Figure 21 - uploaded by Andrew Adamatzky
Content may be subject to copyright.
Source publication
The dynamics of rule 54 one-dimensional two-state cellular automaton (CA) are
a discrete analog of a space-time dynamics of excitations in nonlinear active
medium with mutual inhibition. A cell switches its state 0 to state 1 if one of
its two neighbors is in state 1 (propagation of a perturbation) and a cell
remains in state 1 only if its two neig...
Context in source publication
Context 1
... n is a number of copies of the string. Of course, different parameters will yield a glider with different intervals or a number of gliders. A famous problem established in Conway’s Game of Life was dis- covery of a configuration that will grow permanently, into an infinite evolution space. This problem was solved by Gosper and colleagues at MIT Artificial Intelligence Lab [30]. The same problem can be established in rule 54. Of course, the construction of a glider gun or some other extension is sufficient to demonstrate unlimited growth in rule 54 (Figure 16h). Here we show the production of double glider guns. In [21] McIntosh determined that ECA rule 110 can be studied as a tile problem. What is a largest tile produced via collision between gliders in rule 54? Some answers are given in [8] via studying reactions between gliders. Figure 18(a) shows the construction of a T 16 tile by synchronising multiple collisions between → w , ← w − , and g e gliders. Figure 18(b) shows T 33 tile produced by a chaotic decomposition. Codes to reproduce these reactions are as follows: Rule 54 has been proved to be a ‘universal dynamics rule’ in the ECA memory (ECAM) classification [12]. This means that rule 54 op- erated with some memory functions is able to reach any Wolfram class, including the class IV, to which the memoryless rule 54 belongs [2]. Figure 19 presents evolutions of rule 54 with memory. Each snapshot illustrates four different behaviors. Figure 19(a) shows a uniform evolution with rule φ R 54 maj :6 , Figure 19(b) a periodic behavior with rule φ R 54 maj :10 , Figure 19(c) a chaotic evolution with rule φ R 54 maj :3 , and Figure 19(d) a complex behavior with rule φ R 54 maj :8 . Of course, every memory function represents a different evolution rule but with elements of the original rule. In [11] we show how a number of solitonic collisions can be simulated in rule 54. These solitons can be manipulated to develop some basic computable systems, such as simple substitution systems. In [8] basic logic functions were simulated from basic collisions in rule 54. So far no one has ever implemented an equivalent Turing machine in rule 54. However, taking advantage of codification of gliders in rule 54, we have explored some basic computable functions that could help us to emulate the Turing machine with rule 54 in the future. Some series by reaction gliders are presented in [6]. Here we have three cases. In Figure 20, starting from a collision among three gliders yields an infinite series Z n for n > 2 (without limit boundaries). This sequence is defined by vertical number of T 6 tiles without some perturbation that evolves on each collision. Figure 21 displays an evolution that simulates a parity function 2 k ∀ k ∈ Z . This parity is preserved by number of generations or by number of T 5 tiles ( g o gliders) (without limit boundaries). So, Figure 22 shows a very simple flip-flop configurations that is restricted to limit boundaries. All previous simulations needs more than 1000 generations. Cellular automaton gliders are analogs of optical solitons, kinks in poly- mer chains, excitation in molecular arrays (reaction diffusion computers [32], wave packets used slime mould to communicate information to distant part of the body [33]), and defects in micro-tubules [34]. Also, rule 54 per se is a discrete analog an active nonlinear medium with lateral inhibition between micro-volumes. The lateral inhibition in the nervous system sharpens and strengthens sensor perception and is widely employed in vision and olfactory systems. Thus we can spec- ulate that the rule 54 is a simplest abstract model of the affective nervous system. The gliders then play a role of propagating action po- tential wave packets, and glider guns symbolize activity in the sources of sensorial stimulation. As we can see, there are many analogies of rule 54 behavior in physical and biological systems. And therefore, behavior of these systems can be described by unique subsets of regular expressions, where phase, distance, momentum, position, period, and speed are taken into ...
Similar publications
The statistics of velocities in the cellular automaton model of Nagel and
Schreckenberg for traffic are studied. From numerical simulations, we obtain
the probability distribution function (PDF) for vehicle velocities and the
velocity-velocity (vv) correlation function. We identify the probability to
find a standing vehicle as a potential order par...
