The Clouds in Asynchronous Cellular Automata
aa r X i v : . [ n li n . C G ] J a n The Clouds in Asynchronous Cellular Automata
Souvik Roy ∗ and Sukanta Das † (Dated: January 30, 2020)This article introduces the notion of clouds in asynchronous cellular automata (ACAs). We showthat the cloud behaviour of ACAs has similarity with natural clouds across the sky, election modelof parliamentary democratic system, and electron cloud around nucleus. These systems, therefore,can be modelled by the ACAs. When we look up at the sky, we see the clouds sailingacross the sky. From the global view, the clouds are asso-ciated with multiple attractor basins. It is often unknownwhere a cloud can find her ultimate attractor basin andcreate a downpour of rain on the earth. In this article,we explore natural cloud phenomenon, which is observedin simple asynchronous cellular automata (CAs) mod-els. In such CAs, the cells are considered independentto some extent. A traditional (synchronous) convergentcellular automaton (CA) with an initial configuration al-ways converges to a specific attractor following a specificdeterministic path. On the contrary, a CA with sameinitial configuration may approach to different attractorbasins for different runs when the cells of the CA are up-dated independently. It may also be possible in such asystem that for an initial configuration, the system al-ways converges to a specific point attractor. Obviously,the basin of attraction in the convergence-dynamics of asystem with only one attractor basin includes all the con-figurations for which the system approaches to the onlyattractor. However, the dynamics of the system underconsideration differs from the traditional deterministicone. Under this system, the travelling path towards asingle attractor basin may change in different runs. Inthis perspective, this independent system, itself, breaksthe traditional deterministic concept of basins of attrac-tion in CA [1–3]. x y
FIG. 1. Graphical visualization of clouds in CA.
Now, we can define the notion of clouds for an asyn- ∗ [email protected]; Department of Information Technology, In-dian Institute of Engineering Science and Technology, Shibpur,Howrah, West Bengal, India 711103. † [email protected]; Department of Information Technology,Indian Institute of Engineering Science and Technology, Shibpur,Howrah, West Bengal, India 711103. chronous CA (ACA) system [4–7]. One can assume thatan attractor is associated with a number of orbits. De-pending on the number of travelling time steps to reachto the corresponding attractor, an initial configuration isplaced in an orbit. An initial configuration may simul-taneously exist at multiple orbits for a single attractor,because different time steps may be needed in differentruns to converge to the attractor. It may also be pos-sible that an initial configuration can simultaneously beat multiple orbits for multiple attractors when the sys-tem converges to different attractors from the same initialconfiguration. In Fig. 1, blue initial configuration is si-multaneously present at third and fourth orbits of x (inblack) attractor. Presence of a configuration in orbit i indicates that the CA takes i time steps to reach to theattractor from the configuration. The red initial config-uration in Fig. 1 exists at multiple orbits for both the x and y attractors. Note that, an orbit can be associatedwith multiple initial configurations. Therefore, the con-figurations can be seen as clouds around the attractors.This notion of clouds in the asynchronous CA remindsus the concept of electron cloud.To sum up, an initial configuration for which the CAconverges to a specific attractor for every run or differentattractors for different runs, the concept of cloud is appli-cable to that initial configuration. However, convergenceat different attractors for different runs injects an extraflavour in the system. We name an initial configuration,for which the CA converges to different point attractorsfor different runs, as confused configuration . In Fig. 2,a red initial configuration depicts the graphical visualiza-tion of confused configuration which are linked to boththe attractor basins (in black). In this perspective, de-pending on the presence of confused configuration, weclassify the system into three classes - eccentric cloud , partially eccentric cloud and deterministic cloud system.In an eccentric cloud system, for every possible initialconfiguration, the system converges to different attrac-tors for different runs. Fig. 2(a) shows a two-attractoreccentric