- Email: [email protected]
A structurally dynamic cellular automaton with memory
Chaos, Solitons and Fractals 32 (2007) 1285–1295 www.elsevier.com/locate/chaos
A structurally dynamic cellular automaton with memory Ramo´n Alonso-Sanz
*
ETSI Agro´nomos (Estadı´stica), C. Universitaria, Grupo de Sistemas Complejos—UPM, 28040 Madrid, Spain Accepted 28 December 2005
Abstract Major features of conventional cellular automata (CA) include the inalterability of topology and the absence of memory. The effect of simple memory (memory in cells and links) in a particular structurally dynamic CA is explored in this paper. 2006 Elsevier Ltd. All rights reserved.
1. Cellular automata Cellular automata (CA) are discrete, spatially explicit extended dynamic systems. A CA system is composed of adjacent cells or sites arranged as a regular lattice, which evolves in discrete time steps. Each cell is characterised by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common ðT Þ local transition rule involving only a neighborhood of each cell [1,2]. Thus, if ri is taken to denote the value of cell i at ðT þ1Þ
time step T, the site values evolve by iteration of the mapping: ri
ðT Þ
¼ /ðrj
2 Ni Þ, where / is an arbitrary function
which specifies the cellular automaton rule operating on the neighborhood N of the cell i. In totalistic rules the of P value ðT þ1Þ ðT Þ a site depends only on the sum of the values of its neighbours, and not on their individual values: ri ¼ /ð j2Ni rj Þ. This paper deals with a particular two-dimensional totalistic CA rule with two possible values at each site P ðT þ1Þ ðT Þ (r 2 {0, 1}), the parity rule: ri ¼ j2Ni rj mod 2. In Section 2 (and initially in Sections 3 and 4) it is adopted the von Neumann neighborhood (NSWE), so the sum of neighbor states ranges in the {0, 5} set, and the values that activate a cell state are 1, 3 and 5. Regardless its formal simplicity, the parity rule exhibits a complex behavior [3]. Elementary CA refer to the simplest scenario: one-dimensional CA with two possible values at each site, with rules operating on nearest neighbors. These rules are characterised by a sequence of binary values (b) associated with each of ðT Þ
ðT Þ
ðT Þ
the eight possible triplets ðri1 ; ri ; riþ1 Þ:
*
111
110 101
100 011 010
001 000
ðb1
b2
b4
b7
b3
b5
b6
b8 Þbinary
8 P
bi 28i decimal ¼ R 2 ½0; 255.
i¼1
Fax: +34 91 3365871. E-mail address: [email protected]
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.047
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R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
We consider here (Sections 2 and 4) only elementary legal rules: rules that are reflection symmetric (b2 = b5 and b4 = b7) and quiescent (b8 = 0). These restrictions leave 32 possible legal rules.
2. Cellular automata with memory Standard CA are ahistoric (memoryless): the transition function depends on the cells neighborhood configuration only at the preceding time step. Historic memory can be embedded in the CA dynamics by featuring every cell by a mapping of its states in the previous time steps. Thus, what is here proposed is to maintain the transition rules (/) unalðT þ1Þ ðT Þ ðT Þ ¼ /ðsj 2 Nj Þ, si being a tered, but make them act on the cells featured by a function of their previous states: ri state function of the series of states of the cell i after time-step T. In the approach adopted here, memory becomes operð1Þ
ð1Þ
ð2Þ
ð2Þ
ative after T = 3, with the initial assignations si ¼ ri , si ¼ ri . Cellular automata implementing memory in cells will be termed historic, and the standard ones ahistoric. As stated, the historic and ahistoric evolving patterns are the same up to T = 3. Thus, cells can be featured by a weighted mean value of all their previous states [4–13]:
ðT Þ si
¼
8 1 > > > <
ðT Þ r > i
> > :
0
ðT Þ
if mi
> 0:5 ðT Þ
if
ðT Þ mi
¼ 0:5
ðT Þ
< 0:5
if mi
after
ðT Þ
ð1Þ
ð2Þ
ðT Þ
mi ðri ; ri ; . . . ; ri Þ ¼
PT 1
ðtÞ
aT t ri . Pt¼1 T 1 T t 1 þ t¼1 a
ri
þ
As an example, Fig. 1 shows the evolving patterns of the two-dimensional parity rule with weighted memory starting from a single site live cell. Memory truncates the natural expansive evolution of the parity rule, particularly for high values of the memory factor a in which case small size oscillators of short period appear. But the effect is also dramatic for low values of a. In the interval [0.503, 0.54] memory produces a series of spurious (for the parity rule) snapshots. The evolution patterns for a close to 0.5, the lowest memory operative value, [0.501, 0.502] in Fig. 1, coincide with that of the ahistoric model, but, surprisingly, at T = 10 the pattern reduces to only one 3 · 3 cross and the progression in size turns out truncated. ðT Þ ðT 2Þ ðT 1Þ ðT Þ The trailing memory can also be limited, even to the three last states: si ¼ /ðri ; ri ; ri Þ [14,15]. Fig. 2 shows the effect of elementary legal rules of low number in the scenario of Fig. 1. In the standard (ahistoric) scenario, the 2D parity rule is linear (or additive): i.e., any initial pattern can be decomposed into the superposition of patterns from a single site seed. Each of these configurations can be evolved independently and the results superimposed (module two) to obtain the final complete pattern. The additivity of the 2D parity ðT þ1Þ ðT Þ ðT Þ rule remains when the memory rules are also additive. This is so with the 1D parity rules 90 ðri ¼ ri1 þ riþ1 mod 2Þ, ðT þ1Þ
ðT Þ
ðT Þ ðT Þ
and 150 ðri ¼ ri1 þ ri riþ1 mod 2Þ, as illustrated in Table 1, starting with two adjacent live cells. It should be emphasised that the memory mechanism considered here is different from that of other CA with memory reported in the literature.1 Typically, higher-order-in-time rules incorporate memory into the transition rule, determining the configuration at time T + 1 in terms of the configurations at previous time-steps. Thus, in second order in ðT þ1Þ ðT Þ ðT 1Þ time (memory of capacity two) rules, the transition rule operates as: ri ¼ Uðrj 2 Ni ; rj 2 Ni Þ. Double memðT þ1Þ
ory (in transition rule and in cells) can be implemented as: ri
ðT Þ
¼ Uðsj
the reversible formulation based on the exclusive OR (XOR, noted ): ðT 1Þ ri
¼
ðT Þ /ðrj
ðT þ1Þ 2 Ni Þ ri . To preserve reversibility, ðT þ1Þ ðT Þ ðT 1Þ must be: ri ¼ /ðsj 2 Ni Þ ri [7].
