arxiv:1607.02291v3 [cs.fl] 8 may 2018 · 2 types of cellular automata in this section, we survey...

Noname manuscript No.(will be inserted by the editor)

A Survey of Cellular Automata: Types, Dynamics, Non-uniformityand Applications (Draft version)

Kamalika Bhattacharjee · Nazma Naskar · SouvikRoy · Sukanta Das

Received: date / Accepted: date

Abstract Cellular automata (CAs) are dynamical systems which exhibit complex global be-havior from simple local interaction and computation. Since the inception of cellular au-tomaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchersover various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been in-troduced in update pattern, lattice structure, neighborhood dependency and local rule. In thissurvey, we tour to the various types of CAs introduced till date, the different characteriza-tion tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Specialattention is given to non-uniformity in CAs and especially to non-uniform elementary CAs,which have been very useful in solving several real-life problems.

Keywords Cellular Automata (CAs) · Types · Characterization tools · Dynamics ·Non-uniformity · Technology

Mathematics Subject Classification (2010) 68Q80 · 37B15

1 Introduction

From the end of the first half of 20th century, a new approach has started to come in scien-tific studies, which after questioning the so called Cartesian analytical approach, says thatinterconnections among the elements of a system, be it physical, biological, artificial or anyother, greatly effect the behavior of the system. In fact, according to this approach, knowingthe parts of a system, one can not properly understand the system as a whole. In physics,David Bohm (1980) is one of the advocates of this approach. This approach is adopted in

This research is partially supported by Innovation in Science Pursuit for Inspired Research (INSPIRE) underDept. of Science and Technology, Govt. of India.

Kamalika Bhattacharjee, Souvik Roy and Sukanta Das,Department of Information Technology, Indian Institute of Engineering and Science Technology, Shibpur,Howrah 711103, India; E-mail: [email protected], [email protected] and [email protected]

Nazma NaskarSchool of Computing Engineering, KIIT University, Bhubaneswar, Odisha, IndiaE-mail: [email protected]

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psychology by Jacob Moreno [see as example Moreno et al (1932)], which later gave birthto a new branch of science, named Network Science. During this time, however, a number ofmodels, respecting this approach, have started to be proposed [see, as an example, McCul-loch and Pitts (1943)]. Cellular Automata (CAs) are one of the most important developmentsin this direction.

The journey of CAs was initiated by John von Neumann (1966) for the modelling of bi-ological self-reproduction. A cellular automaton (CA) is a discrete dynamical system com-prising of an orderly network of cells, where each cell is a finite state automaton. The nextstate of the cells are decided by the states of their neighboring cells following a local updaterule. John von Neumann’s CA is an infinite 2-dimensional square grid, where each squarebox, called cell, can be in any of the possible 29 states. The next state of each cell dependson the state of itself and its four neighbors. This CA can not only model biological self-reproduction, but is also computationally universal. The beauty of a CA is that simple localinteraction and computation of cells results in a huge complex behavior when the cells acttogether.

Since their inception, CAs have captured the attention of a good number of researchersof diverse fields. They have been flourished in different directions – as universal construc-tors, see for example Thatcher (1964); Banks (1970); models of physical systems, see asexample Chopard and Droz (1998); parallel computing machine, see Toffoli and Margolus(1987); etc. In recent years, the CAs are highly utilized as solutions to many real life prob-lems. As a consequence, a number of variations of the CAs have been developed. In thissurvey, we will conduct a tour to a few aspects of CAs. Our purpose is not to explore anyspecific aspect of CAs in depth, rather to cover different directions of CAs research alongwith a good number of references. Interested readers may go to the appropriate referencesto explore their interests in detail.

We conduct this survey with respect to six categories. First, we tour to the various typesof CAs, developed since their inception (Section 2). For example, von Neumann’s CA andWolfram’s CA are of different types – first one is defined over 2-dimensional grid having 29states per cell, whereas latter is one-dimensional binary CA. Second, to study the behaviorof CAs, few tools and parameters are developed. They are visited in Section 3. However,the most interesting property of CAs is probably their complex global behavior due to verysimple local computations of the cells. We inspect some of the most explored global behaviorin Section 4.

During last two decades, a new direction in CAs research has been opening up, whichintroduces non-uniformity in CA structure. Practically, CAs are uniform in all respects -uniformity in cell’s update (that is, synchronicity), uniformity in neighborhood dependency,and uniformity in local rule. Though the uniform CAs are very good in modelling physicalsystems, researchers have introduced non-uniformity in CA structure to model some physicalsystems in a better way, and most importantly, to solve some real life problems efficiently.We survey various aspects of non-uniformity in Section 5, and put our deeper attention on aspecial type of non-uniform CAs in Section 6.

Nowadays, CAs are not only modelling tools, but also technologies. Use of CAs indifferent application domains has established this fact in the last two decades. We exploresome of such applications in Section 7.

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications 3

2 Types of Cellular Automata

In this section, we survey various types of CAs proposed till date. Before stating these types,let us define cellular automata formally.

2.1 Basic definition

A cellular automaton consists of a set of cells which are arranged as a regular network. Eachcell of a CA is a finite automaton that uses a finite state set S. The CAs evolve in discretetime and space. During evolution, a cell of a CA changes its state depending on the presentstates of its neighbors. That is, to update its state, a cell uses a next state function, also knownas local rule, whose arguments are the present states of the cell’s neighbors. Collection ofthe states of all cells at a given time is called configuration of the CA. During evolution, aCA, therefore, hops from one configuration to another.

Definition 1 A cellular automaton (CA) is a quadruple (L ,S,N , f ), where

– L ⊆ ZD is the D-dimensional cellular space. A set of cells are placed in the locationsofL to form a regular network.

– S is the finite state set.– N = (−→v1 ,

−→v2 , · · · ,−→vN) is the neighborhood vector of N distinct elements of L whichassociates one cell to its neighbors. Generally, the neighbors of a cell are the nearestcells surrounding the cell. However, when the neighborhood vector N is given, thenthe neighbors of a cell at location −→v ∈ L are at locations (−→v +−→vi ) ∈ L , for all i ∈{1,2, · · · ,N}.

– f : SN → S is called the local rule of the automaton. The next state of a cell is given byf (a1,a2, · · · ,aN), where a1,a2, · · · ,aN are the state of its N neighbors.

A cellular automaton is also identified by its global transition function. Let us considerC represents SL , the set of all configurations. Then, a CA is a function G : C→ C, which iscalled global transition function. Classically, CAs are synchronous (that is, all the cells areupdated simultaneously) and homogeneous (that is, all the cells use single next sates func-tion). For these cases, if a configuration y = (y−→v )−→v ∈L is successor of another configurationx = (x−→v )−→v ∈L , that is, y = G(x), then y is the result of the following application:For each −→v ∈L

y−→v = G(x)−→v = G(x−→v ) = f (x−→v +−→v1,x−→v +−→v2

. · · · ,x−→v +−→vN) (1)

Nevertheless, a CA is described in terms of four parameters –L ,S,N and f . Over theyears, various types of CAs have been proposed varying the properties of these parameters.Even this basic definition of CA (Definition 1) is sometime compromised to get new typesof CAs. In this section, we survey different types of CAs, developed till date, under thefollowing headings –

1. The dimension and neighborhood of a cell.2. The states of the cell.3. The lattice size and boundary condition.4. The local rule.


(a) One way CA (b) Two way CA (c) von Neumannneighborhood

(d) Moore neighbor-hood

Fig. 1: Neighborhood dependencies: 1a and 1b are 1-dimensional CAs, and 1c and 1d are 2-dimensional CAs.The black cell is the cell under consideration, and its neighbors are the shaded cells and the black cell itself

2.2 The dimension and neighborhood of a cell

The neighborhood of CA cells is strongly correlated with the dimension of the CAs. Theoriginal CA, proposed by von Neumann, is of 2-dimension and uses 5-neighborhood (or-thogonal ones and itself) dependency. Fig. 1c shows such a dependency: the neighbors ofthe black cell is itself and the four shaded cells. This neighborhood dependency is tradition-ally known as von Neumann neighborhood.

A natural extension of this neighborhood dependency is the 9-neighborhood depen-dency, where four non-orthogonal cells are additionally considered as neighbors. Fig. 1dshows this kind of neighborhood dependency, which was proposed by Moore (1962), andis traditionally known as Moore neighborhood. This neighborhood structure has been uti-lized to design the famous Game of Life, a CA which was introduced by John Conway andpopularized by Martin Gardner (1971).

Apart from these two popular neighborhood dependencies for 2-dimensional CAs, someother variations, such as Margolus neighborhood (Toffoli and Margolus (1987)) are also re-ported. In this neighborhood, the lattice L is divided into 2× 2 non-overlapping blocks(2×2×2 cubes in three dimensions). The partitioning ofL into blocks is applied on differ-ent spacial co-ordinates on alternate time steps. Fig. 2a clarifies the idea: the smaller boxesare the cells, whereas the squares enclosing 4 cells are the blocks. There are two types ofoverlapping blocks, shown by dark lines and dotted lines. The cells of a block are neighborsof each other, and the type of blocks alternates in alternating time steps. This kind of CAs isalso known as block cellular automata or partitioning cellular automata. In these CAs, thelocal rule, or rather block rule, updates the entire block as a distinct entity rather than anyindividual cell.

However, in the above types of CAs, the cells are considered as squares. In fact, in mostof the CAs, the cells are square in shape. In some of the works, on the other hand, the cellsare considered as hexagons over 2-dimensional space, and as a result, we get a differentneighborhood dependency; see for example Morita et al (1999); Siap et al (2011). Fig. 2bshows such a neighborhood dependency. Not only hexagonal cells, Imai and Morita (2000);Morita (2016) have worked with triangular cells also. Further, CAs have been defined in hy-perbolic plane by Margenstern and Morita (1999, 2001). As a result, we have been a witnessof several types of CAs depending on the variations of neighborhood dependencies. As anextension of uniform neighborhood dependencies of cells of CAs, Automata Networks (seeSection 5.2) have been defined over arbitrary networks, where the neighborhood dependen-cies of different cells may be different; see for example Marr and Hütt (2009); Tomassiniet al (2005); Darabos et al (2007).


(a) Margolus neigh-borhood

(b) Hexagonal CA (c) Partitioned CA

Fig. 2: Neighborhood dependencies: 2a is a block CA, 2b is hexagonal CA, and 2c is a partitioned CA, wherea square cell is partitioned into four triangular parts

Partitioned cellular automata, which are closely related to block cellular automata, arethe CAs where each cell is partitioned into a finite number of parts. Hexagonal CA of Moritaet al (1999), triangular CA of Morita (2016) are the partitioned CAs. Many properties ofthese CAs have been explored by Morita and his co-researchers. Fig. 2c shows the partition-ing of square cells into four parts. Obviously, neighborhood dependencies of the parts aredifferent than that of the cells. The local rules of partitioned CAs consider the states of partsduring the state updates.

Sometimes, neighborhood of a CA is represented by radius. By radius, we mean thenumber of consecutive cells in a direction on which a cell depends on. For example, for1-dimensional CA, neighborhood N can be represented as N = {−r,−(r−1), · · · ,−1,0,1,· · · ,(r− 1),r}, where r is the radius of the CA. In Fig.1b, r = 1. Classically, the radius issame in every direction. However, in some works this restriction is removed; see for exampleJump and Kirtane (1974); Boccara and Fuks (1998). One of such CAs, called one-way CA,has been proposed by Dyer (1980), in which communication is allowed only in one direction,that is, in a 1-dimensional array, the next state of each cell depends on itself and either of itsleft neighbor(s) or right neighbor(s) (see Fig. 1a).

