: - weak:
60, 90, 105, 150.
This is a classification by the evolution of a configuration consisting of an isolated one surrounded by zeros, thatis a bit different from conventional ECA classifications previously displayed. In “Expressiveness of ElementaryCellular Automata”, we can see five ECA kinds of expressivity:35
0, periodic patterns, complex, Sierpinski patterns, finite growth } .See details in [Redeker et al., 2013].
0, 7, 8, 19, 23, 31, 32, 40, 55, 63, 72, 104, 127, 128, 136,160, 168, 200, 232. strong:
7, 128. moderate:
8, 32, 40, 72, 104, 136, 160, 168. weak:
0, 19, 23, 200, 232. periodic patterns:
13, 28, 50, 54, 57, 58, 62, 77, 78, 94, 99, 109, 122,156, 178. strong:
54, 57, 58, 62, 94, 122, 178. moderate:
13, 28, 73, 77, 78, 156. weak: complex:
30, 45, 73, 75, 110. strong:
30, 45, 110. moderate: weak: - Sierpinski patterns:
18, 22, 26, 60, 90, 105, 126, 146, 150, 154. strong:
18, 22, 26, 126, 146, 154. moderate: - weak:
60, 90, 105, 150. finite growth:
1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 24, 25, 27, 29, 33, 34,35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 56, 59, 71, 74,76, 103, 106, 107, 108, 111, 130, 132, 134, 138, 140, 142,152, 162, 164, 170, 172, 184, 204. strong:
2, 9, 10, 11, 15, 24, 25, 34, 35, 41, 42, 46, 56, 106, 108, 130,152, 162, 170, 184. moderate:
1, 3, 4, 5, 6, 14, 27, 29, 33, 37, 38, 43, 44, 74, 140, 142, 164,172. weak:
12, 36, 51, 76, 204.
Normalised compression classification in “Asymptotic Behaviour and Ratios of Complexity in Cellular AutomataRule Spaces”, establishes two ECA classes: { C , , C , } .See details in [Zenil & Zapata]. 36 , :
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44,46, 50, 51, 56, 57, 58, 72, 74, 76, 77, 78, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 154, 156, 160, 162, 164,168, 170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 26, 34, 35, 42, 46, 56, 57, 58, 108,128, 130, 138, 152, 154, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 132, 134, 136, 140, 142, 156, 160. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232. C , :
18, 22, 30, 41, 45, 54, 60, 62, 73, 90, 94, 105, 106, 110, 122,126, 146, 150. strong:
18, 22, 30, 41, 45, 54, 62, 94, 106, 110, 122, 126, 146. moderate: weak:
60, 90, 105, 150.
Expressivity classification in “Emergence of Surface Dynamics in Elementary Cellular Automata”, establishesthree ECA classes: { type A, type B, type C } .See details in [Mora et al.]. type A:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 46,50, 51, 56, 57, 58, 72, 74, 76, 77, 78, 94, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 156, 160, 162, 164, 168,170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56, 57, 58, 128,130, 152, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 108, 132, 134, 136, 140, 142, 156, 160,164, 168, 172. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232. type B:
18, 22, 26, 30, 41, 45, 60, 90, 105, 106, 122, 126, 146, 150,154. strong:
18, 22, 26, 30, 45, 106, 122, 126, 146, 154. moderate: - weak:
18, 22, 26, 30, 45, 106, 122, 126, 146, 154. type C:
54, 62, 73, 110. strong:
54, 62, 110. moderate: weak: - 37 .2.15 Spectral classification (2013)
Spectral classification in “A Spectral Portrait of the Elementary Cellular Automata Rule Space”, establishesfour ECA classes: { DE/SFC, DE/SFC SFC, EB, S } .See details in [Ruivo & de Oliveira, 2013]. DE/SFC:
0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 14, 19, 22, 23, 24, 25, 26,27, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 50,54, 56, 57, 58, 62, 72, 73, 74, 76, 77, 94, 104, 108, 110, 128,130, 132, 134, 136, 138, 140, 142, 152, 160, 162, 164, 168,172, 178, 184, 200, 232. strong:
2, 7, 9, 10, 11, 22, 24, 25, 26, 29, 34, 35, 41, 42, 46, 54, 56,57, 58, 62, 94, 108, 110, 128, 130, 138, 152, 162, 178, 184. moderate:
1, 5, 6, 8, 14, 27, 32, 33, 37, 38, 40, 43, 44, 72, 73, 74, 77,104, 132, 134, 136, 140, 142, 160, 164, 168, 172. weak:
0, 12, 19, 23, 36, 50, 76, 200, 232.
