self-replicating patterns in 2d linear cellular automata

January 29, 2014 6:59 WSPC/S0218-1274 1430002

International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1430002 (24 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021812741430002X

Self-Replicating Patterns in 2D LinearCellular Automata

Selman Uguz∗Department of Mathematics, Arts and Science Faculty,

Harran University, Sanliurfa 63120, [email protected]

[email protected]

Ugur SahinRochester Institute of Technology,

Multi Agent Biorobotic Laboratory, Rochester, NY, [email protected]

Hasan AkinDepartment of Mathematics, Faculty of Education,

Zirve University, Gaziantep 27260, [email protected]

Irfan SiapDepartment of Mathematics, Arts and Science Faculty,

Yıldız Technical University, Istanbul 34210, [email protected]

Received March 27, 2013; Revised July 23, 2013

This paper studies the theoretical aspects of two-dimensional cellular automata (CAs), it classi-fies this family into subfamilies with respect to their visual behavior and presents an applicationto pseudo random number generation by hybridization of these subfamilies. Even though thebasic construction of a cellular automaton is a discrete model, its macroscopic behavior at largeevolution times and on large spatial scales can be a close approximation to a continuous system.Beyond some statistical properties, we consider geometrical and visual aspects of patterns gen-erated by CA evolution. The present work focuses on the theory of two-dimensional CA withrespect to uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) condi-tions. In total, there are 512 linear rules over the binary field Z2 for each boundary conditionand the effects of these CA are studied on applications of image processing for self-replicatingpatterns. After establishing the representation matrices of 2D CA, these linear CA rules are clas-sified into groups of nine and eight types according to their boundary conditions and the numberof neighboring cells influencing the cells under consideration. All linear rules have been found tobe rendering multiple self-replicating copies of a given image depending on these types. Multiplecopies of any arbitrary image corresponding to CA find innumerable applications in real lifesituation, e.g. textile design, DNA genetics research, statistical physics, molecular self-assemblyand artificial life, etc. We conclude by presenting a successful application for generating pseudonumbers to be used in cryptography by hybridization of these 2D CA subfamilies.

Keywords : Cellular automata; representation matrix; self-replicating patterns; image processing;cryptography by hybridization.

∗Author for correspondence

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1. Introduction

Cellular automata (CA for brevity) are discretedynamical systems and their behavior is completelyspecified in terms of local rules. CAs introduced byUlam and von Neumann [von Neumann, 1966] inthe early 1950’s, have been studied by Hedlund froma mathematical perspective [Hedlund, 1969]. Sincethen many researchers have taken interest in thestudy of CA. One-dimensional CA has been inves-tigated to a large extent. However, little interest hasbeen given to two-dimensional cellular automata(2D CA). von Neumann [1966] showed that a cel-lular automaton could be universal, a CA wasorganized, each cell of which had a state space of29 states, and it was obtanied that the organizedCA could operate any computable operation. Mean-while, because of its complexity, von Neumann ruleswere never operated on a computer. In the earlyeighties, Wolfram [1983] studied in detail a fam-ily of one-dimensional (1D) CA rules and showedthat even some simplest rules are capable of com-plex behavior. He studied also 1D CA with thehelp of polynomial algebra. Inokuchi & Sato [2000]investigated the behaviors of 1D CA generated bythe local rule 156. Das [1990] studied the char-acterization of 1D CA by making use of linearalgebra. Recently, 2D CA has attracted much inter-est. Some basic and precise mathematical modelsusing matrix algebra over the binary field that char-acterize the behavior of 2D nearest neighborhoodlinear CA with null and periodic boundary condi-tions have been seen in the literature [Choudhuryet al., 2005; Das, 1990; Dihidar & Choudhury, 2004;Khan et al., 1997, 1999; Packard & Wolfram, 1985].Khan et al. [1997] obtained an analytical approachto study all the possible nearest neighborhood 2DCA linear transformations. They proposed a newrule convention by dividing the 2D linear CA andstudied the characterization of that 2D CA for dif-ferent rules. von Neumann started and sketched apath that begins with self-replicating CAs and endswith self-replicating physical machines. Packardand Wolfram [Packard & Wolfram, 1985; Wolfram,1983] made much progress within self-replicatingphysics of CA. Some other studies on CA up tonow are listed in the references. Everyone agreeswith von Neumann that at some point along thispath, it is necessary to move away from discretespace models to continuous space models.