Although CAs are originally defined over 2-dimensional space, a large number of re-searches have been dedicated to explore one-dimensional CAs; Amoroso and Patt (1972);Pries et al (1986); Sutner (1991); Sikdar et al (2002) are to name a few. Wolfram (1983,1994) has introduced a class of very simple CAs, called Elementary cellular automata orECAs, which are one-dimensional, two-state, and having three-neighborhood dependency(like Fig. 1b). In last three decades, the major share of cellular automata research has goneto ECAs and their variations.

However, several fundamental properties and parameters of 2-dimensional CAs are ex-plored by Packard and Wolfram (1985); Kari (1990); Durand (1993); Terrier (2004); de Oliveiraand Siqueira (2006); Uguz et al (2014), and many others. Higher than 2-dimensional CAs arealso proposed, see for example Gandin and Rappaz (1997); Sarkar and Barua (1998); Millerand Fredkin (2005); Mo et al (2014). For instance, in Miller and Fredkin (2005) two-statethree-dimensional reversible CA (RCA) is described and is shown to accomplish universalcomputation and construction. However, two or higher dimensional CAs sometime behavedifferently than 1-dimensional CAs. For example, reversibility, an well addressed problemof 1-D CA, is undecidable for 2 or higher dimensional CAs, see Kari (1990). In fact, mostof the decision problems related to two or higher dimensional CAs are undecidable, see forexample Dennunzio et al (2014b).


2.3 The states of the cell

A cell can be in any of the states of a finite state set S at any point of time. The number ofstates of von Neumann’s CA is 29. Initially, many researchers targeted to simplify von Neu-mann’s CA by reducing the number of states. For instance, state count of CAs was reducedto eight by Codd (1968). Thatcher (1964) has shown construction and computational uni-versality as well as self-reproducing ability of von Neumann’s cellular space, whereas Arbib(1966) depicted a simple self-reproducing CA capable of universality. Then, Banks (1971)has proved the universal computability of 2-state CA and provided a description of self-reproducing CA using only 4 states. However, all these constructions are for 2-dimensionalCAs, and use 5-neighborhood dependency (see Fig. 1c). Conway’s Game of Life, on theother hand, uses 9-neighborhood dependency (see Fig. 1d) and two states per cell. Gardner(1971) has proved that this binary CA is also computationally universal. In fact, many CAs,including the ECAs, are binary; that is, the cells use only two states. Smith (1971a) hasshown that neighborhood size and state-set cardinality of a CA are interrelated. A CA withhigher neighborhood size can always be emulated by another CA that has lesser neighbor-hood size but higher number of states per cell, and the contrariwise.

Generally, the cellular spaceL is infinite. Hence, the configurations which are the col-lections of states of all cells ofL at different time, are also infinite. As is in von Neumann’sCA, sometime a restriction is imposed on CAs: a quiescent state q ∈ S is considered sothat f (q,q, · · · ,q) = q, where f is the local rule (see Definition 1). This means, a cell whoseneighbors are in quiescent states, remains in quiescent state. As a result, we get a new classof configurations, called finite configurations.

Definition 2 Let q ∈ S be a quiescent state of a CA (L ,S,N , f ), that is, f (q,q, · · · ,q) = q,and the cellular spaceL be infinite. A configuration of the CA is called a finite configurationif all but a finite number of cells are in q.

Thus, if the initial configuration of the CA is a finite configuration, at every time step thenew configuration remains as a finite configuration. Let us consider CF represents the set ofall finite configurations. Obviously, CF ⊂ C, where C = SL , the set of all configurations.Then, the global transition function of the CA over CF , GF : CF → CF is a restriction of G.This restriction is at least needed to simulate a CA in computer.

In a number of works, such as Martin et al (1984); Chaudhuri et al (1997), states ofa CA are considered as elements of a finite field. Sikdar et al (2000, 2002) have furtherconsidered the states as elements of an extension field. Itô et al (1983); Cattaneo et al (2004)have considered the state set as Zm (the integers modulo m) in their works.

In almost all cases, however, every cell of a CA uses the same state set. In polygeneousCAs, on the contrary, a CA may use different state sets, see Sarkar (2000).

2.4 The boundary condition

As the cellular space L is usually infinite, there is no question of boundary, and boundarycondition. However, in a few works, L is assumed as finite, which is obviously havingboundaries. These CAs are finite CAs.

Definition 3 A CA is called as a Finite Cellular Automaton if the cellular spaceL ⊆ ZD isfinite. Otherwise, it is an Infinite Cellular Automaton.


00

(a) Null boundary (b) Periodic boundary

(c) Adiabatic boundary (d) Reflexive boundary

(e) Intermediate boundary

Fig. 3: Boundary conditions of 1-D finite CAs. Arrows pointing to a cell indicate the dependencies of the cell.Here all the CAs use 3-neighborhood dependency

Finite CAs are really important if the automata are to be implemented. Two boundary con-ditions for finite CAs are generally used – open boundary condition and periodic boundarycondition. In open (fixed) boundary CAs, the missing neighbors of extreme cells (leftmostand rightmost cells in case of 1-D CAs) are usually assigned some fixed states. Among theopen boundary conditions, the most popular is null boundary, where the missing neighborsof the terminal cells are always in state 0 (Null). Null boundary CAs have received a goodattention by the researchers, who have used CAs for VLSI (Very Large Scale Integration)design and test. Hortensius et al (1989b); Tsalides et al (1991); Chaudhuri et al (1997); Sik-dar et al (2002); Das and Sikdar (2010) are some examples of such works. Fig. 3a explainsnull boundary condition of a 1-dimensional CA.

In case of periodic boundary condition, on the other hand, the boundary cells are neigh-bors of some other boundary cells. For instance, for 1-D CAs, the rightmost and leftmostcells are neighbors of each other (see Fig. 3b). Higher dimensional CAs under periodicboundary conditions have also been explored by some researchers, such as Sarkar and Barua(1998); Jin and Wu (2012); Uguz et al (2013) etc. However, the periodic boundaries of a 2-DCA can be visualized by folding right and left sides of a rectangle to form a tube, and thentaping top and bottom edges of the tube to form a torus.

Periodic boundary conditions sometime relate finite CAs and infinite CAs through peri-odic configurations.

Definition 4 A configuration x ∈ C of a D-dimensional infinite CA is periodic if the con-figuration is invariant under D linearly independent translations. That is, there exist D nat-ural numbers n1,n2, · · · ,nD so that x is same as ni times shift of x in dimension i, for eachi ∈ {1,2, · · · ,D}.

Let CP ⊂ C denote the set of all periodic configurations of a CA. Then, GP : CP → CP is arestriction of G. However, the members of CP preserve periodicity of configurations, whichis also preserved in periodic boundary condition. So, periodic configurations are sometimecalled as the periodic boundary conditions of finite CAs.

Some other variations of boundary conditions also exist, such as adiabatic (or reflecting)boundary, reflexive boundary and intermediate boundary. In adiabatic (or reflecting) bound-ary condition, the boundary cells assume the missing neighbors as their duplicates. Fig. 3cshows this boundary condition where the left (right) neighbor of the leftmost (rightmost)cell is the cell itself. In reflexive boundary condition, shown in Fig. 3d, the left and right


neighbors of boundary cells are the same cell. Finally, Fig. 3e explains the intermediateboundary condition for a 1-D CA, where a missing neighbor of a boundary cell is the neigh-bor’s neighbor of the boundary cell. Nandi and Chaudhuri (1996) have used the intermediateboundary condition to improve quality of pseudo-random patterns, generated by CAs.

As another special case of open boundary condition, stochastic boundary condition isproposed, where the boundary cells assume some states stochastically. Cheybani et al (2000)have used this boundary condition to model traffic flow on a one-dimensional lattice withfinite number of cells. In this traffic model, the authors have considered that the cars areinjected at the leftmost cell with a probability, and the cars are removed from the rightboundary with another probability.

2.5 Local rule

A cell of a CA changes its state following a next state function f : SN → S where S is theset of states and N is the size of the neighborhood. The map f , which is generally knownas the local rule of the CA, can be conveyed in different ways. As an instance, in Conway’sGame of Life, the local rule is f : {0,1}9→{0,1}, where state 0 represents a dead cell andstate 1 represents an alive cell. It is stated as following in Li et al (2010):

– “Birth: a cell that is dead at time t, will be alive at time t +1, if exactly 3 of itseight neighbors were alive at time t.

– Death: a cell can die by:– Overcrowding: if a cell is alive at time t and 4 or more of its neighbors are

also alive at time t, the cell will be dead at time t +1.– Exposure: If a live cell at time t has only 1 live neighbor or no live neigh-

bors, it will be dead at time t +1.– Survival: a cell survives from time t to time t + 1 if and only if 2 or 3 of its

neighbors are alive at time t.”

However, the local rule f can also be represented in a tabular form. In this form, thetable contains entries for the next state values corresponding to each of the possible neigh-borhood combinations according to the local rule. This representation has been popularizedby Wolfram (1983), and used to name his Elementary CAs (ECAs). Table 1 shows sevenrules of ECAs, which are named after the decimal equivalents (last column of Table 1) of8-bit binary sequence. Obviously, the tabular form is good if size of neighborhood (N) andthe state set (S) are very small.

Against a binary map f : {0,1}N → {0,1}, an N-tuple (s1,s2, · · · ,sN) ∈ SN can beviewed as a Min Term of N-variable switching function. So, each of the tuples of the map fis named as Rule Min Term (RMT). Table 1 shows eight RMTs against a rule. In the worksof present authors, the RMTs of rules have been utilized. This concept is further extendedby Bhattacharjee and Das (2016) to use it by non-binary local rules. However, all the aboveworks that use RMTs are about 1-dimensional CAs.

Definition 5 Let f : SN → S be a local rule of a CA. A tuple (s1,s2, · · · ,sN) ∈ SN is calledas a Rule Min Term (RMT) of the rule f , and is usually represented by it decimal equivalent.If S = {0,1, · · · ,d− 1}, then the RMT is r = s1.dN−1 + s2.dN−2 + · · ·+ sN−1.d + sN . Thenext state against this RMT of the rule, that is f (s1,s2, · · · ,sN), is also represented as f [r].

Therefore, a rule can be seen as a collection of RMTs along with their next states. In thatsense, RMTs are atomic elements of a rule. The RMTs along with their next state values are


Table 1: ECAs rules. Here, PS and NS represent present state and next state respectively

PS 111 110 101 100 011 010 001 000Rule

(RMT) (7) (6) (5) (4) (3) (2) (1) (0)

0 1 0 1 1 0 1 0 901 0 0 1 0 1 1 0 1500 0 0 1 1 1 1 0 30

NS 0 0 0 0 0 1 0 1 5

0 1 0 0 1 0 0 1 731 1 0 0 1 0 0 0 2000 1 0 1 0 0 0 0 80

sometime called as transitions. In case of ECAs, where f : {0,1}3→{0,1}, a transition is aquadruplet (x,y,z, f (x,y,z)). A transition of an ECA rule is active if f (x,y,z) 6= y; otherwisethe transition is passive. A rule f can also be represented by its active transitions only.To present a rule, in general, at least its active transitions are to be noted down. The ruleof Game of Life (see above), for example, shows all active transitions. In case of ECAs,although the numbering style of Wolfram (1983) (as shown in Table 1) is widely used, theyare alternatively identified by their active transitions only, known as transition codes. In theworks of Fatès et al (2006); Fatès (2013), for example, transition codes of ECAs rules havebeen used.