DE/SFC SFC:
3, 4. strong: - moderate:
3, 4. weak: - EB:
13, 18, 28, 78, 122, 126, 146, 156. strong:
18, 122, 126, 146. moderate:
13, 28, 78, 156. weak: - S:
15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170, 204. strong:
15, 30, 45, 106, 154, 170. moderate: - weak:
51, 60, 90, 105, 150, 204.
In this section, we have just bijective and surjective classification (personal communication, Harold V. McIntoshand Juan C. Seck Tuoh Mora): { bijective, surjective } .See details in [McIntosh, 1990], [McIntosh, 2009]. bijective:
15, 51, 170, 204. strong:
15, 170. moderate: - weak:
51, 204. 38 urjective:
30, 45, 60, 90, 105, 106, 150, 154. strong:
30, 45, 106, 154. moderate: - weak:
60, 90, 105, 150.
Creativity classification in “On Creativity of Elementary Cellular Automata”, establishes four ECA classes: { creative, schizophrenic, autistic savants, severely autistic } .See details in [Adamatzky & Wuensche]. creative:
3, 5, 11, 13, 15, 35. strong:
11, 15, 35. moderate:
3, 5, 13. weak: - schizophrenic:
9, 18, 22, 25, 26, 28, 30, 37, 41, 43, 45, 54, 57, 60, 62, 73,77, 78, 90, 94, 105, 110, 122, 126, 146, 150, 154, 156. strong:
9, 18, 22, 25, 26, 30, 41, 45, 54, 57, 62, 110, 122, 126, 146,152, 154. moderate:
28, 37, 43, 73, 77, 78, 156. weak:
60, 90, 105. autistic savants:
1, 2, 4, 7, 8, 10, 12, 14, 19, 32, 34, 42, 50, 51, 76,128, 136, 138, 140, 160, 162, 168, 170, 200, 204. strong:
2, 7, 10, 34, 42, 128, 138, 162, 170. moderate:
1, 4, 8, 14, 32, 136, 140, 160, 168. weak:
12, 19, 50, 51, 76, 200, 204. severely autistic:
23, 24, 27, 29, 33, 36, 40, 44, 46, 56, 58, 72, 74, 104,106, 108, 130, 132, 142, 152, 164, 172, 178, 184, 232. strong:
24, 46, 56, 58, 106, 108, 130, 152, 178, 184. moderate:
27, 29, 33, 40, 44, 72, 74, 104, 132, 142, 164, 172. weak:
23, 36, 232.
After checking that memory has similar effect for every rule in the same equivalence class (please see a full de-scription of them in [Wuensche & Lesser, 1992]), we will deal here for simplicity with the canonical representativerule of every one of the 88 equivalence classes, and not explicitly with the 256 rules.In what follows, we enumerate the most important relations. • Transition of uniform to uniform . unif orm φ CAm : τ −−−−−→ unif orm (8)this is transition from ECA ϕ R to ECAM φ R maj :3 .39 Transition of uniform to periodic . unif orm φ CAm : τ −−−−−→ periodic (9)this is transition rom ECA ϕ R to ECAM φ R par :5 . • Transition of uniform to chaos . unif orm φ CAm : τ −−−−−→ chaos (10)this is transition from ECA ϕ R to ECAM φ R par :2 . • Transition of uniform to complex . unif orm φ CAm : τ −−−−−→ complex (11)this is transition from ECA ϕ R to ECAM φ R par :4 . • Transition of periodic to uniform . periodic φ CAm : τ −−−−−→ unif orm (12)this is transition from ECA ϕ R to ECAM φ R maj :4 . • Transition of periodic to periodic . periodic φ CAm : τ −−−−−→ periodic (13)this is transition from ECA ϕ R to ECAM φ R maj :3 . • Transition of periodic to chaos . periodic φ CAm : τ −−−−−→ chaos (14)this is transition from ECA ϕ R to ECAM φ R par :3 . • Transition of periodic to complex . periodic φ CAm : τ −−−−−→ complex (15)this is transition from ECA ϕ R to ECAM φ R par :2 . • Transition of chaos to uniform . chaos φ CAm : τ −−−−−→ unif orm (16)this is transition from ECA ϕ R to ECAM φ R maj :10 .40 Transition of chaos to periodic . chaos φ CAm : τ −−−−−→ periodic (17)this is transition from ECA ϕ R to ECAM φ R maj :4 . • Transition of chaos to chaos . chaos φ CAm : τ −−−−−→ chaos (18)this is transition from ECA ϕ R to ECAM φ R par :2 . • Transition of chaos to complex . chaos φ CAm : τ −−−−−→ complex (19)this is transition from ECA ϕ R to ECAM φ R maj :4 . • Transition of complex to uniform . complex φ CAm : τ −−−−−→ unif orm (20)this is v from ECA ϕ R to ECAM φ R maj :6 . • Transition of complex to periodic . complex φ CAm : τ −−−−−→ periodic (21)this is transition from ECA ϕ R to ECAM φ R par :2 . • Transition of complex to chaos . complex φ CAm : τ −−−−−→ chaos (22)this is transition from ECA ϕ R to ECAM φ R min :3 . • Transition of complex to complex . complex φ CAm : τ −−−−−→ complex (23)this is transition from ECA ϕ R to ECAM φ R maj :8 .Therefore, from transitions 8–23 we we can reach a class from any other class with some kind of memory atleast once.ECAM preserves main characteristics of the original evolution rule and they can be found in both ECA andECAM rules. As was detailed in ECA rule 126, a glider that is found in ECAM φ R maj :4 already there isin the conventional ahistoric formulation rule (section 3.6). This way, dynamics in ECA move around of thememory effect in ECAM. As a consequence from this systematic analysis, we have that: Proposition 4.