ðT 1Þ
2 Ni ; sj ðT þ1Þ ri
2 Ni Þ. Particularly interesting is ðT Þ
¼ /ðrj
ðT 1Þ
2 Ni Þ ri
, reversed as
inherent in the XOR operation, the reversible formulation
with memory Some authors, for example Wolf-Gladrow [18], define rules with memory as those with dependence in / on the state ðT þ1Þ
ðT Þ
ðT Þ
of the cell to be updated. So elementary rules with no memory, such as rule 90, take the form: ri ¼ /ðri1 ; riþ1 Þ. Our use of the term memory is not this. CA with memory in cells are cited by Wuensche and Lesser, [19, p. 15], who refuse to enter into its study and state that ‘‘CA with memory in cells would result in a qualitatively different behavior’’. Let us point here that the mechanism of implementation of memory adopted in this work, keeping the transition rule unaltered but applying it to a function of previous states, can be adopted in any dynamical system. In an earlier work 1
See, for example, [16, p. 118; 1, p. 43] or class CAM [17, p. 7].
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
1287
NO MEMORY
α = 0.501-0.502
α = 0.503-0.504
α = 0.505
α = 0.510
α = 0.52-0.53
α = 0.54
α = 0.55-0.60
α = 0.70
α = 0.80
α = 0.90 α = 1.00 Fig. 1. Effect of weighted memory of factor a on the two-dimensional parity rule when starting from a single site seed up to T = 15. Von Neumann neighborhood.
[4,6,15] we explored the effect of embedding this kind of memory into discrete dynamical systems: xT+1 = f(xT) by means of xT+1 = f(mT) with mT being a mean value of past states. We have studied this approach in, perhaps, the canonical example: the logistic map, which becomes with memory xT+1 = mT + kmT(1 mT). In [14], we studied the effect of memory in a particular Markovian stochastic process (the random walk), p0T þ1 ¼ p0T M by means of p0T þ1 ¼ p0T M with pT being a weighted mean of the probability distributions up to T.
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+ 4 (00000100)
+ 18 (00010010)
+ 22 (00010110)
+ 36 (00100100)
+50 (00110010)
+ 54 (00110110)
+ 76 (01001100)
+ 90 (01011010)
+ 150 (10010110) PARITY
Fig. 2. Effect of elementary, legal memory rules of low number on the two-dimensional parity rule in the scenario of Fig. 1. Patterns from T = 4 up to T = 15.
Table 1 The 2D parity rule remains additive with the 1D parity (also additive) memory rules 90 (left) and 150 (right)
As an example, the evolution patterns starting with two adjacent live cells as shown in this table coincide with the XOR superposed configuration of those evolved independently from a single seed, shown in Fig. 2. Evolution up to T = 5.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
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Fig. 3. The HC cellular automaton described in text (Section 3) with no memory. Evolution up to T = 6.
3. Structurally dynamic cellular automata Structurally dynamic cellular automata (SDCA) were suggested by Ilachinski and Halpern [20]. The essential new feature of this model was that the connections between the cells, are allowed to change according to rules similar in nature to the state transition rules associated with the conventional CA. This mean that given certain conditions, specified by the link transition rules, links between rules may be created and destroyed; the neighborhood of each cell is now dynamic rather than fixed throughout the automaton. Thus, state and link configurations of an SDCA are both dynamic and are continually altering each other. In the Ilachinski and Halpern model, an SDCA consists of a finite set of binary-valued cells numbered 1 to N whose connectivity is specified by a N · N connectivity matrix in which kij equals 1 if cells i and j are connected; 0 otherwise. ðT Þ
Thus, now: Ni
ðT Þ
ðT þ1Þ
¼ fj=kij ¼ 1g and ri
ðT Þ
¼ /ðrj
ðT Þ
2 Ni Þ. The distance between two cells i and j, dij, is defined as the
number of links in the shortest path between i and j. We say that i and j are direct neighbors if dij = 1, and that i and j are ðT Þ
next-nearest neighbors if dij = 2, so NNi
ðT Þ
¼ fj=dij ¼ 2g. There are two types of link transition functions in an SDCA:
couplers and decouplers, the former add new links, the later remove links. The set of coupler and decoupler determines ðT þ1Þ
ðT Þ
ðT Þ
ðT Þ
the link transition rule: kij ¼ wðlij ; ri ; rj Þ. Instead of introducing the formalism of the SDCA, we deal here with just an example, in which the decoupler rule ðT Þ
ðT þ1Þ
ðT Þ
ðT Þ
¼ 0 iff ri þ rj ¼ 0Þ and the couremoves all links connected to cells in which both values are zero ðkij ¼ 1 ! kij ðT Þ ðT þ1Þ pler rule adds links between all next-nearest neighbor sites in which both values are one ðkij ¼ 0 ! kij ¼ ðT Þ ðT Þ ðT Þ 1 iff ri þ rj ¼ 2 and j 2 NNi Þ. All of these totalistic rules are applied at the same time (in parallel). The SDCA with these transition rules for connections, together with the parity (or mod 2) rule for mass values, is considered by Halpern and Caltagirone [21]; so we will refer to it as the HC cellular automaton. Let us consider the case of Fig. 3, in which the initial Euclidean lattice2 is seeded with a 3 · 3 block of ones surrounded by zeros. After the first iteration, at time-step T = 2, most of the lattice structure has decayed as an effect of the decoupler rule, so that the active value cells and links are confined into a small region. After T = 6, the link and value structure become periodic, with a periodicity of two.