When the rule f of a CA is linear, then the CA is also linear. That is, the global transitionfunction G of the CA is a linear function. In case of linear rule f : SN → S, the rule can be

expressed as f (a1,a2, · · · ,aN) =N∑

i=1ci.ai, where ci ∈ S is a constant. And, the set S forms a

commutative ring with identity. There are seven linear ECAs for rules 60, 90, 102, 150, 170,204 and 240 (excluding rule 0, which is also trivially linear). Apart from these ECAs, there isno additional additive ECAs. So, these rules are presented by some authors as linear/additiverules.

In case of block cellular automata and partitioned cellular automata, which have beenbriefly discussed in Section 2.2, presentation of local rules is altogether different. The rule(block rule) of block CA does not change individual cells of the CA, rather it looks atthe content of the block and updates the whole block. Toffoli and Margolus (1987) havedescribed such rule in their book. For partitioned CAs, on the other hand, each cell is dividedinto several parts and the next state of each cell is determined by the states of the adjacentparts of the neighbor cells. Fig. 4 shows a triangular partitioned CA and its rule, which isexplored by Morita (2016). Like block CA, the local rule of a partitioned CA updates all theparts of a cell. Unlike traditional CA, however, the local rule does not consider the presentstates of the neighbors of a cell, but adjacent parts of neighboring cells (Fig. 4b).

Classically, the CAs are uniform, synchronous and deterministic – that is, all cells areupdated together with the same local rule. A new class of non-classical CAs have beenproposed where cells can follow different rules. These CAs are known as hybrid or non-uniform CAs. Historically, this class of CAs assume that the rules are the ECAs rules only.Pries et al (1986); Hortensius et al (1989b); Chaudhuri et al (1997), and many others haveexplored these CAs, which are one-dimensional and finite. In this case, one needs to definethe local rules for individual cells; hence she needs a rule vector R.


(a) Triangular Partitioned CA (b) A local rule

Fig. 4: Triangular partitioned CA and its rule (as given by Morita (2016)). The rule is rotation symmetric

Definition 6 Let L = Z/nZ be the cellular space, where n, a natural number, representsthe number of cells in the space. A rule vector of a non-uniform CA over L is R =〈R0,R1, · · · ,Rn−1〉, where Ri is a rule used by cell i, i ∈L .

As a proof of concept, let us consider a 5-cell non-uniform CA with rule vector R =〈90,150,90,5,150〉. This implies that the first and third cells use ECA rule 90, second andfifth cells use rule 150, and the fourth cell uses rule 5. This class of CAs are specially uti-lized in VLSI design and test. Another type of CA, called a programmable CA (with respectto VLSI design), exists, where a cell can chose a distinct local rule at every time instant,see for example Nandi et al (1994). Nevertheless, a special kind of CAs have also been in-troduced where a CA updates itself in asynchronous way. In short, non-classical CAs havegained popularity in recent times. Section 5 and Section 6 are dedicated to discuss thesenon-classical CAs.

3 Characterization tools of Cellular Automata

In A New Kind of Science, Wolfram (2002) has argued that, to find out how a particular CAwill behave, one has to observe what is happening just by running the CA. Predicting behav-ior of a system by means of (mathematical) analysis and without running it is only possible,according to Wolfram, for special systems with simple behavior (page 6 of Wolfram (2002)).In spite of this observation of some CAs researchers, a few characterization tools and pa-rameters have been proposed in different time to analyze and predict the behavior of someCAs. Needless to say, all kind of dynamic behaviors of a CA may not be analyzed by a tool,but tools are used to discover some specific properties of a CA.

In this section, we survey the characterization tools and parameters, developed till dateto analyze the CAs. Tools are mainly developed for one-dimensional CAs, and for two orhigher dimensional CAs, “run and watch” is the primary technique to study the behavior.Though, few parameters are proposed which can be used to guess the behavior of two orhigher dimensional CAs.

3.1 De Bruijn graph

In the different development phases of CAs, graph theory has played an important role.One of the roles is describing the evolution of an automaton, and another is relating local


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(a) The de Bruijn graph B(2,{0,1})

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(b) De Bruijn graph for rule 90

Fig. 5: The de Bruijn Graph of CA with rule 90 (2nd row of Table 1)

properties to global properties. As an automaton has states which are mapped to anotherstates using overlapping neighborhood sequence, it is very obvious to treat it by shift register.So, de Bruijn graph is considered as an alternative characterization tool.

Definition 7 Let Σ be a set of symbols, and s≥ 1 be a number. Then, the de Bruijn graph isB(s,Σ) = (V,E), where V = Σs is the set of vertices, and E = {(ax,xb)|a,b ∈ Σ,x ∈ Σs−1}is the set of edges.

Fig. 5a shows the de Bruijn graph B(2,{0,1}). A de Bruijn graph has ms vertices, con-sisting of all possible length-m sequences of Σ, where m is the number of symbols in Σ.This graph is balanced in the sense that each vertex has both in-degree and out-degree s.

A one-dimensional CA can be represented by a de Bruijn graph. Let us consider s =N − 1 and Σ = S, where N is the size of neighborhood and S is the state-set of a 1-DCA (see Definition 1). The edges (ax,xb) of B(N − 1,S) show the overlap of nodes and(axb) ∈ SN . Now label each edge (ax,xb) of B(N− 1,S) by f (axb) ∈ S, where f is thelocal rule. This labelled graph represents a one-dimensional CA with rule f , state set S, andneighborhood size N. Since this graph does not relate to the lattice size, de Bruijn graph canbe used to study finite and infinite 1-D CAs.

Fig. 5b is the de Bruijn graph for ECA rule 90 (whereas Fig. 5a is a graph for any ECA).This graph shows that if the left, self and right neighbors of a cell are all 0s, then next stateof the cell (that is, f (0,0,0)) is 0, if the neighbors are 0,0 and 1 respectively, the next state is1, and so on. In his work, Sutner (1991) has defined s-fusion operation, which is equivalentto labelling an edge in above manner.

Over the years, this graph has been used by various researchers to understand globalbehavior, like surjectivity and reversibility, number conservation, equicontinuity etc. of 1-dimensional CAs, see for example Sutner (1991); Soto (2008); Voorhees (2008). In Martinezet al (2008), ECA 110 has been explored to determine a glider-based regular expressions.Mora et al (2008) have studied cyclic properties and inverse of a CA using pair diagram ofde Bruijn graph. Betel et al (2013) have used de Bruijn graph to solve the parity problem.Recently, Mariot et al (2017) have studied the periods of pre-images of spatially periodicconfigurations in surjective CA using de Bruijn graphs.

The de Bruijn graphs are traditionally used to represent and study classical CAs. How-ever, in recent past, Dennunzio et al (2013b) have shown that de Bruijn graph can also beused to represent non-uniform CAs.


3.2 Matrix algebra

Unlike de Bruijn graph, matrix algebra can represent only finite CAs. In fact, this character-ization tool has been developed by Das et al (1990a) to study the behavior of 1-D hybrid ornon-uniform CA that uses only linear ECA rules (see Section 2.5). A non-uniform CA is lin-ear/additive if all of its local rules are linear/additive. So non-uniform finite linear/additiveCAs can be characterized by matrix algebra. Chaudhuri et al (1997) have provided a goodaccount of works on linear/additive non-uniform finite CAs in their book.

Table 2: Linear and complemented rules

Rule Linear Rule Complement

(With XOR logic) (With XNOR logic)

60: Si(t +1) = Si−1(t)⊕Si(t) 195: Si(t +1) = Si−1(t)⊕Si(t)90: Si(t +1) = Si−1(t)⊕Si+1(t) 165: Si(t +1) = Si−1(t)⊕Si+1(t)102: Si(t +1) = Si(t)⊕Si+1(t) 153: Si(t +1) = Si(t)⊕Si+1(t)150: Si(t +1) = Si−1(t)⊕Si(t)⊕Si+1(t) 105: Si(t +1) = Si−1(t)⊕Si(t)⊕Si+1(t)170: Si(t +1) = Si+1(t) 85: Si(t +1) = Si+1(t)204: Si(t +1) = Si(t) 51: Si(t +1) = Si(t)240: Si(t +1) = Si−1(t) 15: Si(t +1) = Si−1(t)

A binary linear rule f can be expressed as XOR of input variables. For example, ECArule 90 is a linear rule, and f90(x,y,z) = x⊕ z. All the linear ECA rules can similarly beexpressed by XOR logic. Table 2 shows such expressions, which is reproduced directlyfrom Chaudhuri et al (1997). However, if we look at the XORed expression of ECA rule90, it is understood that a cell of ECA 90 depends on left and right neighbors only. Whencharacterized by matrix algebra, a binary finite (uniform/non-uniform) CA is represented bya matrix, and then this neighborhood dependency is taken care of.

Let us consider a binary finite (uniform/non-uniform) CA having n cells. This CA isrepresented by an n×n characteristics matrix operating on GF(2) In this matrix, the ith rowrepresents the dependency of the ith cell to its neighbors. The characteristics matrix (T ), inthis case, is formed as:

“ T [i, j] ={

1 if the next state of the ith cell depends on the present state of the jth cell ”0 otherwise

(2)The non-uniform CAs can use different rules in different cells, hence we need a rule

vector to specify the rules against cells (see Definition 6). Let us take a 4-cell non-uniformCA with rule vector R= 〈150,150,90,150〉 under null boundary condition. Then, the char-acteristics matrix of the CA is:

T =

1 1 0 01 1 1 00 1 0 10 0 1 1


Here, third cell’s rule is 90, hence depends on left and right neighbors only (see Table 2).So, the third row of the T is

[0 1 0 1

]. Since boundary condition is null, left neighboring

cell of first cell and right neighboring cell of last cell are missing.The global transition function of such a CA can be expressed by T . Let us consider x

and y be two length-n configurations of an n-cell linear CA, and x be the successor of y.Then, y = T ·x, where x and y are considered as vectors. The matrix T , however, can be usedto efficiently study the reversibility properties, convergence etc. of a linear (uniform/non-uniform) CA. We briefly discuss this study in Section 6.1.

Originally, the matrix T is defined for 3-neighborhood binary CAs. Latter, Sikdar et al(2000, 2001) have extended this tool to represent 3-neighborhood, non-binary, finite CAs.However, the authors have assumed that the states of the cells are the elements of GF(2p).

These CAs are also linear, and named as hierarchical CAs. Further, CAs over GF(2pqr···

) arealso designed by Sikdar et al (2002).

Apart from 3-neighborhood dependency, 5-neighbor dependent linear CAs are also stud-ied by matrix algebra. For example, Martın del Rey and Rodrıguez Sınchez (2011) havestudied reversibility of such binary CAs with null boundary condition. Whereas, Cinkir et al(2011) have studied reversibility of linear CAs with periodic boundary conditions over Zp,where p ≥ 2 is a prime number. For 2-dimensional linear CAs also, a list of works are re-ported using matrix algebra as characterization tool, see Chattopadhyay et al (1999); Siapet al (2011); Uguz et al (2013, 2014).

3.3 Reachability tree

Like matrix algebra, the reachability tree has been proposed to study 1-D non-uniform finiteCAs that use only ECAs rules in their rule vectors. Unlike matrix algebra, reachability treecan represent non-uniform non-linear CAs. Das et al (2004) have first used this tool tostudy non-uniform finite CAs under null boundary condition. In this section, we first definereachability tree for an n-cell non-uniform CA with rule vector R = 〈R0,R1, · · · ,Rn−1〉under null boundary condition, where each rule Ri, i ∈ {0,1, · · · ,n−1} is an ECA rule.