This note reviews a bio-inspired scheme for aggregating au-tonomous agents in the absence of global communication or coordination, a problem that is known as Decentralized Gathering. We present results on the clustering behavior of the agents, as we vary the main parameter that controls the agents' aggregation. Our observations show that there exis...
Citations
... Finally, we study class 4 CA, which involves a mixture of order and randomness with localized structures that move and interact in complicated ways 51 . A well-studied example of this is ECA Rule 54, defined in Fig. 4a, which can be interpreted as a discrete analog of excitations in an active nonlinear medium with mutual inhibition 52 . ...
Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena, including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer present new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing using light.
... Finally, we study class 4 CA, which involve a mixture of order and randomness, with localized structures that move and interact in complicated ways [46]. A well-studied example of this is ECA Rule 54, defined in Fig. 4(a), which can be interpreted as a discrete analogue of excitations in an active nonlinear medium with mutual inhibition [47]. ...
Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer presents new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing schemes using light.
... partitions of ECAs' space-time diagram are similar to results reported by Crutchfield (1992, 1997). Particles in Rule 54 have been used to implement computations (Boccara et al., 1991;Pivato, 2007;Martinez et al., 2014). Because the presented reduction reduces the size of the grid, it can merge some of those particles, sometimes resulting in ambiguities and gaps. ...
Emergent processes in complex systems such as cellular automata can perform computations of increasing complexity, and could possibly lead to artificial evolution. Such a feat would require scaling up current simulation sizes to allow for enough computational capacity. Understanding complex computations happening in cellular automata and other systems capable of emergence poses many challenges, especially in large-scale systems. We propose methods for coarse-graining cellular automata based on frequency analysis of cell states, clustering and autoencoders. These innovative techniques facilitate the discovery of large-scale structure formation and complexity analysis in those systems. They emphasize interesting behaviors in elementary cellular automata while filtering out background patterns. Moreover, our methods reduce large 2D automata to smaller sizes and enable identifying systems that behave interestingly at multiple scales.
... partitions of ECAs' space-time diagram are similar to results reported by Crutchfield (1992, 1997). Particles in Rule 54 have been used to implement computations (Boccara et al., 1991;Pivato, 2007;Martinez et al., 2014). Because the presented reduction reduces the size of the grid, it can merge some of those particles, sometimes resulting in ambiguities and gaps. ...
... Finally it should be emphasized that rules 50, 54, 58 and, by reflection, 114, 62 and, by reflection, 118 display patterns somewhat far from the sieve, but nevertheless all of them enter the phase transition polygon. Rules 30, 54, 90 and 126 are examined elsewhere [37][38][39][40]. This tree does not contain the subset of rules associated either by conjugation or by conjugation-reflection. ...
... Space-time configurations of ECA derived from these diagrams are illustrated on the left plate of Fig. 5. Position of each mobile localisation and periodic background follows arbitrarily routes into these cycles. Details on these regular expressions for Rule 54 are presented in [16]. ...
... Therefore, we only list the indexes (i, j) of nonzero elements. Y = {(1, 2), (1, 44), (2, 3), (2, 119), (3,4), (4,5), (4, 58), (5, 6), (6, 7), (7, 8), (7, 89), (8,9), (9, 10), (10,11), (10,15), (11,12), (12,13), (12, 31), (13,14), (13,72), (14,1), (15,16), (16,17), (16,153), (17,18), (18,19), (19,20), (20,21), (21,22), (22,23), (23,24), (24,25), (25,26), (26,27), (27,28), (28,29), (29,30), (30,2), (30,44), (31,32), (32,33), (33,34) Denote the elements of Y n as Y n i,j ,1 ≤ i, j ≤ 167. Y n i,j indicates the number of the whole paths from i-th vertex to j-th vertex whose length is n. ...