Dynamics in ECAM also cannot be induced from some previous ECA.41
IIIIIIV
2, 7, 24, 42, 56, 130, 152, 162
2, 7, 9, 11, 15, 34, 35, 46, 56, 57, 58, 94, 108, 130, 138, 152, 154, 162, 170, 178, 1849, 15, 24, 25, 26, 42, 46, 56, 58, 62, 94, 130, 138, 152, 154, 170, 1849, 10, 11, 24, 25, 26, 35, 42, 46, 58, 62, 108, 130, 138, 162, 170, 178, 18418, 22, 146 18, 22, 30, 14622, 30, 45, 122, 126, 14622, 30, 45, 122, 126, 14654 5441, 54, 106, 11041, 54, 106, 110
Figure 17: “Strong” ECAM class is able to reach some other classes. Starting from a Wolfram’s class (rule)and selecting some kind of memory inside strong
ECAM class, one can reach some other class with such a rule.
I IIIIIIV
8, 32, 40, 136, 160, 168 8, 32, 136, 160, 16840401, 3, 4, 5 1, 3, 4, 5, 6, 13, 14, 27, 28, 29, 33, 37, 38, 43, 44, 72, 73, 74, 77, 78, 104, 132, 134, 140, 142, 156, 164, 17213, 14, 28, 29, 37, 43, 73, 74, 104, 134, 142, 156, 1646, 13, 27, 33, 43, 44, 72, 77, 78, 104, 132, 140, 142, 156, 172
Figure 18: “Moderate” ECAM class is able to reach some other classes. Starting from a Wolfram’s class (rule)and selecting some kind of memory inside moderate
ECAM class, one each some other class with such a rule.42
IIIIIIV
Figure 19: “Weak” ECAM class is not able to reach other classes. Starting from a Wolfram’s class (rule) andselecting some kind of memory inside weak
ECAM class, one cannot reach some classes with such a rule.If you have selected a ECA class I, II, III, or IV; you could obtain a ECAM class I, II, III, or IV withoutsome prefix which determines exactly the result. Diagrams displayed in Fig. 17, 18, 19 show how move betweenclasses. If you choice a specific ECA rule (that is in some Wolfram’s class) hence with a kind of memory you can‘move’ to another class if it is the case. You can see these finite machines with respect to ECAM classification,Fig. 17 for strong class, Fig. 18 for moderate class, and Fig. 19 for the weak class.Finally, diagram in Fig. 20 (all memories) shows a directed graph strongly connected due to the transitions8–23. That means than you can reach any class from any class including them self (loops).As outlined in [Culik II & Yu, 1988] in the conventional ahistoric context, it is not possible to determine thebehaviour of a ECAM from that of its conventional ahistoric ECA. This way, it is undecidable determines thebehaviour of a CAM from any CA. Of course, memory can be selected on any dynamical systems useful mainlyfor discover hidden information, such as was studied in excitable CA [Alonso-Sanz & Adamatzky, 2008].