Recall that an ECA rule can be considered as a collection of RMTs (see Definition 5)along with their next state values. Let us denote the set of RMTs of Ri as Zi

8. That is,Zi

8 = {0,1,2,3,4,5,6,7}. However, under null boundary condition, RMTs 4, 5, 6 and 7 ofR0 (similarly, RMTs 1, 3, 5 and 7 of Rn−1) are invalid. So, Z0

8 = {0,1,2,3}, and Zn−18 =

{0,2,4,6} for null boundary condition.

Definition 8 The reachability tree of an n-cell CA with rule vector 〈R0,R1, · · · ,Ri, · · · ,Rn−1〉under null boundary condition is a rooted and edge-labeled binary tree with n+ 1 levels,where Ei.2 j = (Ni. j,Ni+1.2 j, li.2 j) and Ei.2 j+1 = (Ni. j,Ni+1.2 j+1, li.2 j+1) are the edges be-tween nodes Ni. j ⊆ Zi

8 and Ni+1.2 j ⊆ Zi+18 with label li.2 j ⊆ Ni. j, and between nodes Ni. j

and Ni+1.2 j+1 ⊆ Zi+18 with label li.2 j+1 ⊆ Ni. j respectively (0 ≤ i ≤ n− 1, 0 ≤ j ≤ 2i− 1).

Following are the relations which exist in the tree:

1. [For root] N0.0 = Z08 = {0,1,2,3}.

2. ∀r ∈ Ni. j, RMT r of Ri is in li.2 j (resp. li.2 j+1), if Ri[r] = 0 (resp. 1). That means,li.2 j ∪ li.2 j+1 = Ni. j (0≤ i≤ n−1, 0≤ j ≤ 2i−1).

3. ∀r ∈ li. j, RMTs 2r (mod 8) and 2r+ 1 (mod 8) of Ri+1 are in Ni+1. j (0 ≤ i ≤ n− 3,0≤ j ≤ 2i+1−1).

4. [For level n−1] ∀r ∈ ln−2. j, RMT 2r (mod 8) of Rn−1 is in Ni+1. j (0≤ j ≤ 2n−1−1).


N0.0

2,3,6,7

N1.0

E0.0

0,1,4,5

N1.1

E0.1

4,5,6,7

N2.0

E1.0

4,5,6,7

N2.1

E1.1

0,1,2,3

N2.2

E1.2

0,1

2.3N

E1.3

0,2 4,6 0,2 4,6 0,2,4 6 0,2

0,1,2,3

(1,3)

0

(0,2)

1

(2,7)

0(3,6)

1(1,4,5)

0

(0)

1

0

(6,7)

10

(6,7)

1(4,5) (4,5) (0,1,2)

0

(3)

1

(0,1)

0 1

0 1 0 1

(4,6)

0 0 1

(4,6)

0 1

(4)

0 1

(6)

0 1 0 1

(0,2) (0,2) (0,2)

1

(0,2)

Fig. 6: Reachability Tree for null boundary 4-cell non-uniform CA with rule vector R= 〈5,73,200,80〉

5. [For level n] Nn, j = /0, for any j, 0≤ j ≤ 2n−1.

Informally, we can state the reachability tree construction in the following way: Formthe root with RMTs 0, 1, 2 and 3 of R0. Out of the 4 RMTs, the RMTs having next statevalue 0 are in the label of left edge of the root, and the rest RMTs are in the right edge. Nowwe can get the nodes of level 1: if an RMT r is in a label of an edge, then put the RMTs 2r(mod 8) and 2r+1 (mod 8) of R1 in the node which is connected by the edge. Similarly,we can get next edges and nodes of level 2, and of level 3, etc. However, for the last rule,that is Rn−1, all RMTs are not present in null boundary condition. So, only even RMTs ofRn−1 can be in the nodes of level n−1. Finally, we get the leaves, which are empty.

Fig. 6 shows an example of reachability tree of a 4-cell non-uniform CA with rule vectorR = 〈5,73,200,80〉 under null boundary condition. The root is a set {0, 1, 2, 3}, and theleaves are empty. It can be observed in the figure that the edges between the nodes of twoconsecutive levels are formed after applying the corresponding rule on the RMTs. However,an edge Ei.2 j (resp. Ei.2 j+1) is called as 0-edge (resp. 1-edge). In Fig. 6, 0-edges and 1-edgesare marked.

Reachability tree represents reachable configurations of a CA. A configuration is reach-able if it has at least one predecessor; otherwise it is a non-reachable (or Garden-of-Eden)configuration. A sequence of edges from root to a leaf represents a reachable configurationwhen 0-edges and 1-edges are replaced by 0 and 1 respectively. For example, the configura-tion 0011 of the CA of Fig. 6 is reachable. In Fig. 6, some edges (and nodes) are shown bydotted lines. These edges do not exist in the tree. So, a configuration 1110, for example, isnon-reachable.

Reachability tree has been utilized to discover many properties of above class of non-uniform CAs, such as reversibility, convergence, etc., see for example Das (2007); Adaket al (2016). We discuss some of these discoveries in Section 6.2.

When boundary condition changes, the structure of reachability tree also changes. Dasand Sikdar (2009) have first developed this structure to study reversibility of non-uniformCAs under periodic boundary condition. Recently, reachability tree is extended by Bhat-tacharjee and Das (2016) to study 1-D non-binary finite CAs.


3.4 Z-Parameter, λ -Parameter, Θ-Parameter

There are some parameters that can be used to characterize some aspects of one-dimensionalCAs, such as the λ parameter, the Z parameter, and the obstruction (Θ) parameter. Theλ , Z, and Θ parameters are proposed respectively by Langton (1990), Wuensche (1994,1998) and Voorhees (1997). For a d-state CA with radius r, if m out of the total d2r+1

neighborhood configurations map to a non-quiescent state, then λ is defined as: λ = md2r+1 ,

that is, the percentage of all the entries in a rule table which maps to non-quiescent states.This parameter can be compared with temperature in statistical physics, or the degree ofnon-linearity in dynamical systems, although these are not equivalent (Li et al (1990)).

To track behavior of binary CAs, Z parameter is proposed as an alternate. It takes intoaccount the allocation of rule table values to the sub-categories of related neighborhoods andpredicts the convergence of global behavior, extremes of local behavior between order andchaos, surjectivity etc. From a given partial pre-image of a CA state, values of the successivecells can be deduced using this parameter. Two probabilities – Zle f t and Zright of the nextunknown cell being deterministic are calculated from the rule-table. The Z parameter is thegreater of these values and varies between 0 & 1. A derivation of the Z parameter in termsof rule table entries is also given in Wuensche (1998). High value of Z implies that numberof pre-images of an arbitrary CA state is relatively small.

In Voorhees (1997), the Θ parameter is defined and shown to characterize the degreeof non-additivity of a binary CA rule. It is shown that the λ parameter and Θ parameterare equal respectively to the area and volume under certain graphs. These parameters provetheir utilization in classification of one-dimensional binary CAs.

3.5 Space-time diagram and Transition diagram

Although neither of space-time diagram and transition diagram (or state-transition diagram)is a characterization tool, but these two diagrams have been used to observe and predict thebehavior and dynamics of a CA. These diagrams are finite in size. So, for the CAs withinfinite or big sizes, only a part of the whole systems can be viewed through these diagrams.

Space-time diagram is the graphical representation of the configurations of a CA ateach time t. Here, the configuration lies on x-axis and y-axis represents time. Each of theCA states are generally depicted by some color (see Fig. 7a). So, the evolution of the CAcan be visible from the patterns generated in the state-space diagram. This has been usedto study the nature of CA in a set of papers, see for example Wolfram (2002); Dennunzioet al (2014a). Some packages are available in public domain which can be used to observespace-time diagrams of CAs - Fiatlux (see Fatés (2017)) is one of them.

Transition diagram, also called as state transition diagram, of a CA is a directed graphwhose vertices are the configurations of the CA, and an edge from a configuration y toanother configuration x represents that x is the successor of y (see Fig. 7b). Sometime thedirections are omitted if they can be understood from the context. Transition diagrams areheavily used in the works of Chaudhuri et al (1997). It can be noted that, the dynamicbehavior of any CA can also be visualized and studied using its transition diagram. DiscreteDynamics Lab, an online laboratory (Wuensche (2017)), is a good place to get transitiondiagrams of CAs.


(a) Space-time diagram

015

129

63

10

52

78

13

1

4

1114

(b) Transition diagram

Fig. 7: Space-time diagram and transition diagram of n-cell ECA 90 (2nd row of Table 1) under periodicboundary condition. For (a), cell length n = 51, and black implies state 1 and white implies state 0. For (b),n = 4, and the configurations are in their decimal form.

4 Global behavior of Cellular Automata

The most exciting aspect of CAs is their complex global behavior, which is resulted fromsimple local interaction and computation. The CAs have many elementary properties of thereal world, such as reversibility and conservation laws, chaotic behavior, etc. These prop-erties motivate researchers to simulate physical and biological systems by CAs. Vichniac(1984) has targeted to simulate physics by CAs. Lattice gases (Frisch et al (1986)), Ising spinmodels (Grinstein et al (1985)), traffic systems (Kerner (2004); Matsukidaira and Nishinari(2003)) etc. are also simulated using CAs. The book of Chopard and Droz (1998) showsabout these works. Apart from the ability of modelling physical world, the CAs are capableof performing several computational tasks. In this section, we survey some of the globalbehaviors of CAs such as universality, reversibility, conservation laws, computability, etc.

4.1 Universality

The first feature that has attracted the researchers is the capability of CAs in performinguniversal computation. An arbitrary Turing machine (TM) can be simulated by a CA, souniversal computation is possible by CA. The universality of CAs has been first studied byThatcher (1964); Arbib (1966); Banks (1970). However, the idea of simulating TM doesnot use the parallelism property of a CA. In Smith (1971b), the existence of computation-universal cellular spaces with small neighbor-state product is proved. Later, Morita andHarao (1989); Dubacq (1995) have shown the procedure of simulating a TM by a reversibleCA. Computation universality of one-way CAs and totalistic CAs are also reported by Albertand Culik II (1987). Computational universality is also shown following very simple rules,such as Game of Life (Gardner (1971)), the billiard ball computer (block CAs) (Durand-Lose (1998)). The existence of computation-universal one-dimensional CA with 7-statesper cell for radius r = 1 and 4-states per cell for r = 2 is proved in Lindgren and Nordahl


(1990). A very simple automaton, ECA 110 of Wolfram (1984) is shown as computationallyuniversal by Cook (2004).

Apart from simulating a Turing machine, a CA can simulate another CA. If a CA cansimulate all CAs of the same dimension, it is called intrinsically universal (Kari (2005b)).The smallest intrinsically universal CA in one-dimension is reported in Ollinger (2003). Fortwo dimension, nonetheless, the same is a 2-state and 5-neighborhood CA (Banks (1970)).Some other notable related works are Martin (1994); Ollinger (2002).

4.2 Invertibility and reversibility

For long, the questions of invertibility have been the major focus of research in CAs. If theglobal transition function G of a CA is one-to-one, the CA is termed as an injective CA.However, it is called surjective if the function is onto and bijective if G is both onto and one-to-one. The study of reversibility of CA was started with Hedlund (1969) and Richardson(1972). From Hedlund (1969), we get that a function G : SZD → SZD

is the global transitionfunction of a D-dimensional (infinite) CA if and only if G is continuous, and it is shift-equivalent (this result is known as Curtis - Hedlund - Lyndon Theorem). As a consequenceof this result, we get that a CA G is reversible if and only if it is a bijective map. In fact,Hedlund and Richardson independently proved that all injective CAs are reversible.