4. A structurally dynamic cellular automaton with memory Memory can be embedded in links in a similar manner as in state values, so the link between any two cells is featured ðT Þ ð1Þ ðT Þ by a mapping of its previous link values: lij ¼ lðkij ; . . . ; kij Þ. The distance between two cells in a historic model (dij), is defined in terms of the l instead of the k values, so that i and j are direct neighbors if dij = 1, and are next-nearest neighðT Þ ðT Þ ðT Þ ðT Þ bors if dij = 2; N i ¼ fj=d ij ¼ 1g, and NN i ¼ fj=d ij ¼ 2g. In our approach here, the memory rule for links (l) is the same that of state values (s). Generalizing the approach to embedded memory introduced in Section 2, the unchanged ðT þ1Þ ðT Þ ðT þ1Þ transition rules (/ and w) operate now on the featured link and mass values: ri ¼ /ðsj 2 N i Þ; kij ¼ ðT Þ
ðT Þ
ðT Þ
wðlij ; si ; sj Þ. Fig. 4 shows the effect of geometric discounted memory, with a low value of the memory factor (with no effect at T = 4) and with full memory. The initial scenario is that of Fig. 3. Three different simulations are shown under every 2
Von Neumann neighborhood in CA terminology, with next-nearest neighborhood:
.
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R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
Fig. 4. The HC cellular automaton with weighted memory of factor a. Evolution from T = 4 up to T = 9 starting as in Fig. 3. The first row of evolving patterns applies to memory only in mass, the second one to memory only in links, the third one to memory both in mass and in links. The last row shows the evolving patterns of the featuring values of mass and links of its immediate above row of last patterns, with si = 1 represented as h, and lij = 1 as !.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
1291
a label in Fig. 4: the first row of evolving patterns applies to memory only in mass, the second one to memory only in links,3 the third one to memory in both in mass and links. The last row shows the evolving patterns of the featuring ðT Þ ðT Þ values of mass and links (si , and lij ) of its row immediate above, the row of last patterns which corresponds to full ðT Þ
ð1Þ
ðT Þ
ðT Þ
ð1Þ
ðT Þ
memory (a = 1.0), so si ¼ modeðri ; . . . ; ri Þ, and lij ¼ modeðkij ; . . . ; kij Þ.4 All the pattern dynamics in Fig. 4 converge fairly soon to different period-two oscillators, except in the a = 1.0 with memory only in the links scenario, in which case it is generated a period four oscillator. Fig. 5 shows the effect of geometric discounted memory starting at random in an euclidean lattice of size 15 · 15, with the same memory factors as in Fig. 4. After T = 3, the lattice has so many connections that it is difficult to discern which sites are connected, so only the links of the central cell are shown. Fig. 6 shows the evolution of mass density, the average number of nearest neighbors and next-nearest neighbors per site, and the effective dimension: the average ratio of the number of next-nearest to nearest neighbors per site5 (a discrete analogue to the continuous Hausdorff dimension), in the simulations of Fig. 5. In the historic model with a = 0.6, the effective dimension tends to oscillate around 1 after an initial sharp increase (up to 4.3 at T = 4) followed by a similar decrease soon after; in the full memory model this decrease is less dramatic, so the effective dimension tends to oscillate around 2. The simulations for a in the (0.6,0.9] interval (not shown in Fig. 6) resemble that of a = 0.6, there seems to be no a gradual transition to the more defined curves of a = 1.0. The ahistoric simulation is very erratic in the first time-steps; it reaches a peak of mass density (0.6) at T = 14, followed by a pattern with only 12 live cells as shown in Fig. 5, then the mass pattern extinguishes at T = 16. This abrupt extinction after a dramatic explosion is fairly frequent in the ahistoric model, and can be explained to some extent by the erratic behavior of the mass parity rule. This kind of helter-skelter behavior is avoided with (geometric) memory, which tends to produce an inertial effect. Fig. 7 shows the effect of the elementary legal memory rules of Fig. 2, embedded both in cells and links: ðT Þ
si
ðT 2Þ
¼ /ðri
ðT Þ
ðT 1Þ
; ri ; ri
ðT Þ
ðT 2Þ
Þ, and lij ¼ /ðkij
ðT Þ
ðT 1Þ
; kij ; kij
Þ, in the scenario of Fig. 3. Rules 4, 36 and 76 induce early
extinction, but the remaining rules generate patterns of a great diversity, and sometimes visually quite appealing. The pattern with the memory rule 18 is empty at T = 7: no cell nor link is alive; but this does not mean extinction: the pattern at T = 8, presents 21 alive cells and a rich web of connections. This odd cataleptic phenomenon (which appeared already in Fig. 2 with rule 18) is unfeasible in the ahistoric formulation, but fairly frequent with elementary rules of low number. Another example in Fig. 7 is that of rule 50 at T = 8. Elementary rules of low number acting as memory tend to generate fairly soon pattern oscillators of not long period. This is so with rules 18, 22, 50, and 54 in Fig. 7. Rule 54 generates a period two oscillator (the empty pattern followed by a very sophisticated pattern); rules 18 and 50 generate a period four oscillator with two non-consecutive patterns empty; rule 22 generates a period eight oscillator with three patterns empty, two6 of them consecutive.
5. Discussion The effect of memory embedded in cells and links on a particular structurally dynamic cellular automaton is qualitatively (pictorically) studied in this work. As a rule, memory notably alters the ahistoric dynamics. A complete analysis of the effect of memory on structurally dynamic cellular automata (SDCA) is left for future work which will develop a phenomenology of SDCA with memory, i.e. the full analysis of the rule space based on the morphological classification of patterns formed, the intrinsic parameters (e.g. Langton’s lambda, Wuensche’s Z parameter), the structure of global transition graphs (this would be feasible only in the 1D case), the entropy and other dynamics-related issues. Potential fractal features are also to come under scrutiny [25].