Characterization and applications of some spe-cific uniform and hybrid 2D CA linear rules are

reported in the literature [Siap et al., 2011a; Uguzet al., 2013a; Ying et al., 2009]. Due to differ-ent important applications of CA in many researchareas (e.g. mathematics, computer science, physics,chemistry, cryptography, etc) with many differentpurposes (e.g. simulation of some natural phe-nomena, pseudo random number generation model,image processing), CA has been of important atten-tion in the last few decades [Adamatzky et al.,2006; Akın, 2005; Akın & Siap, 2007; Blackburnet al., 1997; Chua & Yang, 1988; Dogaru & Chua,1999, 2000; Das, 1990; Dihidar & Choudhury, 2004;Khan et al., 1997, 1999]. Due to its structure,CA can be modeled to understand many behav-iors in nature easily. Most of the studies in CAinvestigates the one-dimensional case [Inokuchi &Sato, 2000]. Note that two-dimensional (2D) CA hasfound applications in traffic modeling. For instancemultivalue (including ternary) cellular automatonmodels for traffic flow are proposed in [Nishi-nari & Takahashi, 2000]. CA have found appli-cations in cryptography [Blackburn et al., 1997],recently multistate CA have also found applica-tions in cryptography [Mihaljevic et al., 1999] andespecially two-dimensional CA has been proposedfor multisecret sharing scheme for colored images[Alvarez et al., 2008]. The set of papers [Chattopad-hyay et al., 1999; Dihidar & Choudhury, 2004; Khanet al., 1997, 1999] deals with the behavior of the uni-form 2D CA over binary fields. In [Julian & Chua,2002], the replication properties of additive CA areanalyzed to extend the existing results.

Recently the studies related to the emergence ofself-replicating structures in 1D and 2D CA spaceappeared in [Bilotta et al., 2005; Chou & Reggia,1997; Gravner & Griffeath, 2011; Mitra & Kumar,2005; Chua & Mainzer, 2011; Reggia et al., 1998].For example, Gravner et al. [2011] studied the the-ory and application of additive Rule 22 1D CellularAutomata to explain the reason for frequent replica-tions and present a method for collecting the small-est periodic seeds. Further, in [Mitra & Kumar,2005], the fractal replication in time-manipulated1D CAs was investigated to exhibit complex fractalreplication behaviors.

In this paper, we study the theory of two-di-mensional uniform periodic, adiabatic and reflexiveboundary CA (2D PCA, ACA, RCA) of all linearrules (e.g. von Neumann, Moore neighborhood andthe others) and applications of image process-ing for self-replicating patterns (see Figs. 3–18).

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Self-Replicating Patterns in 2D Linear Cellular Automata

In other words, we are interested in cases when aCA replicates under some special number of evolu-tion (i.e. iteration number), that is, makes copiesof a finite collection of finite configurations, calledself-replicating patterns. Our aim is to investigateself-replicating patterns for some special bound-aries with their effects. We investigate their clas-sification under periodic, adiabatic and reflexiveboundary conditions over the binary field Z2. Weobtain the rule matrix (or representation matrix)of 2D finite CA for all rules and boundary condi-tions. Further, applications of image processing asself-replicating patterns corresponding to the lin-ear rules of 2D uniform CA with periodic, adia-batic and reflexive boundary conditions over Z2 arealso studied. We present some illustrative exam-ples and figures. Using the rule matrices obtainedin this work, the present paper contributes fur-ther to the algebraic structure of these CA andrelates the results to applications studied by dif-ferent authors previously (i.e. [Choudhury et al.,2010; Packard & Wolfram, 1985; Wolfram, 1983]).The paper introduces the 2D nine-neighborhoodCA and the rules by which the dependencies of acell are governed. The application of such linearrules on designed matrices is demonstrated whichforms the basis of self-replicating image processing.The paper also studies the effect of these 512 linearrules on a particular image. Especially, these rulesare used for image transformations as translation,multiplication of one image into several replicatingimages. While the translation and multiplicationcan be carried out on any arbitrary image, struc-tured images are needed for others. Some rules yieldweird and wonderful patterns as seen in the nextsections.

The self-replicating implications of von Neu-mann for research in nanotechnology, theoreticalbiology and artificial life were discussed in [Smithet al., 2003]. Self-replication is the process inwhich an object or structure produces a copy ofitself. Of course there is some difference betweenself-replication and self-reproduction. In the self-replication side, an exact duplicate is made (seeFigs. 2–12). Since multiple copies of an arbitraryimage have innumerable applications in real life sit-uations (for example, textile design, DNA geneticsresearch, etc., see [Chua & Mainzer, 2011; Dogaru &Chua, 1999, 2000; Schadschneider & Schreckenberg,1993; Rubio et al., 2004; Smith et al., 2003]), thefindings of this study are interesting.