In their seminal paper, Amoroso and Patt (1972) have shown an effective way to decidereversibility of 1-dimensional infinite CA, on the basis of the local rule. In Di Gregorio andTrautteur (1975), a decision algorithm is reported for CAs with finite configurations. An ele-gant scheme based on de Bruijn graph for deciding the reversibility of a one dimensional CAis presented in Sutner (1991). However, Kari (1990) has shown that no algorithm can decidewhether or not an arbitrary 2-D CA is reversible. This result can also be extended to higherdimensional CAs. An interesting result has been reported by Toffoli (1977), which says, a(D+1)-dimensional reversible CA can simulate any D-dimensional CA. Later, Morita andHarao (1989); Morita (1995) have shown that a 1-D reversible CA can simulate reversibleTuring machines as well as any 1-D CA with finite configurations. Since reversible Turingmachines can be computationally universal, we can find universal 1-D reversible CA. Someother notable works on reversible CAs are Maruoka and Kimura (1979); Culik (1987); Du-rand (1993); Kari (1994, 2005a); Moraal (2000); Mora et al (2006); Soto (2008); Morita(2008).

However, all the works reported above deal with infinite CAs. In case of finite CAs,things are bit different. For example, the algorithms of Amoroso and Patt (1972) can notsuccessfully work on finite CAs. Martın del Rey and Rodrıguez Sınchez (2011); Cinkir et al(2011) have studied reversibility of 1-D linear finite CAs having only two states. Recently,Bhattacharjee and Das (2016) have reported an algorithm to decide the reversibility of 1-Dfinite CAs. The reversibility of 1-D non-uniform CAs has also deeply studied. We discussabout these studies in Section 6.

4.3 Garden-of-Eden

One of the earliest discovered results on CAs was the Garden-of-Eden theorems by Moore(1962) and Myhill (1963). Injectivity and surjectivity properties of CA are correlated bythese theorems. A configuration is named as a Garden-of-Eden configuration, if it does nothave a predecessor; that is, if it is a non-reachable configuration. In 1962, Moore has shown


that the existence of mutually erasable configurations in a two-dimensional CA is sufficientfor the existence of Garden-of-Eden configurations (Moore (1962)). He has also claimed thatexistence of a configuration with more than one predecessors ensures existence of anotherconfiguration without any predecessor. However, in Myhill (1963), the reverse was proved.He has shown that the extant of mutually indistinguishable configurations is both necessaryand sufficient for the extant of Garden-of-Eden configurations.

In Amoroso and Cooper (1970), the equivalence between the existence of mutuallyerasable configurations and mutually indistinguishable configurations has been established.This implies that the converse of Moore’s result is true as well. It has also been shown inthe paper that, for finite configurations (Definition 2) both of the above conditions remainsufficient, but neither is then necessary. For finite configurations, a CA is irreversible if andonly if a Garden-of-Eden configuration exists. And, a CA is surjective, if and only if it isbijective (Amoroso and Cooper (1970)). So, the existence of Garden-of Eden configurationsviolates the injectivity property. The relation between global function of infinite CA and itsrestriction to finite configurations has been established in Richardson (1972). Some otherimportant results are listed in Maruoka and Kimura (1976); Sato and Honda (1977); Toffoliand Margolus (1990); Kari (2005b).

Garden-of-Eden theorems for CAs have further been extended to Cayley graphs ofgroups by Machì and Mignosi (1993); Ceccherini-Silberstein et al (1999). In Capobianco(2007), the surjectivity and surjunctivity of CA in Besicovitch topology is recoded. More-over, in Margenstern and Morita (1999, 2001), CA is defined in hyperbolic plane. In Mar-genstern (2009), it is depicted that, the injectivity and surjectivity properties, proved byMoore and Myhill, are no longer valid for CAs in the hyperbolic plane.

4.4 Topology, dynamics and chaotic behavior of CAs

The global transition function G of an infinite CA is a continuous map on the compact metricspace SZD

according to the topology induced by the metric d. This metric d is usually con-sidered as the Cantor distance. Therefore, a CA can be viewed as a discrete time dynamicalsystem 〈SZD

,G〉. So, the (infinite) CAs can be studied by topological dynamics and chaostheory.

Many definitions of chaos, however, use the notion of sensitivity to initial condition. ACA is sensitive to initial condition, or simply sensitive if and only if there exists a δ > 0 suchthat, ∀x ∈ SZD ∀ε > 0 ∃y ∈ SZD ∃n ∈ N : d(x,y) < ε and d(Gn(x),Gn(y)) ≥ δ . Thatis, in a sensitive CA, a (small) change in initial condition would greatly affect the CA infuture. A popular definition of chaos is due to Devaney (1986), which says that a dynamicalsystem is chaotic if and only if it is transitive, has dense periodic points, and sensitive toinitial conditions. However, a CA is transitive if and only if for any two non-empty opensubsets U and V of SZD

, there exists an n ∈ N so that Gn(U)∩V 6= /0. Further, a CA hasdense periodic points if and only if the set {x ∈ SZD |∃k ∈ N : Gk(x) = x} of all periodicpoints is a dense subset of SZD

. It has been proved by Codenotti and Margara (1996) thatfor CAs, transitivity implies sensitivity to initial condition. So, a CA is chaotic in Devaney’ssense if and only if it is transitive and has dense periodic points.

Apart from the Devaney’s definition, there are more restrictive definitions of chaos,such as Knudsen chaos, positively expansive chaos, etc. A CA is chaotic according tothe definition of Knudsen if and only if it has a dense orbit and sensitive to initial con-ditions. The CA G has a dense orbit if and only if there exists a configuration x so that


∀y ∈ SZD ∀ε > 0 ∃n ∈ N : d(Gn(x),y)< ε . On the other hand, a CA is positively expan-sive chaotic if and only if it is transitive, has dense periodic points, and positively expansive.Positive expansivity is a stronger form of sensitivity. A CA is positively expansive CA ifand only if there exists a δ > 0, such that for any two different configurations x and y,d(Gn(x),Gn(y)) ≥ δ for some n ∈ N. However, these definitions of chaos are related. Thechaotic behavior of CAs are studied by a number of works; some examples are Mitchell et al(1993); Cattaneo et al (1999, 2000); Acerbi et al (2009).

Based on their dynamic behavior, CAs are classified into different classes. One suchclassification, depending on their degree of equicontinuity, is due to Kurka (1997). A con-figuration x ∈ SZD

is an equicontinuity point of the CA if for every ε > 0, there is a δ > 0such that ∀y ∈ SZD

: d(x,y) < δ implies d(Gn(x),Gn(y)) < ε for all n ∈ N. Now, a CA iscalled equicontinuous if all configurations are equicontinuity points. There is a connectionbetween sensitivity and equicontinuity points of a CA: a CA G is sensitive to initial condi-tions, if the CA does not have any equicontinuity points. Following is the classification ofKurka:

(1) equicontinuous CAs,(2) CAs with some equicontinuity points,(3) sensitive but not positively expansive CAs, and(4) positively expansive CAs.

Durand et al (2003b) have studied the decision problems to determine whether a given CAbelongs to a given class, and they have shown that most of the problems are undecidable.However, for the additive CAs, Dennunzio et al (2009) have studied the directional dynam-ics, and then classified them.

Apart from the above work, there are many other works, specially for the 1-D case,that target to classify CAs depending on their dynamical behavior. Wolfram (1984) hasreported a classification of ECAs without considering a precise mathematical definition.Culik and Yu (1988) have formalized Wolfram’s classification. In fact, different parameters,discussed in Section 3.4, have been developed to classify the CAs depending on their chaoticbehavior. For the two-dimensional space, the generalization of the parameters – sensitivity,neighborhood dominance and activity propagation, is reported in de Oliveira and Siqueira(2006).

Another studied behavior of CAs is nilpotency. A CA is nilpotent if for each configu-ration x, Gn(x) = c is a singleton set for sufficiently large n. Obviously, c is a fixed point.Culik et al (1990) have shown that for two or more dimension, the nilpotency of CAs is un-decidable. The same result for 1-D CAs has been proved by Kari (1992). Some more worksto study the dynamical properties of CAs, using expansivity, subshift, homomorphism auto-morphisms and endomorphisms are reported in Blanchard and Maass (1997); Ward (1994).Shereshevsky (1992) has defined the left and right Lyapunov exponents for 1-dimensionalCAs. Finelli et al (1998) generalized the theory of Lyapunov exponents for D-dimensionalCAs and proved that all expansive CAs have positive Lyapunov exponents for almost all thephase space configurations.

4.5 Randomness

Stephen Wolfram (1985) has introduced CAs as an excellent source of pseudo-randomness.Massive parallelism, simplicity and locality of interactions of CAs, offer many benefits overother techniques, specially in case of hardware implementation. These benefits along with


pseudo-randomness have made CAs as an area of extensive research in VLSI circuit testing(see for example Hortensius et al (1989b); Chaudhuri et al (1997); Das and Sikdar (2010)),Monte-Carlo simulations (see Saraniti and Goodnick (2000)), Field Programmable GateArrays (see as example Comer et al (2012)), cryptography (see as example Wolfram (1986a);Das and Chowdhury (2013, 2011); Formenti et al (2014); Leporati and Mariot (2014)) etc.In fact, the most appealing application of pseudo-randomness of CAs is in the domain ofcryptography. We briefly discuss these applications in Section 7. However, most of the workson pseudo-randomness have been divided in mainly two directions:

– Most of the research have been going on in the first direction, to generate pseudo-random numbers using CAs. Here, generally an integer Xi is generated between zero andsome number m (word size of the computer), where the fraction Ui =

Xim is the real num-

ber, uniformly distributed between 0 and 1. This number can be generated in several ways.For example, in Wolfram (1986b), the sequence is generated by ECA 30 from the single cellwith initial state 1 among all cells, initiated with state 0. In a recent work, Bhattacharjee et al(2017) have used a 3-state 3-neighborhood 1-D CA, and considered a small window of cells.The base-3 numbers, observed through the window, are considered the pseudo-random num-bers. Sometime whole configurations of a finite CA are also considered as numbers. Someother works of generating pseudo-random numbers are Tomassini et al (2000); Alonso-Sanzand Bull (2009). In Sipper and Tomassini (1996); Wang et al (2008), some optimizationalgorithms are applied to CAs, whereas in Guan and Zhang (2003); Guan and Tan (2004)dynamic behavior is allowed in the cells to generate the pseudo-random numbers.

– In the second direction, pseudo-random patterns are generated using finite CAs. Here,pattern means the configuration of a CA of length n, where each cell can take any of theCA states. Note that, in a pattern, individual cell values have significance. For example, asequence 〈0110〉 can be a treated as a number 6, but in case of pattern, it is “0110”. Somenotable works on pseudo-random pattern generation using non-uniform CAs are in Das et al(2003b); Das (2007). Here finding of the minimum cell length is important. For example, inDas (2007), a 45-cell CA based pseudo-random pattern generator (PRPG) is designed whichbeats all existing PRPGs.

4.6 Conservation law

Apart from reversibility, there exist other conservation laws, which are equally important inphysics. Various direction of conservation (invariants) laws in CAs are reported in Fredkinand Toffoli (1982); Pivato (2002); Boccara and Fuks (2002). Among them number conserv-ing CAs (NCCAs) are the most studied and used concept. Let us interpret the states of cellsas numbers. Then, a CA is an NCCA if the sum of the states remains invariant during evolu-tion of the CA. NCCAs are defined with respect to the spatially periodic configurations andfinite configurations. Boccara and Fuks have given the necessary and sufficient conditionsof a 1-D CA to be NCCA, initially for 2 states per cell in Boccara and Fuks (1998) andthen for any arbitrary number of states in Boccara and Fuks (2002). In Pivato (2002), a gen-eral treatment to conserved quantities in 1-D CA is reported. Fuks has shown that motionrepresentations can be constructed by 1-D binary NCCAs. Fuks (2004) has also extendedthe work to probabilistic CAs. NCCAs in higher dimensions are also studied by Durandet al (2003a). Further, universality and other dynamics of NCCAs are explored in Moreira(2003); Formenti and Grange (2003).