cloud system where for every initial configura-tion (in red), the system may converge to any one of thetwo point attractors (in black). The dynamics of thissystem reminds us the clumsy clouds in the sky whichdo not really have a predictable destination. Obviously,a CA with point attractor as initial configuration alwaysconverges to itself in a deterministic manner.In a partially eccentric cloud system, some initial con- (a) (c)(b) FIG. 2. Two attractor (in black) (a) eccentric; (b) partially eccentric; and (c) deterministic cloud system where red initialconfigurations are confused configuration and blur configurations are associated with deterministic destination. figurations are confused configuration and some are not.In a two-attractor system of Fig. 2(b), red initial con-figuration is a confused configuration and blue configu-ration is with fixed destination. Remark that, partiallyeccentric cloud system may catch the dynamics of elec-tion model of a parliamentary democratic system. In anelection, there are certain or lock votes who are solidlybehind or partisan to a particular candidate and willnot consider changing their minds whatever the oppo-sition says. However, there also exist some swing voterswhose votes are unpredictable. Here, swing voters can bemodeled by confused configurations. In American poli-tics, many centrists, liberal Republicans and conservativeDemocrats are considered ‘swing voters’ since their vot-ing patterns can not be predicted with certainty.Every initial configuration is always linked to a spe-cific point attractor in the deterministic cloud system.In Fig. 2(c), every blue initial configuration is associatedwith a deterministic destination attractor. One can imag-ine this type of system as electron cloud model where wecannot know exactly where an electron is at any giventime, but the electrons are more likely to be in specificareas.Here, we work in the rule space of binary threeneighbourhood 1-D CA (known as elementary cellularautomata (ECA)) under periodic boundary condition,where the cells are arranged as a ring. The next state ofeach CA cell is defined as S t +1 i = f ( S ti − , S ti , S ti +1 ), where f is the local rule and S ti − , S ti and S ti +1 are the presentstates of left, self and right neighbours respectively. Acollection of (local) states of cells at time t is referred toas a configuration of the CA. The local rule f : { , } , } can be expressed as a look-up table (see Table I).We call each argument of f as Rule Min Term (RMT) [8].An RMT ( x, y, z ) is generally represented by its decimalfrom r = 4 ∗ x + 2 ∗ y + z (row 2 of Table I). However, thedecimal counterpart of the eight next state is referred as“rule” [9]. There are 2 (256) ECA rules, out of which 88are minimal representative rules and the rests are theirequivalents [10]. The CAs are updated asynchronouslywhere a single cell is selected uniformly at random for up-date. This update scheme also implies a scheme whereall but two neighbouring cells may act simultaneously.Such CAs are referred as fully asynchronous CA (ACA)[8]. ( x, y, z ) 111 110 101 100 011 010 001 000(RMT) (7) (6) (5) (4) (3) (2) (1) (0) Rule0 1 1 0 1 0 0 0 104f( x, y, z ) 1 0 1 0 1 0 0 0 1681 0 1 0 1 0 1 0 170 TABLE I. Look-up table for ECA 104, 168 and 170.
An RMT r of a rule R is active if a CA cell flips its state(1 to 0 or 0 to 1) on r ; otherwise, the RMT r is passive.An ACA configuration is a point attractor if the RMTcorresponding to any three consecutive bits of the con-figuration is passive. That is, if an ACA reaches a pointattractor, the ACA remains in that particular configura-tion forever. In the CAs under consideration, we haveidentified six special configurations which we call homo-geneous configurations - , , , , and [11]. In and , all cells are in state 0 and 1 respec-tively. Similarly, in , cells are in states alternate 0 and1. And, so on. The corresponding RMT sets of homoge-neous configurations are referred as primary RMT sets - { } , { } , { } , { } , { } and { } . An ACAconfiguration is formed using the RMTs of one or moreof these six sets. We have identified that 50 minimal rep-resentative ACAs, out of 88, converge to point attractor[8]. Out of this 50 convergent ACAs, 18 ACAs with onlyone point attractor always belong to deterministic cloudACA. Now, the target is to explore the convergence dy-namics of rest 32 ACAs with multiple point attractors(see Table II).
128 130
136 138
146 152 154 160162
168 170
178 184
200 204 232
TABLE II. List of 32 ACAs with multiple point attractors.