The organization of the paper is presented asfollows. Section 2 discusses the technical preliminar-ies of the subject and their details. A mathemati-cal model for CA with the earlier works is given inSec. 3. In Sec. 4, the rule matrices of 2D finite peri-odic CA with linear rules and their properties arepresented. The application of 2D CA in image pro-cessing for the self-replicating patterns are given inSec. 5. The classifications are presented in Sec. 6.In Sec. 7, an application to cryptography obtainedby the hybridization of these families of CA is pre-sented. Finally a conclusion is drawn in Sec. 8.

2. Technical Preliminaries

In this section, we introduce 2D CA over the binaryfield Z2 = 0, 1 by using the uniform linear localrules. We recall the definition of a CA. Further, weconsider the two-dimensional lattice Z

2 and the con-figuration space Ω = 0, 1Z

2with elements

σ : Z2 → 0, 1.

The value of σ at a point v ∈ Z2 will be denoted

by σv. Let u1, . . . , us ∈ Z2 be a finite set of distinct

vectors and f : 0, 1s → 0, 1, 2 be a function.CA with local rule f is defined as a pair (Ω, Ff ),

where the global transition map Ff : Ω → Ω isgiven by

(Ffσ)v = f(σv+u1 , . . . , σv+us), v ∈ Z2.

The function f is called the local rule. The spaceΩ is assumed to be equipped with a (metrizable)Tychonoff topology; it is easily seen that the globaltransition map Ff introduced above and the shiftoperator Uv are continuous in this topology. The2D finite CA consists of m × n cells arranged in mrows and n columns, where each cell takes one ofthe values of 0 or 1. A configuration of the systemis an assignment of states to all the cells. Everyconfiguration determines a next configuration viaa linear transition rule that is local in the sensethat the state of a cell at time (t + 1) dependsonly on the states of some of its neighbors at time tusing modulo 2. For 2D CA nearest neighbors, there

Table 1. Local rule number conventionsfor 2D finite CA.

26 = 64 27 = 128 28 = 256

25 = 32 20 = 1 21 = 2

24 = 16 23 = 8 22 = 4

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Fig. 1. The nearest neighborhood comprises eight cellswhich surround center xij with the coefficients ci ∈ Z2 fori = 1, 2, 4, . . . , 128. The values of ci will be restricted to thecase of being zero or nonzero (i.e. ∈ Z2). According to thisassignment to the coefficients ci in Eq. (1) the rule numberwill be defined. The linear combination of the neighboringcells on which each cell value is dependent is called the rulenumber of the 2D CA over Z2.

are nine cells arranged in a 3 × 3 matrix center-ing that particular cell (see [Chattopadhyay et al.,1999; Das, 1990; Dihidar & Choudhury, 2004] forthe details). For 2D CA there are many types ofneighborhoods, but in this work we restrict our-selves to all linear rules from Rule 1 to Rule 512.So, we can define the (t + 1)th state of the (i, j)thcell as follows;

x(t+1)(i,j) = c1x

(t)(i,j) + c2x

(t)(i,j+1) + c4x

(t)(i+1,j+1)

+ c8x(t)(i+1,j) + c16x

(t)(i+1,j−1) + c32x

(t)(i,j−1)

+ c64x(t)(i−1,j−1) + c128x

(t)(i−1,j)

+ c256x(t)(i−1,j+1) (mod 2), (1)

where the values of ci will be restricted to the caseof being zero or nonzero i.e. binary. According tothis assignment to the coefficients ci in equation (1)the rule number will be defined. The linear combi-nation of the neighboring cells on which each cellvalue is dependent is called the rule number of the2D CA over the field Z2. Regarding the neighbor-hood of the extreme cells, there exist four differentapproaches.

Table 2. 2D finite CA3×3 with the centerelement x(i,j).

x(i−1,j−1) x(i−1,j) x(i−1,j+1)

x(i,j−1) x(i,j) x(i,j+1)

x(i+1,j−1) x(i+1,j) x(i+1,j+1)

• A null boundary (NB) CA is where the extremecells are connected to 0-state.

• A periodic boundary (PB) CA is where theextreme cells are adjacent to each other (seeTable 3 or proof of Lemma 2).

• A adiabatic boundary (AB) CA duplicates thevalue of the cell in an extra virtual neighbor (seeTable 4).

• A reflexive boundary (RB) CA is designed suchthat the value of left and right neighbors are thesame with respect to the boundary cell.