NCCAs have widely appeared as the models of highway traffic. The notable works inthis regard are Nagel and Schreckenberg (1992); Nagel (1996); Matsukidaira and Nishinari


(2003); Kerner (2004). Kohyama (1989, 1991) has used NCCAs for particle conservation.In 2-dimension, the Margolus CA is a number conserving CA; see Margolus (1984). Apartfrom NCCAs, additive conserved quantities in CAs are introduced and investigated in Hat-tori and Takesue (1991). In Takesue (1995), a necessary and sufficient condition for a givenCA rule to profess a staggered invariant is studied. In Morita and Imai (1998); Morita et al(1999), the computational universality of partitioned number-conserving (and reversible)CA is shown by simulating a universal counter machine. However, these CAs are not ex-actly the same as NCCAs, because reducing a partitioned CA to a non-partitioned one isnot number preserving. The concept of extending conserved quantities to that of mono-tone quantities was presented in Kurka (2003), considering CAs with vanishing particles.Number conserving property of CAs has also been studied by applying the communicationcomplexity approach; see for example Goles et al (2011).

In Das (2011), non-uniform NCCAs are characterized. This paper has reported O(n)time algorithms for verification and synthesis of an n-cell non-uniform NCCA. Dennunzioet al (2014a) have given a generalized definition of number conservation of non-uniformCAs. Further, number conservation property of ECA under asynchronous update has beenstudied in Hazari and Das (2014).

4.7 Computational tasks

In the CAs literature, two domains related to computation are mostly studied: density classi-fication task and synchronization problems. The problem statement of density classificationcan be summarized as follows - any initial configuration having more 1s (0s) than 0s (1s)must converge to all 1s (0s) configuration. However, Land and Belew (1995) have provedthat it is impossible to solve this problem with 100% accuracy. Because of the impossibilityof solving the standard density classification task, research efforts have shifted towards find-ing the best rule which can solve the problem almost perfectly – for one-dimensional CA byKari and Le Gloannec (2012), for two-dimensional CA by Morales et al (2001); de Oliveiraand Siqueira (2006), for non-uniform CA by Maiti et al (2006) and stochastic CA by Fatès(2013). However, Fuks (1997) has shown that the density classification task is solvable byrunning in sequence the trivial combination of elementary rules 184 and 232. A good surveyabout the problem can be found in de Oliveira (2013).

The synchronization problems, like firing squad, queen bee, firing mob, etc. are studiedby CAs. The goal of firing squad synchronization problem is designing a CA, which ini-tially has one active cell and evolves to a state, where every cell is simultaneously active. Agood number of works on this problem are found in literature; some examples are Noguchi(2004); Umeo et al (2005); Manzoni and Umeo (2014). The firing mob problem, which is ageneralization of firing squad problem, is solved by Culik and Dube (1991). However, thequeen bee and leader election problems are considered as the inverse problem of firing squadproblem. Here, initially states of all cells are identical and by choosing a proper rule, a cellcomes to a special state. Smith (1976) has first introduced the problem. Latter the problemhas been explored by a number of researchers, such as Mazoyer et al (1999); Beckers andWorsch (2001); Stratmann and Worsch (2002); Banda et al (2015).

Another computation problem, named early bird problem has been defined and investi-gated first by Rosenstiehl et al (1972). Here, any cell in the quiescent state may be excitedby the outside world. These excitations result in special “bird” states instead of the quies-cent states. The task is to give a transition function such that, after a certain time the firstexcitation(s) can be distinguished from the latter ones. This problem has been studied by


Vollmar (1977); Legendi and Katona (1981, 1986). A variation of this problem is the dis-tributed mutual exclusion problem, where some cells of initial configuration are in a specialstate (called critical section requesting state) and during evolution of the CA, these cellswill be in another special state (called critical section executing state) one-by-one. Recentlythis problem is explored by Roy and Das (2017). Some other computation problems, such asFrench flag problem (Herman and Liu (1973)), shortest path problem (Adamatzky (1996)),generating discretized circles and parabolae in real time (Delorme et al (1999)) etc. are alsodiscussed in literature.

5 Non-uniformity in cellular automata

Conventionally, all the variants of CAs possess basic three properties - uniformity, syn-chronicity and locality. The uniformity refers to that each of the CA cells are updated bythe identical local rule. The synchronicity implies that all the cells are updated simultane-ously; whereas locality refers to that the rules act locally and neighborhood dependencies ofeach cell is uniform. Note that, the cells perform computation locally, and the global behav-ior of CA is received due to this local computation only. However, synchronicity is a specialtype of uniformity, where all the cells are updated simultaneously and uniformly. In fact,uniformity is everywhere in CA, in local rule, cell update and in lattice structure. We cansummarize this in the following way:

– Uniformity in update: all cells are updated simultaneously in each discrete time step.– Uniformity in lattice structure and neighborhood dependency: lattice structure is uni-

form and each cell follows similar neighborhood dependency.– Uniformity in local rule: each of the cells updates its state following the same rule.

Over the years, researchers have successfully been using classical CA as a modellingtool. However, it has become apparent that many phenomena, such as chemical reactionsoccurring in a living cell, are found in nature which are not uniform. These new modellingrequirements led to a new variant of the CAs. As a result, non-uniformity in CAs has beenintroduced. Following are the main three variants of non-uniformity in CAs, which we getafter relaxing above mentioned constraints of uniformity.

1. Asynchronous cellular automata (ACAs): the cells are not updated at the same (discrete)time step and can be independently updated - breaks the uniform update constraint (dis-cussed in Section 5.1).

2. Automata Network: the CA is on a network and the states of the node evolve with neigh-borhood defined by the network - breaks uniform neighborhood constraint (discussed inSection 5.2).

3. Hybrid or non-uniform cellular automata: cells can assume different local transitionfunctions - breaks uniform local rule constraint (discussed in Section 5.3).

5.1 Asynchronous Cellular Automata (ACAs)

Like other synchronous systems, a CA also assumes a global clock which forces the cellsto get updated simultaneously. This assumption of global clock is not very natural, andis relaxed in ACAs. The concept of ACAs and their computational ability has first beendeveloped by Nakamura (1974), it has further been studied by Golze (1978); Nakamura


(1981); Hemmerling (1982); Ingerson and Buvel (1984); Le Caër (1989). ACAs have beendeveloped on two-dimensional grid by Cori et al (1993) to report the concurrent situationsemerged in distributed systems.

The word ‘asynchronism’ means that the parts of the system do not share the same time.In asynchronous CAs, cells are independent and so, during the evolution of the system,the cells are updated independently. There are several interpretations on the way of applyingasynchronism. By simplifying, it can be said that asynchronism is to break the perfect updatescheme. The main asynchronous updating schemes found in literature are fully asynchronousupdating and α-asynchronous updating.

– Under fully asynchronous updating scheme, a cell is chosen uniformly and randomly ateach time step to update. That is, at each step, only one cell is updated.

– In α-asynchronous updating scheme, each cell is updated with probability α . This im-plies that a cell does not apply the rule with probability 1−α , and stays in its old state.

The parameter α is known as synchrony rate. Obviously, when α = 1, the CA becomes syn-chronous. In that sense, the classical CAs are special case of α-asynchronous CAs. Latter,Bouré et al (2012) have used other asynchronous updating schemes: β - and γ-asynchronism.Then, Dennunzio et al (2013a) have developed an m-asynchronous CA and generalized thevarious updating methods used so far. A survey on asynchronous update schemes can befound in Fatès (2014).

In one of the pointing work, Hemmerling (1982) has shown the computation equiva-lence of synchronous and asynchronous cellular space. He has also shown that, any d-statedeterministic rule can be simulated by a 3d2 state asynchronous rule with same neighbor-hood dependency. From a different point of view, Golze (1978) has shown that, a (D+ 1)-dimensional asynchronous rule can simulate a D-dimensional synchronous rule. Dennunzioet al (2012a) showed how fully asynchronous CA can simulate universal Turing machine.

In an early work, Golze and Priese (1982) have shown that ACA can implement Petrinet. A Petri net is a directed bipartite graph which is used to model distributed systems.Cori et al (1993) have shown latter that ACAs can be used as models of concurrency anddistributed systems. The computing abilities of these ACAs have further been investigatedby Pighizzini (1994); Droste et al (2000). As shown by Manzoni and Umeo (2014), ACAscan be used to address firing squad synchronization problem.

In Blok and Bergersen (1999), the change that occurs in the Game of Life, when thesites get updated with a given probability, are identified. Ruxton and Saravia (1998) haveanalyzed the sensitivity of ecological system modelled by simple stochastic cellular au-tomata to spatio-temporal ordering. In Tomassini and Venzi (2002); Fatès (2013), asyn-chronous rules are used to address the density classification problem. Suzudo (2004) hasstudied the usage of genetic algorithms for determining the mass-conservative (also callednumber-conserving) asynchronous models that would generate nontrivial patterns.

It is thought that reversibility and asynchronism are two opposite terms. However, thequestion of reversibility in ACAs has been studied by Sarkar et al (2012); Sethi et al (2014).Their approach is to study the possibility of returning back to the initial configuration aftera finite number of time steps. These so-called reversible ACAs have been used by Das et al(2012) to generate patterns with specific Hamming distance. These reversible ACAs havefurther been utilized by Sethi and Das (2016) in symmetric-key cryptography. Mariot (2016)has introduced the notion of asynchrony immunity for CAs that could be used for crytog-raphy. In Manzoni (2012), the dynamical properties of CAs, such as injectivity, surjectivity,permutivity, sensitivity, expansivity, transitivity, dense periodic orbits and equicontinuity arere-defined for the asynchronous CAs.


Convergence of ECAs to fixed points under fully asynchronous update has been studiedby Fatès et al (2006). The convergence time of these ACAs have also been explored in thiswork. Recently, Sethi and Das (2015); Sethi et al (2016) have further studied the convergenceof these ACAs, and have used the convergent ACAs in designing an efficient two classpattern classifier.

5.2 Automata Network

Traditionally, cellular automata consist of a regular network with local uniform neighbor-hood dependency. However, in automata network (also called as cellular automata network),this uniform local neighborhood dependency is relaxed. Here, cellular automata rules allowa cell to have an arbitrary number of neighbors, and thus can be set to work on any givennetwork topology, as shown, for example, in Marr and Hütt (2009). As a matter of fact, therules of automata networks can not be always local. And, since the non-local and local rulesare different, it is, therefore, expected that the non-local rules may lead to different behaviorfrom the conventional local rule-based CAs. Some example works in this regard are Boccaraand Roger (1994); Newman and Watts (1999); Yang and Yang (2007).

The earliest version of such non-uniformity in neighborhoods are found in Jump andKirtane (1974); Smith (1976). From the 1990s, networks have been used as an importantmodel for solving different complex problems; see as example Adami (1995); Watts andStrogatz (1998). In fact, after the work of Watts and Strogatz (1998), researchers becomemore interested on automata networks. As noted by Tomassini (2006), we get a wider classof generalized automata networks by extending standard lattice cellular automata and ran-dom Boolean networks. Tomassini et al (2005); Darabos et al (2007) have shown that au-tomata networks with arbitrary topologies perform better than the regular lattice structuresfor the majority and synchronization problems. The work of Cori et al (1993), which is apioneering work on ACAs and which models concurrency and distributed systems, also usesautomata network. Yang and Yang (2007) have developed a new type of small-world cellularautomata by combining local updating rules with a probability of long-range short-cuts tosimulate the interactions and behavior of a complex system.