3 ðT Þ aij
4
¼
ðT Þ kij
þ
1þ
PT 1
T t ðtÞ kij t¼1 a PT 1 T t a t¼1
!
ðT Þ lij
8 > 1 > < Þ ¼ kðT ij > > : 0
ðT Þ
if aij > 0:5 ðT Þ
if ai
¼ 0:5 ;
T > 2;
ð1Þ
ð1Þ
lij ¼ kij ;
ð2Þ
ð2Þ
lij ¼ kij .
ðT Þ
if aij < 0:5
Note that some s = 1 cells are not appreciated in the last row of patterns because of the concurrence of arrows on them. This happens for example with the central cell after T = 6 (third pattern) whose square is hidden; in fact the central cell is the only live cell after T = 6, connected with the 21 cells alive at T = 7. 5 Starting from an Euclidean lattice, the effective dimension is: 8/5 = 1.6. 6 The maximum length of catalepsy with memory of length three: three consecutive time steps empty means extinction.
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Fig. 5. The HC cellular automaton starting at random in an euclidean lattice of size 15 · 15. Ahistoric model and patterns with geometric discounted memory. After T = 3, only the links of the central cell are shown.
0.7
120
0.6
100 Nearest Neighbors
Mass Density
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
0.5 0.4 0.3 0.2
0
50
100 Time
150
60 40
0
200
120
6
100
5
Effective Dimension
Next Nearest Neighbors
80
20
0.1 0
80 60 40 20 0
1293
0
50
100 Time
150
200
4 3 2 1
0
50
100 Time
150
200
0
0
50
100 Time
150
200
Fig. 6. Evolution of mass density, average number of nearest neighbors and next-nearest neighbors per site, and effective dimension in simulations of Fig. 5 up to T = 200. Ahistoric simulations in red, full memory (a = 1.0) in blue, and simulations with a = 0.6 in green. (For interpretation of colour in this figure, the reader is referred to the Web version of this article.)
Some critics may argue that memory is not in the realm of CA (or even of Dynamic Systems), but we believe that the subject is worth studying. At least CA with memory can be considered as a promising extension of the basic paradigm. A major impediment to modeling with CA stems from the difficulty of utilizing the CA complex behavior to exhibit a particular behavior or perform a particular function: embedding memory in states and links broadens the spectrum of CA as a tool for modeling. It is likely that in some contexts, a transition rule with memory could match the ‘‘correct’’ behavior of the CA system of a given complex system (physical, biological, social and so on). The SDCA seems to be particularly appropriate for modelling the human brain function (links/synapses connect cells/neurons) in which the relevant role of memory is apparent. Models similar to SDCA have beeen adopted to build a dynamical network approach to quantum space–time physics [26]. Reversibility is an important issue at such a fundamental physics level; a generalisation of the Fredkin’s reversible construction is feasible in the SDCA scenario, which ðT þ1Þ
ðT Þ
ðT Þ
ðT 1Þ
ðT þ1Þ
ðT Þ
ðT Þ
ðT Þ
ðT 1Þ
can be endowed with memory as: ri ¼ /ðsj 2 N i Þ ri ; kij ¼ wðlij ; si ; sj Þ kij . Anyway, besides their potential applications, SDCA with memory have an aesthetic and mathematical interest on their own. The study of the effect of memory on CA has been rather neglected and there have been only limited investigations of SDCA since its introduction in the late eighties.7 Nevertheless, it seems plausible that further study on SDCA (and Lattice Gas Automata with dynamical geometry [27]) with memory8 should turn out to be profitable.
7 To the best of our knowledge, the relevant references on SDCA are [21–24], together with a review chapter in [1] and a section in [17]. 8 Not only in the basic paradigm scenario, but also in SDCA with random but value dependent rule transitions (which relates SDCA to genetic networks), and/or in SDCA with the extensions considered in [24], such as unidirectional links.
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R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
Fig. 7. The HC cellular automaton in the scenario of Fig. 3 with the elementary, legal memory rules of Fig. 2 in cells and links. Evolution from T = 4 up to T = 8.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
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Acknowledgements Dr. Andrew Adamatzky (UWE) encouraged me to pay attention to the fascinating world of SDCA. Thank you, Andy. I’ve seen things you would believe, . . . SDCAM evolving on my computer screen . . . All those results will not be lost in the sea of scientific literature, like tears in rain, but very much cited (I hope). Time to pursue. This work was supported by CICYT Grant AGL2002-04003-C03-02 AGR.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
Ilachinski A. Cellular automata. A discrete universe. World Scientific; 2001. Wolfram S. A new kind of science. Wolfram Media; 2001. Julian P, Chua LO. Replication properties of parity cellular automata. Int J Bifurcat Chaos 2002;12:477. Alonso-Sanz R. Phase transitions in an elementary probabilistic cellular automaton with memory. Physica A 2005;347:383. Alonso-Sanz R. One-dimensional, r = 2 cellular automata with memory. Int J Bifurcat Chaos 2004;14:3217. Alonso-Sanz R, Martin M. Three-state one-dimensional cellular automata with memory. Chaos, Solitons & Fractals 2004;21:809. Alonso-Sanz R. Reversible cellular automata with memory. Physica D 2003;175:1. Alonso-Sanz R, Martin M. Elementary cellular automata with memory. Complex Syst 2003;14:99. Alonso-Sanz R, Martin M. Cellular automata with accumulative memory: legal rules starting from a single site seed. Int J Modern Phys C 2003;14:695. Alonso-Sanz R, Martin M. One-dimensional cellular automata with memory: patterns starting with a single site seed. Int J Bifurcat Chaos 2002;12:20. Alonso-Sanz R, Martin M. Two-dimensional cellular automata with memory: patterns starting with a single site seed. Int J Modern Phys C 2002;13:49. Alonso-Sanz R, Martin MC, Martin M. Historic life. Int J Bifurcat Chaos 2001;11:1665. Alonso-Sanz R, Martin MC, Martin M. The effect of memory in the continuous-valued prisoner’s dilemma. Int J Bifurcat Chaos 2001;11:2061. and the references therein. Alonso-Sanz R. Elementary rules with elementary memory rules: the case of linear rules. J Cellular Automata 2006;1:71. Alonso-Sanz R, Martin M. One-dimensional cellular automata with memory in cells of the most frequent recent value. Complex Syst 2005;15:203. Wolfram S. Universality and complexity in cellular automata. Physica D 1984;10:1. Adamatzky A. Identification of cellular automata. Taylor and Francis; 1994. Wolf-Gladrow DA. Lattice-gas cellular automata and lattice Boltzmann models. Springer; 2000. Wuensche A, Lesser M. The global dynamics of cellular automata. Addison-Wesley; 1992. Ilachinsky A, Halpern P. Structurally dynamic cellular automata. Complex Syst 1987;1:503. Halpern P, Caltagirone G. Behaviour of topological cellular automata. Complex Syst 1990;4:623. Halpern P. Sticks and stones: a guide to structurally dynamic cellular automata. Am J Phys 1989;57:405. Hillman D. Combinatorial spacetimes. PhD thesis (mathematics). University of Pitssburg, 1995. Available from hep-th/9805066. Majercik SM. Structurally dynamic cellular automata. University of Southern Maine (MsC); 1994. Sanchez JR, Alonso-Sanz R. Multifractal properties of R90 cellular automaton with memory. Int J Modern Phys C 2004;15:1461. Requardt M. Cellular networks as models for Plank-scale physics. J Phys A 1998;31:7797. Available from http://arxiv.org/abs/ gr-qc/0308089. Love PJ, Boghosian BM, Meyer DA. Lattice gas simulations of dynamical geometry in one dimension. Phil Trans R Soc London A 2004;362:1667.