Let us start the discussion with the dynamics of two-attractor system where the point attractors are and (see Table II, in bold). Remark that, in a two-attractorsystem, active RMTs 1 and 4 (resp. 3 and 6) are re-sponsible for the travelling path towards point attrac-tor (resp. ). That is, active RMTs 1 and 4 areanswerable for increasing number of 1’s. For example,consider a configuration 001100 with sequence of RMTs h i for which the possible transitions are 001100 → / (a) (b)FIG. 3. Convergence dynamics of 8-cell two attractor (a) unbiased system (ACA 170); and (b) biased system (ACA 162). Theleft (resp. right) point attractor is referred as (resp. ). swerable for decreasing number of 1’s. For example con-figuration, the possible transitions are 110011 h i→ / { , } and { , } in the active RMT set identifiestwo attractor eccentric cloud system. Here, we have iden-tified ACAs , , , as eccentric cloud sys-tems. Let us now explore the dynamics of two-attractoreccentric cloud ACA 170 where active RMTs are 1 and 6(see Table I). Here, the participation of RMT sets { , } and { , } is equal in the active RMT set. Therefore,the probability of increasing number of 1’s and decreas-ing number of 1’s at the next time step starting fromany arbitrary initial configuration is equal. In details,the probability of convergence to point attractor (resp. ) starting from initial configuration x is P x ( ) = a/n (resp. P x ( ) = b/n ) where number of 0’s (resp. 1’s) in x initial configuration is a (resp. b ) and a + b = n . There-fore, P ( ) ≈ P ( ), where P ( ) (resp. P ( )) isthe overall probability of convergence to point attractor (resp. ). As an evidence, let us draw the conver-gence dynamics of ACA 170. In Fig. 3, if for a confusedconfiguration, the ACA has high probability to reach toa point attractor, the configuration is placed closer tothe point attractor. Here, the cloud of confused config-urations, in Fig. 3(a), is not biased towards any specificpoint attractor. It may also possible that the cloud ofconfused configurations is biased towards a specific pointattractor, see Fig. 3(b).In this context, we classify the eccentric cloud systemsinto two classes - unbiased system and biased system.For the two-attractor system, an ACA δ is said to beunbiased system if P δ ( ) ≈ P δ ( ), whereas ACA δ issaid to be biased system if P δ ( ) ≫ P δ ( ) or P δ ( ) ≪ P δ ( ). From the RMT point of view, equal participa-tion of RMT sets { } and { } in the active RMT setidentifies unbiased system (ACAs , , ), oth- FIG. 4. Convergence dynamics of 8-cell two-attractor par-tially eccentric cloud system (ACA 168). The left and rightpoint attractor is referred as and respectively. . . . . / P / ( ) FIG. 5. P / ( ) as a function of number of 01/011 in the homo-geneous initial configuration / for ACA 168. erwise the system reflects the biased nature (ACA ).Fig. 3(b) depicts the dynamics of biased ACA 162 where P ( ) ≫ P ( ).In this context, the notion of unbiased and biased na-ture is not applicable to the configurations leading to spe-cific destination in a partially eccentric system. However, FIG. 6. Convergence dynamics of 7-cell multiple attractorpartially eccentric cloud system (ACA 12). the notion is valid for confused configurations in the sys-tem. Remark that, participation from only one set { , } or { , } in the active RMT set identifies two-attractorpartially eccentric cloud ACAs , . To understandthe dynamics of two attractor partially eccentric cloudsystem, we consider ACA 168 (see Table I). Here, it isnot possible to break more than one consecutive 0’s asRMTs 1 and 4 are passive. That is, a configuration withmore than one consecutive 0’s, · · · · , can be viewedas a sequence of RMTs h· · · ·i . Therefore, an ini-tial configuration with primary RMT set { , , } and/or { , , , } is associated with deterministic point attrac-tor . However, the notion of unbiased or biased natureis true for a confused configuration with primary RMTsets { , } and/or { , , } where the system can be ableto converge at both the point attractors and . Duringthe evolution of the confused configuration, if a single cellin state 1 moves to state 0, then the system converges topoint attractor . That is, the configuration is associatedwith more than one (two) consecutive 0’s when a singlecell in state 1 moves to state 0. As an example, one of thepossible transition is 011011 → , the target is to update cell(s) instate 0 at every time steps which is almost impossible forlarge CA size. That is, P / ( ) → P ( ) and P ( )starting from homogeneous initial configurations and respectively. Therefore, P ( ) → P ( ) → (left one).We have also identified simple two-attractor determin-istic cloud system, where the system can not be ableto converge at point attractor for any initial configu-ration. Remark that, in a two-attractor system, pointattractor (resp. ) is not reachable from any ini- . . . . P ( ) FIG. 7. P ( ) as a function of number of 011 in the homogeneousinitial configuration for ACA 104. tial configuration if RMT 2 (resp. 5) is passive. Theexample transitions are 00100 h i and 11011 h i respectively. Here, deterministic cloudACAs , , , , , , are associ-ated with passive RMT 5.Let us now start the discussion on the convergence dy-namics of multiple attractor (more than two) system withthe dual property of RMTs. This dual property of ac-tive RMTs 0 and 7 indicates that these RMTs are respon-sible for both fixed and confused convergence journeywhen primary RMT sets { } , { } and { } , { } are responsible for point attractors respectively. As anexample, RMT 0 is accountable for fixed convergencejourney 010001 → → / , , , , , .The observations made on the convergence of partiallyeccentric cloud system are probably the most interest-ing result of the study. Fig. 6 shows multiple attractorpartially eccentric cloud system for ACA 12 where pri-mary RMT sets { } , { , } and { , , } are responsiblefor point attractors. Here, every attractor is associatedwith some confused configurations, and some initial con-figurations with deterministic destination. This systemcan catch the complex vote sharing political alliance dy-namics of Indian politics where large number of regionalpolitical parties under federal structure play role in thedynamics of vote sharing. Under this CA system, a con-fused configuration is not associated with every attrac-tors. Therefore, an alliance of some attractors may cover(or optimize) all possible destination of a confused con-figuration. Similarly, in Indian politics, the formationof alliance by regional parties target to cover the swingvoters based on language, religion, caste etc.We should also mention the surprising behaviour ofmultiple attractor partially eccentric cloud ACA 104where the size of the CA matters. Here, passive pri-mary RMT sets { } and { , , , } are responsible forpoint attractors (see Table I). For ACA 104, the proba-bility of convergence to point attractor starting fromconfused configurations depends on the CA size. As anevidence, Fig. 7 shows the P ( ) starting from homo-geneous confused configuration where P ( ) = 0for odd CA size. In details, during the evolution of thehomogeneous confused configuration , an intermedi-ate configuration can be viewed as a 1-block (i.e. consec-utive cells in state 1) which is surrounded by cell(s) instate 0, see example configurations ‘0111111’ (even size1-block) and ‘011111’ (odd size 1-block). Active RMTs2 and 5 play the reason behind this. The size of the 1-block is either even or odd depending on the CA size.Hereafter, the 1-block is separated into two parts by acell in state 0 as RMT 7 is active. As an example, one ofthe possible transition starting from even size 1-block is0111111 → { , , , } ).Therefore, point attractor is not reachable startingfrom even size 1-block depending on the CA size. How-ever, an odd size 1-block may always produce two oddsize 1-block which may finally converges to point attrac-tor . As an example, one of the possible transition start-ing from odd size 1-block is 011111 → { } and { , , , } , the restriction condition to avoid con- fused journey is active RMT 2 and passive RMT 5. Here,active RMT 2 is accountable to reach the point attrac-tor. As an example, one of the possible transition is010110 h i → h i . Whereas activeRMT 5 may responsible for confusion, see example tran-sition 010110 h i → h i . For otherset of point attractors, it can be shown by similar logic.Here, we have identified ACAs , , , (resp. , , , , , ) as eccentric (resp. partiallyeccentric) cloud system. Whereas, ACAs , , are deterministic cloud system.To sum up, we have already classified two-attractorand multiple attractor ACAs of Table II into eccentric,partially eccentric and deterministic cloud system. Fi-nally, we state our main theorem for 32 convergent ACAsof Table II. Theorem 1 (Main result) Under asynchronous updat-ing scheme, among the minimal representing con-verging ACAs with multiple point attractors, , and rules are eccentric, partially eccentric, deterministiccloud ACAs respectively. Eccentric cloud ACA
Partially eccentric cloud ACA
Deterministic cloud ACA [1] A. Wuensche and M. Lesser,
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