Domosi and Nehaniv (2005) have investigated automata network as algebraic structuresand developed their theory in line with other algebraic theories, such as semi-groups, groups,rings and fields. They have also shown a new method for the emulation of the behavior ofany (synchronous) automata network by the corresponding asynchronous one. Kayama andImamura (2011) have shown the network representation of Game of Life, where the char-acteristics is like one of Wolfram’s class IV rules. In Kayama (2012), the network derivedfrom ECAs and five neighbor totalistic CA rules are further reviewed. However, the studiesin this area are still at a very early stage.

5.3 Non-uniform CAs or Hybrid CAs

Among the above mentioned models, the most popular and studied model is Hybrid CAor Non-uniform CA, where the cells can use different local rules. The study of the non-uniform CA has been started with Pries et al (1986), where the authors have studied thegroup properties of 1-dimensional finite CAs under null and periodic boundary conditions.In that work, a special type of non-uniform CAs have been investigated, where the cells useWolfram’s CAs rules (see Table 1). Since then, however, the major thrust of non-uniform


. . . . . .IN IN IN IN IN

logicCombinational

OUT OUT OUT OUT OUT

Cell 0 Cell i−1 Cell i+1 Cell n−1Cell i

. . .. . . fi

null boundary null boundary

0 0

(FF) (FF) (FF) (FF)(FF)

f0 fn−1

Fig. 8: Implementation of an n-cell non-uniform ECA under null boundary condition

CAs research has been on this class of CAs, see for example Hortensius et al (1989a); Daset al (1992); Chaudhuri et al (1997); Das and Sikdar (2009). We dedicate the next section tosurvey this class of non-uniform CAs.

In recent years, the generalized definition of non-uniform CAs has been given by Cat-taneo et al (2009); Dennunzio et al (2012b), where the cells may follow different rules withdifferent neighborhood dependencies. Formenti and his colleagues have been investigatingthis class of non-uniform CAs, and they have identified various sub-classes of these CAs.Some basic global properties of non-uniform CAs, such as surjectivity, injectivity, equicon-tinuity, decidability, structural stability etc. have also been explored by Cattaneo et al (2009);Dennunzio et al (2012b, 2014a); Salo (2014).

6 Non-uniform ECAs

Nowadays, research on non-uniform CAs has gained a popularity. The researchers havebeen exploring them from various directions, and proposing generalized definition of non-uniform CAs. However, since late 1980s until today, the primary focus of non-uniform CAsresearch has been on a special class of 1-dimensional CAs, where the cells follow Wolfram’srules. The main reason of choosing this class of CA is two-fold – (1) Wolfram, in early1980s, showed the efficacy of 3-neighborhood binary CAs in modelling physical systemsand in producing complex global behavior, and (2) ease of implementing Wolfram’s CAsrules in hardware. We call these CAs as non-uniform ECAs, to differentiate them from oth-ers. Since the early days, these CAs are explored targeting some hardware-related problems.Obviously, these CAs are finite. In this section, we discuss only about finite non-uniformECAs.

Fig. 8 shows hardware implementation of an n-cell non-uniform ECA under null bound-ary condition. Here, each cell consists of a flip-flop (FF) to store the state of a cell and a com-binational logic circuit to find the next state of the cell. Due to similarity of this structurewith finite state machine, many authors, who have been working with finite non-uniformECAs, consider these CAs as finite state machine (FSM). As a consequence, these authorscall the configurations of CAs as states.

As we have discussed, however, different cells of a non-uniform ECA may use differentrules, so we need a rule vector R = 〈R0,R1, · · ·Rn−1〉 to define these CAs (see Defini-tion 6). Here, cell i uses rule Ri, and Ri is presented as a decimal number as shown inTable 1. Obviously, uniform or classical CAs are special cases of non-uniform CAs whereR0 =R1 = · · ·=Rn−1. In this section, we will survey the research organized to understand


the behavior of non-uniform ECAs, with respect to two categories: additive/linear CAs andnon-linear CAs.

6.1 Linear/additive CAs

A CA is linear if G, the global transition function of the CA, is linear. There are sevenECAs having rules 60,90,102,150,170,204 and 240 which satisfy the linearity conditions(we have excluded rule 0 from the list, which also satisfies the conditions but a trivial one).These are linear ECAs, and the rules are linear rules (see also Section 2.5). However, thereis no additive ECAs other than these ECAs. So in literature, the terms “linear CA” and“additive CA” are used interchangeably.

A (finite) non-uniform ECA is linear/additive if and only if each of the rules in rulevector is linear/additive. Since there are 7 linear/additive rules, the rule vector R of a lin-ear/additive non-uniform ECA is to be designed with only these 7 rules. The linear ECAsrules can be expressed by XOR logic (see Table 2, which is reproduced directly from Chaud-huri et al (1997)). A linear/additive non-uniform ECA can also be expressed by characteristicmatrix, T . The matrix T is a tri-diagonal matrix, where all elements except the elements ofmain, upper and lower diagonals are always zero (see Section 3.2). In Das et al (1990a,1992), the matrix algebraic tool has been used for analyzing state transition behavior of thisclass of CAs. From the matrix algebraic tool and characteristic polynomial, several interest-ing features of the non-uniform ECAs are derived.

Apart from the seven linear/additive rules, there are another seven rules which are com-plement of the seven. These rules are named as complemented rules, see Table 2. The CAswith these complemented rules can also be characterized by matrix algebra. In fact, a CAthat uses the fourteen rules, seven linear and seven complemented, can be efficiently char-acterized by algebraic tools, see for example Chaudhuri et al (1997); Ganguly et al (2008)1.However, for these CAs, we additionally need an Inversion Vector along with the character-istics matrix. An inversion vector F for such an n-cell CA is defined as following:

Fi ={

1, if the ith cell uses a complemented rule0 otherwise

(3)

If a configuration y of the CA is successor of an configuration x, then y = T.x+F .Linear/additive CAs are broadly classified as group CAs and non-group CAs, in litera-

ture. Next, we briefly discuss about them.

6.1.1 Group cellular automata

A non-uniform ECA is called a group CA if and only if the determinant det(T ) = 1. Thenaming of the subclass of linear/additive non-uniform ECAs as group CAs comes from thefact that, these CAs form cyclic group “under the transformation of operation with T ”, seeChaudhuri et al (1997). Group CAs are reversible CAs. In a group CA, therefore, all theconfigurations are reachable from some other configurations of the CA.

In Pries et al (1986), it is stated that if Ri is a group rule then its complement Ri (thatis, 255−Ri) is also a group rule, which is proved in Das (1990). A subclass of group CAs is

1 Some authors call the non-uniform CAs with linear/additive and complemented rules as additive CAs.The reason may be that, these CAs can be characterized using the tools of linear CAs. However, strictlyspeaking, these CAs are not additive, in general.


maximal length CAs, in which all non-zero configurations lie in the same cycle. As shownby many authors, such as Hortensius et al (1989a); Bardell (1990); Serra et al (1990), theseCAs produce pseudo-random patterns having high randomness quality, and are utilized inelectronic circuit testing.

In case of maximal length CAs, the characteristics polynomial is primitive. Cattell andMuzio (1996) have given us a scheme of synthesizing a maximal length CA from a givenprimitive polynomial over GF(2). Rules 90 and 150 are used to construct such non-uniformECAs, and no single rule can produce a maximal length CA. In Cattell and Zhang (1995),the maximal length CAs are synthesized up to size 500, where one or two cells follow rule150 and the rest follow rule 90. Boundary condition of these maximal length CAs is null.However, Nandi and Chaudhuri (1996) have shown that, for a periodic boundary CA, thecharacteristic polynomial is factorizable; therefore, there exists no maximal length CA underperiodic boundary condition.

6.1.2 Non-group cellular automata

These are irreversible linear/additive non-uniform ECAs. Here, the characteristics matrixT is singular, whereas the T of a group CA is non-singular. Non-group CAs are exploredin different areas, see for example Bhattacharjee et al (1995); Chakraborty et al (1996);Chattopadhyay et al (2000); Roncken et al (2000). Any non-group CA is characterized bythe following terms -

– attractors: cyclic states form attractors. If an attractor contains only a single state, it issaid to be in graveyard state or a point state attractor or fixed point.

– α-basin or α-tree: the set of state(s) rooted at any attractor state α , is termed as α-basinor α-tree.

– depth or height of a CA represents the number of single-step evolution, required by theCA to reach to the nearest cyclic state from a non-reachable (that is, Garden-of-Eden)state.

Some important findings about number of predecessors of the all-zero configurationand depth of non-uniform ECAs are reported in Chaudhuri et al (1997). A fraction of allreachable/non-reachable configurations of a uniform CA have been identified by Martinet al (1984). However, in general for any additive CA (uniform/hybrid), the fraction ofreachable/non-reachable configurations can be numerated from the knowledge of the num-ber of predecessors of a reachable state. Another scheme has been developed by Das andSikdar (2008) to identify and count reachable and non-reachable configurations of a finite(uniform/non-uniform) ECA, which can be linear and as well as non-linear.

Some fascinating classes of non-group CAs are also explored, like - multiple attractorCAs (MACAs) by Chattopadhyay et al (2000); Maji et al (2003b), depth-1∗ CAs (D1 ∗ CAs)by Chowdhury (1994) and single attractor CAs (SACAs) by Das et al (1992). These CAshave been employed in a broad range of purposes like hashing, see Ganguly et al (2000),classification, see Ganguly et al (2002a); Maji et al (2003b), designing easy and fully testableFSM, see Chowdhury et al (1993), etc.

6.2 Non-Linear CAs

During the early age of non-uniform ECAs, the non-linear non-uniform ECAs have not beenwidely explored, due to absence of proper characterization tool. In the past, several attempts


Table 3: First and Last Rule Tables

(a) First rule table

Rules for Class ofR0 R1

3, 12 I5, 10 II6, 9 III

(b) Last rule table

Rule class Rule setfor Rn−1 for Rn−1

I 17, 20, 65, 68II 5, 20, 65, 80III 5, 17, 68, 80IV 20, 65V 17, 68VI 5, 80

have been made to study the characteristics of uniform CAs (linear or nonlinear) qualita-tively and quantitatively in terms of parameters, such as λ parameter (Langton (1990)), Zparameter (Wuensche (1994, 1998)) etc. De Bruijn graph is also considered as a characteri-zation tool of CAs (linear/nonlinear) (Sutner (1991); Provillard et al (2011)).

A characterization tool, named reachability tree (see Section 3.3), has been proposedto characterize (finite) non-uniform ECAs Das (2007); Das and Sikdar (2008, 2009). It hasbeen shown that, out of 256 ECAs rules, only 62 rules can design a rule vector of non-uniform reversible ECA under null boundary condition, whereas for periodic boundary con-dition, additional 8 rules, that is, total 70 rules can take part in a rule vector of reversibleautomaton. However, the position of these rules in a rule vector is not arbitrary. For the pur-pose of getting a rule vector of reversible CA, the 62 rules are classified into six classes,which are related to each other. Table 4 shows the classes and their relations. To get a rulevector of a reversible non-uniform ECA under null boundary condition, one needs to choosea rule from Table 3a, then the subsequent rules from Table 4, and finally the last rule fromTable 3b. It can also be observed that the group CAs under null boundary condition can bedecided from the tables.

In Naskar (2015), reachability tree has been used to distinguish cyclic/acyclic configu-rations from the configuration space of non-linear non-uniform ECA. In the same work, 1-Dnon-linear non-uniform ECA with point state attractors are also explored using reachabil-ity tree. However, Maiti et al (2010); Ghosh et al (2011) have developed rule vector graph(RVG) for studying invertibility of 3-neighborhood CA with null boundary condition.