A structurally dynamic cellular automaton with memory Ramo´n Alonso-Sanz
*
ETSI Agro´nomos (Estadı´stica), C. Universitaria, Grupo de Sistemas Complejos—UPM, 28040 Madrid, Spain Accepted 28 December 2005
Abstract Major features of conventional cellular automata (CA) include the inalterability of topology and the absence of memory. The effect of simple memory (memory in cells and links) in a particular structurally dynamic CA is explored in this paper. 2006 Elsevier Ltd. All rights reserved.
1. Cellular automata Cellular automata (CA) are discrete, spatially explicit extended dynamic systems. A CA system is composed of adjacent cells or sites arranged as a regular lattice, which evolves in discrete time steps. Each cell is characterised by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common ðT Þ local transition rule involving only a neighborhood of each cell [1,2]. Thus, if ri is taken to denote the value of cell i at ðT þ1Þ
time step T, the site values evolve by iteration of the mapping: ri
ðT Þ
¼ /ðrj
2 Ni Þ, where / is an arbitrary function
which specifies the cellular automaton rule operating on the neighborhood N of the cell i. In totalistic rules the of P value ðT þ1Þ ðT Þ a site depends only on the sum of the values of its neighbours, and not on their individual values: ri ¼ /ð j2Ni rj Þ. This paper deals with a particular two-dimensional totalistic CA rule with two possible values at each site P ðT þ1Þ ðT Þ (r 2 {0, 1}), the parity rule: ri ¼ j2Ni rj mod 2. In Section 2 (and initially in Sections 3 and 4) it is adopted the von Neumann neighborhood (NSWE), so the sum of neighbor states ranges in the {0, 5} set, and the values that activate a cell state are 1, 3 and 5. Regardless its formal simplicity, the parity rule exhibits a complex behavior [3]. Elementary CA refer to the simplest scenario: one-dimensional CA with two possible values at each site, with rules operating on nearest neighbors. These rules are characterised by a sequence of binary values (b) associated with each of ðT Þ
ðT Þ
ðT Þ
the eight possible triplets ðri1 ; ri ; riþ1 Þ:
*
111
110 101
100 011 010
001 000
ðb1
b2
b4
b7
b3
b5
b6
b8 Þbinary
8 P
bi 28i decimal ¼ R 2 ½0; 255.
i¼1
Fax: +34 91 3365871. E-mail address: [email protected]
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.047
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We consider here (Sections 2 and 4) only elementary legal rules: rules that are reflection symmetric (b2 = b5 and b4 = b7) and quiescent (b8 = 0). These restrictions leave 32 possible legal rules.