The 2D infinite CA generated by the local rule Rulewith periodic boundary is defined as

FRule=PB : ZZ

2

2 → ZZ

2

2 where

(FRule=PBx)(t)(i,j)

= c1x(t)(i,j) + c2x

(t)(i,j+1) + c4x

(t)(i+1,j+1)

+ c8x(t)(i+1,j) + c16x

(t)(i+1,j−1) + c32x

(t)(i,j−1)

+ c64x(t)(i−1,j−1) + c128x

(t)(i−1,j)

+ c256x(t)(i−1,j+1) (mod 2)

= x(t+1)(i,j)

. (2)

In this paper, FPB denotes a 2D finite CA gen-erated by the rule with periodic boundary (PB)(as above), similarly FAB with adiabatic boundary(AB) and FRB with reflexive boundary (RB). It iswell known that these CA are discrete dynamicalsystems formed by a finite two-dimensional arraym × n composed by identical objects called cells.Let

I : Mm×n(Z2) → Zmn2 .

I takes the tth state [Xt] given by

x11 x12 · · · x1n

x21 x22 · · · x2n

...... · · · ...

xm1 xm2 · · · xmn

→ (x11, x12, . . . , x1n, . . . , xm1, . . . , xmn)T . (3)

Therefore, the local rules will be assumed toact on Z

mn2 rather than Mm×n(Z2). The binary

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information matrix of order m × n denoted by

C(t) =

x(t)11 . . . x

(t)1n

.... . .

...

x(t)m1 . . . x

(t)mn

is called the configuration of the 2D finite CA ata particular time t. From (3), we can define asfollows,

(TRule=PB )mn×mn.

x(t)11

...

x(t)1n

...

x(t)m1

...

x(t)mn

=

x(t+1)11

...

x(t+1)1n

...

x(t+1)m1

...

x(t+1)mn

where

x(t+1)ij = c1x

(t)i(j) + c32x

(t)i(j−1) + c2x

(t)i(j+1)

+ c4x(t)i+1(j+1) + c8x

(t)(i+1)j + c16x

(t)(i+1)j−1

+ c64x(t)(i−1)j−1 + c128x

(t)(i−1)j

+ c256x(t)(i−1)j+1 (mod 2), (4)

where i denotes the ith row of the matrix C(t) andj denotes the jth column of the matrix C(t). Hence,the action on configuration sets is given as

(TRule)mn×mn · [Xt] = [Xt+1].

The matrix (TRule)mn×mn is called the rule matrixwith respect to the 2D finite CAm×n. The conven-tional method of 2D CA with periodic boundarycan be explained in Table 3.

Table 3. Periodic boundary condition in a 2D finite CA3×3.

x(i+1,j−1) x(i−1,j−1) x(i,j−1) x(i+1,j−1) x(i−1,j−1)

x(i+1,j+1) x(i−1,j+1) x(i,j+1) x(i+1,j+1) x(i−1,j+1)

x(i+1,j) x(i−1,j) x(i,j) x(i+1,j) x(i−1,j)

x(i+1,j−1) x(i−1,j−1) x(i,j−1) x(i+1,j−1) x(i−1,j−1)

x(i+1,j+1) x(i−1,j+1) x(i,j+1) x(i+1,j+1) x(i−1,j+1)

3. Mathematical Model for CellularAutomata Rules

CAs are nowadays useful mathematical models forsystems to obtain many different complicated pat-terns of behavior. The extension from one dimen-sion (1D) to two dimension (2D) is significant forcomparisons with many experimental results onpattern formation in physical and many differentsystems. In the literature, there are several possi-ble lattices and neighborhood structures for two-dimensional CAs, i.e. triangular, square, hexagonal,etc. [Packard & Wolfram, 1985; Siap et al., 2011b;Uguz et al., 2013b]. This paper considers primar-ily square lattice, with its neighborhood structuresillustrated in Fig. 1. We consider possible near-est neighborhood structures to construct the rulematrices over two-dimensional CAs. In the cellu-lar automaton iteration, the value of the centercell is updated according to a rule which dependson the values of the possible neighborhood cells.It is seen that despite the simplicity of their con-struction, CAs have very complicated behaviors (seeFigs. 2–18). The application of linear rules men-tioned in the previous section can be realized on aconfiguration matrix, where every entry is either 0or 1 (for the case Z2). It may be mentioned thatinstead of applying the same rule to each entry ofthe configuration matrix, it is admissible to applydifferent rules to different entries at the same time.While the former characterizes the uniform CA,the latter characterizes hybrid CA (an applicationof this type is given in Sec. 7). An illustrativeexample is presented for uniform null boundaryvon Neumann CA corresponding to Rule 170NBbelow.