Non-linear non-uniform ECAs are proved to be efficient in various application fields,like VLSI design and test (Das (2007)), pattern recognition and classification (Maji (2005);Das et al (2009)), etc. Further, designing of pseudo-random pattern generator (PRPG) aroundreversible non-linear non-uniform ECAs are reported in Das et al (2003b); Das (2007); Dasand Sikdar (2010).

In Ganguly et al (2002b), the concept of GMACA (generalized MACA) has been intro-duced for non-linear non-uniform ECAs. The efficiencies of MACA and GMACA have beencompared with respect to pattern recognition. In Maji (2005), both linear and non-linearnon-uniform ECAs have been used for designing a pattern classifier.

Fuzzy CA, a natural extension of boolean CA, is analyzed and synthesized using matrixalgebraic tool by Maji and Chaudhuri (2005, 2004). This CA has also been used to designpattern classifier.


Table 4: Class relationship of Ri and Ri+1

Class of Ri Class ofRi Ri+1I 51, 60, 195, 204 I

85, 90, 165, 170 II102, 105, 150, 153 III

53, 58, 83, 92, 163, 172, 197, 202 IV54, 57, 99, 108, 147, 156, 198,201 V

86, 89, 101, 106, 149, 154, 166, 169 VIII 15, 30, 45, 60, 75, 90, 105, 120, 135, I

150, 165, 180, 195, 210, 225, 240III 15, 51, 204, 240 I

85, 105, 150, 170 II90, 102, 153, 165 III

23, 43, 77, 113, 142, 178, 212, 232 IV27, 39, 78, 114, 141, 177, 216, 228 V

86, 89, 101, 106, 149, 154, 166, 169 VIIV 60, 195 I

90, 165 IV105, 150 V

V 51, 204 I85, 170 II

102, 153 III86, 89, 90, 101, 105, 106, 149, 150, VI

154, 165,166, 169VI 15, 240 I

105, 150 IV90, 165 V

7 Cellular Automata as Technology

Technology refers to the collection of techniques, methods or processes used to providesome services or solutions to problems, or in the accomplishment of an objective, such asscientific invigilation. The CAs have been historically used as a method for simulating bi-ological and physical systems, and utilized to theoretically study such systems. Since late1980s, however, the CAs have been started to be used as solutions to many real-life prob-lems. In this section, we survey some of such solutions.

7.1 Electronic circuit design

The CAs, particularly non-uniform ECAs, have received their popularity as technology inthe era of VLSI. The simplicity, modularity and cascadability of CA have enticed the re-searchers of VLSI domain. Some of these areas are briefly described here.

7.1.1 Early phase developments:

CAs based machines, CAMs (CA Machines), having high degree of parallelism have beendeveloped by Toffoli and Margolus (1987), which are ideally suited for simulation of com-plex systems. Even before the introduction of CAM, CAs have been utilized as parallel mul-tipliers by Atrubin (1965); Cole (1969), parallel processing computers by Manning (1977),


prime number sieves by Fischer (1965), and sorting machine by Nishio (1981). Design offault-tolerant computing machine by Nishio and Kobuchi (1975) and nanometer-scale clas-sical computer by Benjamin and Johnson (1997) are also commendable works.

7.1.2 VLSI Design and Test:

Hortensius et al (1989b,a) has proposed non-uniform ECA based pseudo random patterngenerator (PRPG) for built-in self-test (BIST) in VLSI circuits. Some of the major contri-butions in the research of PRPGs are reported in Das (1990); Tsalides et al (1991); Chowd-hury (1994); Das and Sikdar (2010). The CAs are also proposed as a framework for BISTstructures by Tsalides (1990); Das (1990); Chowdhury (1994); Das (2007); Chakrabortyand Chowdhury (2009) and as a deterministic test pattern generator by Albicki and Khare(1987); Das and Chaudhuri (1989); Das (1990, 2007). Utilizing the scalability of non-uniform ECAs, a test solution for multi-core chips has been proposed in Das and Sikdar(2010). While testing a CUT (Circuit-Under-Test) with pseudo-random patterns, a set ofpatterns may be prohibited to the CUT which may adversely affect the circuit. Non-uniformECAs based solutions to this problem have been proposed by Ganguly et al (2002c); Daset al (2003b); Das (2007) that can generate pseudo-random-patterns without PPS (Prohib-ited Pattern Set). Finally, a universal test pattern generator, termed as the UBIST (UniversalBIST) has been reported in Das and Chaudhuri (1993); Das et al (2003a). This UBIST isable to generate any one of the four types of test patterns - (i) pseudo-random, (ii) pseudo-exhaustive, (iii) pseudo-random without PPS (Prohibited Pattern Set), and (iv) deterministic.

7.1.3 Synthesis of Finite-State Machine (FSM):

Mitra et al (1991); Misra et al (1992) have depicted the design of testable FSM with CAs.In Chowdhury et al (1993); Chakraborty et al (1996), some fascinating properties of a non-group CA and its dual, and their relationship are reported. These papers have also exploreda particular class of non-group CAs, named as D1∗CAs, which have been recommended “asan ideal test machine which can be efficiently embedded in a finite state machine to enhancethe testability of the synthesized design”, see Chakraborty et al (1996).

7.1.4 Security and others

A good number of works have been reported in literature that deal with CAs based en-cryption, cryptography and authentication techniques; see for example Nandi et al (1994);Bao (2004); Seredynski et al (2004); Seredynski and Bouvry (2005); Das and Chowdhury(2011, 2013); Formenti et al (2014). Other notable works, related to electronic circuit designare non-uniform ECAs based error correcting codes by Chowdhury et al (1994); Paul andChowdhury (2000), signature analysis by Hortensius et al (1990); Das et al (1990b), etc. Formore discussion, see Chaudhuri et al (1997).

7.2 Computer vision and machine intelligence

A group of researchers have also explored CAs in the fields of image processing, patternrecognition etc. These fields are roughly named as computer vision and machine intelli-gence.


7.2.1 Image processing:

CAs, specially 2-dimensional CAs, are used for image processing. They can address all thesignificant image processing works like translation, zooming, rotation, segmentation, thin-ning, compression, edge detection and noise reduction, etc., see for example Rosin (2006,2010); Kazar and Slatnia (2011). Khan (1998) has shown that using hybrid CAs, it is pos-sible to rotate any image through an arbitrary angle. In Paul et al (1999), a new GF(2) CAbased transform coding scheme for gray level and color still images has been proposed.Some works regarding edge detection and noise reduction are reported in Wongthanavasuand Sadananda (2003); Sadeghi et al (2012).

7.2.2 Pattern recognition:

CAs have been a popular tool for pattern recognition and classification since long, see forexample Jen (1986); Raghavan (1993); Chaudhuri et al (1997). As shown in the book byChaudhuri et al (1997), MACAs (multiple attractor CAs) can act as a natural classifier.The correlation between MACA and Hamming Hash Family (HHF) is shown in Gangulyet al (2000). Hamming hash family has inherent capacity to address classification task. Thisbasic framework of linear non-uniform ECA is extended to GMACA (generalized MACA)and utilized for modelling the associative memory, see Maji et al (2003a); Maji (2005). InMaji and Chaudhuri (2004), fuzzy CAs have been explored as an efficient pattern classifier.Non-linear non-uniform ECAs based pattern classifiers have further been developed by Daset al (2009). Recently, an asynchronous CA based pattern classifier is reported by Sethi et al(2016).

7.2.3 Compression and others:

In Bhattacharjee et al (1995), some methods are proposed to perform text compression usingCA as a technology. In Lafe (1997), cellular automata transforms are proposed for digitalimage compression and data encryption. It is shown that, CAs can generate orthogonal,semi-orthogonal, bi-orthogonal and non-orthogonal bases. Non-uniform ECA based trans-forms have been presented by Paul et al (2000); Shaw et al (2004b, 2006) for developing ef-ficient schemes of image and document compression. Some other related works are reportedin Lafe (2002); Ye and Li (2008). A technology, called Encompression, where encryptionand compression are married, has been reported in Shaw et al (2003, 2004a).

7.3 Medical science

Since the seminal work of von Neumann in 1950, CA has attracted attention of computerscientists and biologists as an excellent tool to model self-replicating system. The reasonof choosing CA for biological modelling is – (1) it is fast and easily implementable and(2) the visual result of the simulation provides remarkable resemblance with the originalexperiment. In Burks and Farmer (1984), the research on modelling the evolution and role ofDNA sequences within the framework of CA has been initiated. Ermentrout and Edelstein-Keshet (1993) have reviewed a number of biologically motivated CAs, such as deterministicor Eurelian automata, lattice gas models and solidification models, heuristic CA etc. andshown the effectiveness of CAs in understanding a physical process. In Mitra et al (1996),a pioneering work on modelling amino acid using GF(22) CA, called as amino acid CA


(AACA), has been reported. Some other interesting works are recorded in dos Santos andCoutinho (2001); Moreira and Deutsch (2002); Xiao et al (2006); Santos et al (2013).

CpG island detection in DNA sequence is a known problem in biological sequence anal-ysis. Ghosh et al (2007) has addressed this problem. In Ghosh et al (2010), a noteworthyclass of non-uniform ECAs, known as Equal Length Cycle CAs, has been proposed to pre-dict and classify the enzymes. In Ghosh et al (2012), another notable class of CAs, termedas restricted 5-neighborhood CA (R5NCA) has been introduced to predict protein structure,and to develop a protein modelling CA machine (PCAM). Here, a R5NCA rule is used tomodel an amino acid of a protein chain. This PCAM has been used for synthesis of proteinstructure using an organized knowledge base, see Ghosh et al (2014).

8 Conclusion

Cellular automaton has gone through a long journey from the early period of von Neumannto elementary form of Wolfram, and to the modern trends of research using this simple yetbeautiful model. In this survey, various milestones of development regarding CAs are brieflydepicted, such as the variety of types, different characterization tools, global behavior andnon-uniformity.

During this journey of CAs, different types of automata have been developed by varyingthe parameters of CAs. Based on neighborhood dependencies of cells, various types of CAshave been developed. Similarly, varying dimension of cellular space, states per cell, localrules, we get various types of CAs.

In this survey we have discussed about characterization tools, such as de Bruijn graph,matrix algebra, reachability tree, etc, which have been used to understand the behavior ofCAs. The global behavior of CAs, which are results of simple local interactions, are veryinteresting. We have surveyed some of the behavior, such as universality, reversibility, num-ber conservation, computational ability, etc. Still, there are few more behavior of CAs, suchas language recognition, which are not covered in this survey.

Then, we have focused our discussion on the non-classical CAs - asynchronous CAs, au-tomata networks, and non-uniform CAs. We have noted that, in recent past, a good attentionhave been put on these non-classical CAs. Among them, the most explored non-classicalCAs are (finite) non-uniform ECAs, where the local rules are always ECAs rules. We havededicated a section for these CAs.

Lastly, we have discussed about the potential of CAs as technology. CAs have beenused as solutions to many real-life problems. In fact, in this era of VLSI, the CAs, whichcan efficiently be implemented in hardware, can take a lead role to solve many real-lifeproblems.

This survey is a brief history of cellular automata. We have tried to arrange a tour todifferent corners of the developments. As a result, many concepts are stated informally,and the writing style remains primarily narrative. However, a good number of referenceshave been added in support of the presented concepts. Interested readers may go to thesereferences to explore their interests.

Acknowledgements The authors gratefully acknowledge the anonymous reviewers for their comments andsuggestions, which have helped to improve the quality and readability of the paper.


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