2. Cellular automata with memory Standard CA are ahistoric (memoryless): the transition function depends on the cells neighborhood configuration only at the preceding time step. Historic memory can be embedded in the CA dynamics by featuring every cell by a mapping of its states in the previous time steps. Thus, what is here proposed is to maintain the transition rules (/) unalðT þ1Þ ðT Þ ðT Þ ¼ /ðsj 2 Nj Þ, si being a tered, but make them act on the cells featured by a function of their previous states: ri state function of the series of states of the cell i after time-step T. In the approach adopted here, memory becomes operð1Þ
ð1Þ
ð2Þ
ð2Þ
ative after T = 3, with the initial assignations si ¼ ri , si ¼ ri . Cellular automata implementing memory in cells will be termed historic, and the standard ones ahistoric. As stated, the historic and ahistoric evolving patterns are the same up to T = 3. Thus, cells can be featured by a weighted mean value of all their previous states [4–13]:
ðT Þ si
¼
8 1 > > > <
ðT Þ r > i
> > :
0
ðT Þ
if mi
> 0:5 ðT Þ
if
ðT Þ mi
¼ 0:5
ðT Þ
< 0:5
if mi
after
ðT Þ
ð1Þ
ð2Þ
ðT Þ
mi ðri ; ri ; . . . ; ri Þ ¼
PT 1
ðtÞ
aT t ri . Pt¼1 T 1 T t 1 þ t¼1 a
ri
þ
As an example, Fig. 1 shows the evolving patterns of the two-dimensional parity rule with weighted memory starting from a single site live cell. Memory truncates the natural expansive evolution of the parity rule, particularly for high values of the memory factor a in which case small size oscillators of short period appear. But the effect is also dramatic for low values of a. In the interval [0.503, 0.54] memory produces a series of spurious (for the parity rule) snapshots. The evolution patterns for a close to 0.5, the lowest memory operative value, [0.501, 0.502] in Fig. 1, coincide with that of the ahistoric model, but, surprisingly, at T = 10 the pattern reduces to only one 3 · 3 cross and the progression in size turns out truncated. ðT Þ ðT 2Þ ðT 1Þ ðT Þ The trailing memory can also be limited, even to the three last states: si ¼ /ðri ; ri ; ri Þ [14,15]. Fig. 2 shows the effect of elementary legal rules of low number in the scenario of Fig. 1. In the standard (ahistoric) scenario, the 2D parity rule is linear (or additive): i.e., any initial pattern can be decomposed into the superposition of patterns from a single site seed. Each of these configurations can be evolved independently and the results superimposed (module two) to obtain the final complete pattern. The additivity of the 2D parity ðT þ1Þ ðT Þ ðT Þ rule remains when the memory rules are also additive. This is so with the 1D parity rules 90 ðri ¼ ri1 þ riþ1 mod 2Þ, ðT þ1Þ
ðT Þ
ðT Þ ðT Þ
and 150 ðri ¼ ri1 þ ri riþ1 mod 2Þ, as illustrated in Table 1, starting with two adjacent live cells. It should be emphasised that the memory mechanism considered here is different from that of other CA with memory reported in the literature.1 Typically, higher-order-in-time rules incorporate memory into the transition rule, determining the configuration at time T + 1 in terms of the configurations at previous time-steps. Thus, in second order in ðT þ1Þ ðT Þ ðT 1Þ time (memory of capacity two) rules, the transition rule operates as: ri ¼ Uðrj 2 Ni ; rj 2 Ni Þ. Double memðT þ1Þ
ory (in transition rule and in cells) can be implemented as: ri
ðT Þ
¼ Uðsj
the reversible formulation based on the exclusive OR (XOR, noted ): ðT 1Þ ri
¼
ðT Þ /ðrj
ðT þ1Þ 2 Ni Þ ri . To preserve reversibility, ðT þ1Þ ðT Þ ðT 1Þ must be: ri ¼ /ðsj 2 Ni Þ ri [7].
ðT 1Þ
2 Ni ; sj ðT þ1Þ ri
2 Ni Þ. Particularly interesting is ðT Þ
¼ /ðrj
ðT 1Þ
2 Ni Þ ri
, reversed as
inherent in the XOR operation, the reversible formulation
with memory Some authors, for example Wolf-Gladrow [18], define rules with memory as those with dependence in / on the state ðT þ1Þ
ðT Þ
ðT Þ
of the cell to be updated. So elementary rules with no memory, such as rule 90, take the form: ri ¼ /ðri1 ; riþ1 Þ. Our use of the term memory is not this. CA with memory in cells are cited by Wuensche and Lesser, [19, p. 15], who refuse to enter into its study and state that ‘‘CA with memory in cells would result in a qualitatively different behavior’’. Let us point here that the mechanism of implementation of memory adopted in this work, keeping the transition rule unaltered but applying it to a function of previous states, can be adopted in any dynamical system. In an earlier work 1
See, for example, [16, p. 118; 1, p. 43] or class CAM [17, p. 7].
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
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NO MEMORY
α = 0.501-0.502
α = 0.503-0.504
α = 0.505
α = 0.510
α = 0.52-0.53
α = 0.54
α = 0.55-0.60
α = 0.70
α = 0.80
α = 0.90 α = 1.00 Fig. 1. Effect of weighted memory of factor a on the two-dimensional parity rule when starting from a single site seed up to T = 15. Von Neumann neighborhood.
[4,6,15] we explored the effect of embedding this kind of memory into discrete dynamical systems: xT+1 = f(xT) by means of xT+1 = f(mT) with mT being a mean value of past states. We have studied this approach in, perhaps, the canonical example: the logistic map, which becomes with memory xT+1 = mT + kmT(1 mT). In [14], we studied the effect of memory in a particular Markovian stochastic process (the random walk), p0T þ1 ¼ p0T M by means of p0T þ1 ¼ p0T M with pT being a weighted mean of the probability distributions up to T.
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+ 4 (00000100)
+ 18 (00010010)
+ 22 (00010110)
+ 36 (00100100)
+50 (00110010)
+ 54 (00110110)
+ 76 (01001100)
+ 90 (01011010)
+ 150 (10010110) PARITY
Fig. 2. Effect of elementary, legal memory rules of low number on the two-dimensional parity rule in the scenario of Fig. 1. Patterns from T = 4 up to T = 15.
Table 1 The 2D parity rule remains additive with the 1D parity (also additive) memory rules 90 (left) and 150 (right)
As an example, the evolution patterns starting with two adjacent live cells as shown in this table coincide with the XOR superposed configuration of those evolved independently from a single seed, shown in Fig. 2. Evolution up to T = 5.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
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Fig. 3. The HC cellular automaton described in text (Section 3) with no memory. Evolution up to T = 6.