In this section, we present a kind of complex CA derived since ECA rule 22 with memory. Again, we haveselected the majority memory and we focus on τ = 4, generating a new ECAM rule, φ R maj :4 .Figure 21 displays a typical random evolution of ECAM φ R maj :4 . There we witness emergence of non-trivialtravelling patterns and outcomes of their collisions.The main and most interesting characteristic is that this complex ECA with memory has only two gliders,maybe we can tell only one with its respective reflection. With these gliders G φ R maj :4 = { g L , g R } we can designcomputing circuits (this is a partial result of our research detailed in [Mart´ınez et al. a]).We should start with basic logic gates derived from simple binary collisions. A logic gate performs a logicoperation on one or more logic inputs yielding just one logic output. Normally a logic gate is a Boolean function,such that for some positive integer n we have that f : Σ n → Σ for Σ = { , } , and therefore it can be representedby a truth table that describes the behaviour of a logic gate [Minsky, 1967].Figure 22 displays implementation of not and and gates with gliders G φ R maj :4 and a symmetric delay element. 43 IIIIIIVI IIIIIIVI IIIIIIVWEAKSTRONG MODERATEALL MEMORIESI IIIIIIV
Figure 20: Every ECAM class has rules with behaviour class I, II, III, or IV. If you take one ECA rule witha kind of memory hence you can change to another class. “All memories” diagram show that it is possible toreach any class from some ECA enriched with memory, thus some ECAM is able to reach any class.44igure 21: Typical random evolution of φ R maj :4 from an initial configuration where 37% of cells take state’1’. The automaton is a ring of 767 cells. Evolution is recorded for 372 generations.Generally a problem to implement computations in injective CA is related to the synchronisation of collisionsbetween gliders and accurate positioning of gliders in initial configuration.A majority gate and and gate as is represented in the Fig. 23. A not gate is aggregated to get a nand gate (Fig. 23b).To implement a majority gate we must represent binary values across of gliders (Fig. 24(a)). Later, let usethis construction to implement a nand gate by gliders collisions as Fig. 24(b) display. This way, we use a g R glider that works as an operator processing three input values at the same time. A not gate is represented by asecond g R glider inverting the final result. Also, we can utilise this scheme to design a modified chip related to7400 chip but with four majority and not gates instead of four nand gates, working with three independentinputs per gate on 18 pins as in the Fig. 24(c).Figures 25 and 26 show the implementation of nand gate with φ R maj :4 . As illustrated in diagram (Fig. 24c)a glider works as an operator of the majority gate and this operator is reused in the next majority gate. Wepresent all stages where the nand gate works, thus proving functionality of the design. We demonstrated that a memory is a ‘universal’ switch which allows us to change dynamics of a complexspatially extended systems and to guide the system in a ‘labyrinth’ of complexity classes. Memory allows us tomake complex systems simple and to simple ones complex.The memory implementation mechanism studied here constitute a simple extension (of straightforwardcomputer codification) of the basic CA paradigm allowing for an easy systematic study of the effect of memoryin cellular automata (and other discrete dynamical systems). This may inspire some useful ideas in using cellular National semiconductor web site. Device 5400/DM5400/DM7400 Quad 2-Input
NAND
Gates b ¬ b ¬ aa b (a) a ⇥ ¬ b (b) ¬ a ⇥ b (c) a b (d) delay a (e) Figure 22: We implement basic logic functions as not and and gates via collisions of gliders and a delay element. Single or pair of particles represent bits 0’s or 1’s respectively.
MAJ cbaba
NOT a AND (a)
MAJ
NOT
NAND gate a b c maj nand (b)
Figure 23: A nand gate based in majority and not gates.46 alue 1value 0 (a)
NAND gate M A J a b c operatorNOT (b) MAJ MAJMAJ MAJ G ND V CC
18 1091 (c)
Figure 24: (a) binary values by gliders codification, (b) scheme of a nand gate from majority and not gateswith glider reaction, and (c)circuit based on four nand gates like a modified 7400 chip but now with 18 pins(for the majority gate). 47 (a)
000 1 1 (b)
Figure 25: nand gate implemented from majority and not gates in φ R maj :4 . Inputs (a) 000 and (b) 010.48 (a)
00 111 (b)
Figure 26: nand gate implemented from majority and not gates in φ R maj :4 . Inputs (a) 100 and (b) 110.49utomata as a tool for modeling phenomena with memory. This task has been traditionally attacked by meansof differential, or finite-difference, equations, with some (or all) continuous component. In contrast, full discretemodels are ideally suited to digital computers. Thus, it seems plausible that further study on cellular automatawith memory should prove profitable, and may be possible to paraphrase T. Toffoli [Toffoli, 1984] in presentingcellular automata with memory as an alternative to (rather than an approximation of ) integro-differentialequations in modeling phenomena with memory. Besides their potential applications, cellular automata withmemory have an aesthetic and mathematical interest on their own, so that we believe that the subject is worthto studying.Last but not least, other memories are possible. In this study we have implemented an explicit dependencein the dynamics of the past states in the manner: first summary then rule. But the order summary-rule may beinverted, i.e., the rule is first applied and a summary is then presented as new state (for details see [Alonso-Sanz,2013]. This alternative memory implemention enriches the potential use of memory in discrete systems as atool for modeling, and, again, in our opinion deserves attention on its own. References [1] Amoroso, S. & Cooper, G. [1970] “The Garden-of-Eden theorem for finite configurations,”
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Appendix and full paper is available in http://eprints.uwe.ac.uk/21980/http://eprints.uwe.ac.uk/21980/