... Therefore, we only list the indexes (i, j) of nonzero elements. Y = {(1, 2), (1, 44), (2, 3), (2, 119), (3,4), (4,5), (4, 58), (5, 6), (6, 7), (7, 8), (7, 89), (8,9), (9, 10), (10,11), (10,15), (11,12), (12,13), (12, 31), (13,14), (13,72), (14,1), (15,16), (16,17), (16,153), (17,18), (18,19), (19,20), (20,21), (21,22), (22,23), (23,24), (24,25), (25,26), (26,27), (27,28), (28,29), (29,30), (30,2), (30,44), (31,32), (32,33), (33,34) Denote the elements of Y n as Y n i,j ,1 ≤ i, j ≤ 167. Y n i,j indicates the number of the whole paths from i-th vertex to j-th vertex whose length is n. ...
We overview networks which characterise dynamics in cellular automata. These networks are derived from one-dimensional cellular automaton rules and global states of the automaton evolution: de Bruijn diagrams, subsystem diagrams, basins of attraction, and jump-graphs. These networks are used to understand properties of spatially-extended dynamical systems: emergence of non-trivial patterns, self-organisation, reversibility and chaos. Particular attention is paid to networks determined by travelling self-localisations, or gliders.
... Space-time configurations of ECA derived from these diagrams are illustrated on the left plate of Fig. 9. Position of each glider and periodic background follows arbitrarily routes into these cycles. Details on these regular expressions for rule 54 are presented in Martínez et al. (2014). ...
... Maximum high in this tree has 32 transients before to reach the attractor. Particularly, if we concatenate the leaf 41,819 on the initial condition its evolution will converge to a meta-glider in ECA rule 54 preserved by multiple collisions between three gliders Martínez et al. (2014). Extended analysis with cycle diagrams implies meta diagrams interconnecting not configuration but basins of attractions, where complex rules display diagrams strongly connected Martínez et al. (2017). ...
... In general, gliders are localized structures of nonquiescent and nonether patterns (ether represents a periodic background) translating along the automaton's lattice. Importantly, as gilders are crucial components of conception and simulation of the universal computing model, the concrete rules with a great variety of gliders have captured special attention in a series of CA research work [Martínez et al., 2010a;Cook, 2004;Freire & Gallas, 2007;Chen et al., 2012;Martínez et al., 2014]. It leads to an intriguing possibility: HCAM(43, 74) is an ideal candidate for the universal computing. ...
... And the whole background of ether can be obtained by splicing this ether unit repeatedly without any gap or overlap. To help visualize this, Fig. 7 provides a schematic diagram of ether matching according to the definition of tile in [Martínez et al., 2014]. ...
In the case of one-dimensional cellular automaton (CA), a hybrid CA (HCA) is the member whose evolution of the cells is dependent on nonunique global functions. The HCAs exhibit a wide range of traveling and stationary localizations in their evolution. We focus on HCA with memory (HCAM) because they produce a host of gliders and complicated glider collisions by introducing the hybrid mechanism. In particular, we undertake an exhaustive search of gliders and describe their collisions using quantitative approach in HCAM(43, 74). By introducing the symbol vector space and exploiting the mathematical definition of HCAM, we present an analytical method of complex asymptotic dynamics of the gliders.
... For rule 54, the starting point was the work by Boccara et al. [4]; they identified the most common particles that arise from random ini-tial configurations, described their interactions and gave them the names that are still used. This research was later continued by the group around McIntosh [5,8,9], who found more complex particles and interactions. ...
This is a study of localized structures in one-dimensional cellular automata, with the elementary cellular automaton rule 54 as a guiding example. A formalism for particles on a periodic background is derived, applicable to all one-dimensional cellular automata. We can compute which particles collide and in how many ways. We can also compute the fate of a particle after an unlimited number of collisions—whether they only produce other particles, or the result is a growing structure that destroys the background pattern. For rule 54, formulas for the four most common particles are given and all two-particle collisions are found. We show that no other particles arise, which particles are stable and which can be created, provided that only two particles interact at a time. More complex behavior of rule 54 requires therefore multi-particle collisions. © 2017, Complex Systems Publications, Inc. All rights reserved.
... It is worth mentioning that Cook proved that ECA rule 110 is universal via simulating a cyclic tag system [20]. In their study of unconventional computation, Martínez et al. highlighted the dynamical characteristics of gliders in ECA rules 110 and 54 [21][22][23][24]. However, as gliders in ECA rule 54 are less complicated than those in ECA rule 110, so far no literature has proven that rule 54 is universal. ...