3. Structurally dynamic cellular automata Structurally dynamic cellular automata (SDCA) were suggested by Ilachinski and Halpern [20]. The essential new feature of this model was that the connections between the cells, are allowed to change according to rules similar in nature to the state transition rules associated with the conventional CA. This mean that given certain conditions, specified by the link transition rules, links between rules may be created and destroyed; the neighborhood of each cell is now dynamic rather than fixed throughout the automaton. Thus, state and link configurations of an SDCA are both dynamic and are continually altering each other. In the Ilachinski and Halpern model, an SDCA consists of a finite set of binary-valued cells numbered 1 to N whose connectivity is specified by a N · N connectivity matrix in which kij equals 1 if cells i and j are connected; 0 otherwise. ðT Þ
Thus, now: Ni
ðT Þ
ðT þ1Þ
¼ fj=kij ¼ 1g and ri
ðT Þ
¼ /ðrj
ðT Þ
2 Ni Þ. The distance between two cells i and j, dij, is defined as the
number of links in the shortest path between i and j. We say that i and j are direct neighbors if dij = 1, and that i and j are ðT Þ
next-nearest neighbors if dij = 2, so NNi
ðT Þ
¼ fj=dij ¼ 2g. There are two types of link transition functions in an SDCA:
couplers and decouplers, the former add new links, the later remove links. The set of coupler and decoupler determines ðT þ1Þ
ðT Þ
ðT Þ
ðT Þ
the link transition rule: kij ¼ wðlij ; ri ; rj Þ. Instead of introducing the formalism of the SDCA, we deal here with just an example, in which the decoupler rule ðT Þ
ðT þ1Þ
ðT Þ
ðT Þ
¼ 0 iff ri þ rj ¼ 0Þ and the couremoves all links connected to cells in which both values are zero ðkij ¼ 1 ! kij ðT Þ ðT þ1Þ pler rule adds links between all next-nearest neighbor sites in which both values are one ðkij ¼ 0 ! kij ¼ ðT Þ ðT Þ ðT Þ 1 iff ri þ rj ¼ 2 and j 2 NNi Þ. All of these totalistic rules are applied at the same time (in parallel). The SDCA with these transition rules for connections, together with the parity (or mod 2) rule for mass values, is considered by Halpern and Caltagirone [21]; so we will refer to it as the HC cellular automaton. Let us consider the case of Fig. 3, in which the initial Euclidean lattice2 is seeded with a 3 · 3 block of ones surrounded by zeros. After the first iteration, at time-step T = 2, most of the lattice structure has decayed as an effect of the decoupler rule, so that the active value cells and links are confined into a small region. After T = 6, the link and value structure become periodic, with a periodicity of two.
4. A structurally dynamic cellular automaton with memory Memory can be embedded in links in a similar manner as in state values, so the link between any two cells is featured ðT Þ ð1Þ ðT Þ by a mapping of its previous link values: lij ¼ lðkij ; . . . ; kij Þ. The distance between two cells in a historic model (dij), is defined in terms of the l instead of the k values, so that i and j are direct neighbors if dij = 1, and are next-nearest neighðT Þ ðT Þ ðT Þ ðT Þ bors if dij = 2; N i ¼ fj=d ij ¼ 1g, and NN i ¼ fj=d ij ¼ 2g. In our approach here, the memory rule for links (l) is the same that of state values (s). Generalizing the approach to embedded memory introduced in Section 2, the unchanged ðT þ1Þ ðT Þ ðT þ1Þ transition rules (/ and w) operate now on the featured link and mass values: ri ¼ /ðsj 2 N i Þ; kij ¼ ðT Þ
ðT Þ
ðT Þ
wðlij ; si ; sj Þ. Fig. 4 shows the effect of geometric discounted memory, with a low value of the memory factor (with no effect at T = 4) and with full memory. The initial scenario is that of Fig. 3. Three different simulations are shown under every 2
Von Neumann neighborhood in CA terminology, with next-nearest neighborhood:
.
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Fig. 4. The HC cellular automaton with weighted memory of factor a. Evolution from T = 4 up to T = 9 starting as in Fig. 3. The first row of evolving patterns applies to memory only in mass, the second one to memory only in links, the third one to memory both in mass and in links. The last row shows the evolving patterns of the featuring values of mass and links of its immediate above row of last patterns, with si = 1 represented as h, and lij = 1 as !.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
1291
a label in Fig. 4: the first row of evolving patterns applies to memory only in mass, the second one to memory only in links,3 the third one to memory in both in mass and links. The last row shows the evolving patterns of the featuring ðT Þ ðT Þ values of mass and links (si , and lij ) of its row immediate above, the row of last patterns which corresponds to full ðT Þ
ð1Þ
ðT Þ
ðT Þ
ð1Þ
ðT Þ
memory (a = 1.0), so si ¼ modeðri ; . . . ; ri Þ, and lij ¼ modeðkij ; . . . ; kij Þ.4 All the pattern dynamics in Fig. 4 converge fairly soon to different period-two oscillators, except in the a = 1.0 with memory only in the links scenario, in which case it is generated a period four oscillator. Fig. 5 shows the effect of geometric discounted memory starting at random in an euclidean lattice of size 15 · 15, with the same memory factors as in Fig. 4. After T = 3, the lattice has so many connections that it is difficult to discern which sites are connected, so only the links of the central cell are shown. Fig. 6 shows the evolution of mass density, the average number of nearest neighbors and next-nearest neighbors per site, and the effective dimension: the average ratio of the number of next-nearest to nearest neighbors per site5 (a discrete analogue to the continuous Hausdorff dimension), in the simulations of Fig. 5. In the historic model with a = 0.6, the effective dimension tends to oscillate around 1 after an initial sharp increase (up to 4.3 at T = 4) followed by a similar decrease soon after; in the full memory model this decrease is less dramatic, so the effective dimension tends to oscillate around 2. The simulations for a in the (0.6,0.9] interval (not shown in Fig. 6) resemble that of a = 0.6, there seems to be no a gradual transition to the more defined curves of a = 1.0. The ahistoric simulation is very erratic in the first time-steps; it reaches a peak of mass density (0.6) at T = 14, followed by a pattern with only 12 live cells as shown in Fig. 5, then the mass pattern extinguishes at T = 16. This abrupt extinction after a dramatic explosion is fairly frequent in the ahistoric model, and can be explained to some extent by the erratic behavior of the mass parity rule. This kind of helter-skelter behavior is avoided with (geometric) memory, which tends to produce an inertial effect. Fig. 7 shows the effect of the elementary legal memory rules of Fig. 2, embedded both in cells and links: ðT Þ
si
ðT 2Þ
¼ /ðri
ðT Þ
ðT 1Þ
; ri ; ri
ðT Þ
ðT 2Þ
Þ, and lij ¼ /ðkij
ðT Þ
ðT 1Þ
; kij ; kij
Þ, in the scenario of Fig. 3. Rules 4, 36 and 76 induce early
extinction, but the remaining rules generate patterns of a great diversity, and sometimes visually quite appealing. The pattern with the memory rule 18 is empty at T = 7: no cell nor link is alive; but this does not mean extinction: the pattern at T = 8, presents 21 alive cells and a rich web of connections. This odd cataleptic phenomenon (which appeared already in Fig. 2 with rule 18) is unfeasible in the ahistoric formulation, but fairly frequent with elementary rules of low number. Another example in Fig. 7 is that of rule 50 at T = 8. Elementary rules of low number acting as memory tend to generate fairly soon pattern oscillators of not long period. This is so with rules 18, 22, 50, and 54 in Fig. 7. Rule 54 generates a period two oscillator (the empty pattern followed by a very sophisticated pattern); rules 18 and 50 generate a period four oscillator with two non-consecutive patterns empty; rule 22 generates a period eight oscillator with three patterns empty, two6 of them consecutive.
5. Discussion The effect of memory embedded in cells and links on a particular structurally dynamic cellular automaton is qualitatively (pictorically) studied in this work. As a rule, memory notably alters the ahistoric dynamics. A complete analysis of the effect of memory on structurally dynamic cellular automata (SDCA) is left for future work which will develop a phenomenology of SDCA with memory, i.e. the full analysis of the rule space based on the morphological classification of patterns formed, the intrinsic parameters (e.g. Langton’s lambda, Wuensche’s Z parameter), the structure of global transition graphs (this would be feasible only in the 1D case), the entropy and other dynamics-related issues. Potential fractal features are also to come under scrutiny [25].
3 ðT Þ aij
4
¼
ðT Þ kij
þ
1þ
PT 1
T t ðtÞ kij t¼1 a PT 1 T t a t¼1
!
ðT Þ lij
8 > 1 > < Þ ¼ kðT ij > > : 0
ðT Þ
if aij > 0:5 ðT Þ
if ai
¼ 0:5 ;
T > 2;
ð1Þ
ð1Þ
lij ¼ kij ;
ð2Þ
ð2Þ
lij ¼ kij .
ðT Þ
if aij < 0:5
Note that some s = 1 cells are not appreciated in the last row of patterns because of the concurrence of arrows on them. This happens for example with the central cell after T = 6 (third pattern) whose square is hidden; in fact the central cell is the only live cell after T = 6, connected with the 21 cells alive at T = 7. 5 Starting from an Euclidean lattice, the effective dimension is: 8/5 = 1.6. 6 The maximum length of catalepsy with memory of length three: three consecutive time steps empty means extinction.
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Fig. 5. The HC cellular automaton starting at random in an euclidean lattice of size 15 · 15. Ahistoric model and patterns with geometric discounted memory. After T = 3, only the links of the central cell are shown.
0.7
120
0.6
100 Nearest Neighbors
Mass Density
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
0.5 0.4 0.3 0.2
0
50
100 Time
150
60 40
0
200
120
6
100
5
Effective Dimension
Next Nearest Neighbors
80
20
0.1 0
80 60 40 20 0
1293
0
50
100 Time
150
200
4 3 2 1
0
50
100 Time
150
200
0
0
50
100 Time
150
200
Fig. 6. Evolution of mass density, average number of nearest neighbors and next-nearest neighbors per site, and effective dimension in simulations of Fig. 5 up to T = 200. Ahistoric simulations in red, full memory (a = 1.0) in blue, and simulations with a = 0.6 in green. (For interpretation of colour in this figure, the reader is referred to the Web version of this article.)
Some critics may argue that memory is not in the realm of CA (or even of Dynamic Systems), but we believe that the subject is worth studying. At least CA with memory can be considered as a promising extension of the basic paradigm. A major impediment to modeling with CA stems from the difficulty of utilizing the CA complex behavior to exhibit a particular behavior or perform a particular function: embedding memory in states and links broadens the spectrum of CA as a tool for modeling. It is likely that in some contexts, a transition rule with memory could match the ‘‘correct’’ behavior of the CA system of a given complex system (physical, biological, social and so on). The SDCA seems to be particularly appropriate for modelling the human brain function (links/synapses connect cells/neurons) in which the relevant role of memory is apparent. Models similar to SDCA have beeen adopted to build a dynamical network approach to quantum space–time physics [26]. Reversibility is an important issue at such a fundamental physics level; a generalisation of the Fredkin’s reversible construction is feasible in the SDCA scenario, which ðT þ1Þ
ðT Þ
ðT Þ
ðT 1Þ
ðT þ1Þ
ðT Þ
ðT Þ
ðT Þ
ðT 1Þ
can be endowed with memory as: ri ¼ /ðsj 2 N i Þ ri ; kij ¼ wðlij ; si ; sj Þ kij . Anyway, besides their potential applications, SDCA with memory have an aesthetic and mathematical interest on their own. The study of the effect of memory on CA has been rather neglected and there have been only limited investigations of SDCA since its introduction in the late eighties.7 Nevertheless, it seems plausible that further study on SDCA (and Lattice Gas Automata with dynamical geometry [27]) with memory8 should turn out to be profitable.
7 To the best of our knowledge, the relevant references on SDCA are [21–24], together with a review chapter in [1] and a section in [17]. 8 Not only in the basic paradigm scenario, but also in SDCA with random but value dependent rule transitions (which relates SDCA to genetic networks), and/or in SDCA with the extensions considered in [24], such as unidirectional links.
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Fig. 7. The HC cellular automaton in the scenario of Fig. 3 with the elementary, legal memory rules of Fig. 2 in cells and links. Evolution from T = 4 up to T = 8.
R. Alonso-Sanz / Chaos, Solitons and Fractals 32 (2007) 1285–1295
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Acknowledgements Dr. Andrew Adamatzky (UWE) encouraged me to pay attention to the fascinating world of SDCA. Thank you, Andy. I’ve seen things you would believe, . . . SDCAM evolving on my computer screen . . . All those results will not be lost in the sea of scientific literature, like tears in rain, but very much cited (I hope). Time to pursue. This work was supported by CICYT Grant AGL2002-04003-C03-02 AGR.
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