Example 3.1. We present a 2D CA with configura-tion of size 3×3 under the null boundary conditionto illustrate the theory. By using the linearity ofrule,

Rule 170NB = Rule(2 + 8 + 32 + 128)NB

= Rule 2NB + Rule 8NB

+ Rule 32NB + Rule 128NB

is applied uniformly to each cell of a problemmatrix of order (3 × 3) with null boundary condi-tion (extreme cells are connected with 0-states). Ifwe take m = 3 and n = 3, then we consider a config-uration of size 3 × 3 with null boundary condition.Now, we apply the Rule 170NB, and we obtain the

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rule matrix T170NB as

T170NB =

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

=

S3 I3 03

I3 S3 I3

03 I3 S3

9×9

where each submatrix is of order 3 × 3, and

S3×3 =

0 1 01 0 10 1 0

.

The evolution of 2D CAs from seeds calledas a first image in Figs. 4–20, consist of a fewnonzero initial sites. Similar cases in one dimen-sion, some CA give regular and self-similar patterns,others give complicated and random patterns (seeFigs. 14–17). Some new features in two dimensionsare the generation of patterns with different bound-aries observed in many natural systems. Detailedanalysis will be given in Sec. 5.

For simplicity, we consider only 2D CA lin-ear rules. The rule convention is organized as fol-lows. When we consider the nearest neighbors of2D CA, there are nine variables which are underconsideration as given in Fig. 2. The center cellmarked xij (see Fig. 1) is the cell under consid-eration. In 2D CA, the state of the cell which isunder consideration depends on its neighboring cellstates. Each of the cells could be considered as avariable and thus for 2D CA there are nine vari-ables to be studied. The number of linear rules canbe realized in the following way. The number ofrules combined by a combination of these nine vari-ables is

(90

)+

(91

)+

(92

)+ · · · +

(99

)= 512 which

includes rules characterizing no dependency. Notethat, these 512 linear rules were previously classi-fied by taking into account the number of cells underjust null boundary consideration [Choudhury et al.,2010] (see Table 5). The classification presented inTables 5 and 6 defines Type n rules. An important

note is that these Type n rules are not related tothe types given by Wolfram [1983].

Remark 3.1. In [Choudhury et al., 2010], these clas-sifications are referred to as Group n rules (we callType n rules just in case they are confused with thealgebraic group structure), the grouping has beenGroup n for n = 1, 2, . . . , 9, including the rules thatrefer to the dependency of current cell on N neigh-boring cells amongst top, bottom, left, right, top-left, top-right, bottom-left, bottom-right and itself(see Fig. 1).

Here we briefly mention how Type 1 rules dif-fer when the boundary conditions change. For thenull and periodic boundary case (see Fig. 5), Type 1includes 1, 2, 4, 8, 16, 32, 64, 128, and 256. Whereasfor the adiabatic and reflexive boundary case (seeFig. 6), Type 1 includes 1, 2, 3, 4, 5, 8, 9, 16,17, 32, 33, 64, 65, 128, 129, 256 and 257. Simi-larly rules belonging to other Type n have beenobtained. It is noted that number of 1’s presentedin the binary sequence of a rule is the same as itstype number. These Type n rules are obtained bythe number n of self-replicating number of the firstimage (see Figs. 2–12). These rules are classified inTables 5 and 6 with respect to the chosen boundaryconditions.

4. Rule Matrices of 2D FiniteCA Rules

In this section, we obtain the rule matrix corre-sponding to a 2D finite CA with the periodic bound-ary generated by the local rules over the field Z2

by presenting the auxiliary matrices that play animportant role in the rule matrix of a 2D cellularautomaton. Further, this will be established for eachboundary case.

The rule matrices of rule number R are denotedby TR. We study the rule matrix TR such that TR

operates on the current CA states [Xt] (the binarymatrix of dimension (m×n)) and generates the nextstate [Xt+1] of (m × n).

4.1. Rule matrices with primaryrules under null boundarycondition

The auxiliary matrices T1 and T2 for the null bound-ary are defined as follows:

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T1 =

0 1 0 0 .

0 0 1 0 .

. . . . .

. . . . 10 0 0 0 0

and T2 =

0 0 0 . 01 0 . . .

0 1 0 . .

. . . . .

0 0 . 1 0

.

Lemma 1 [Choudhury et al., 2005]. The representation of the next state of all primary rules(1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 and 256 ) under the null boundary condition can be given by using the auxil-iary matrices T1 and T2 defined above in the following way:

Rule 1N : [Xt+1] = [Xt] Rule 2N : [Xt+1] = [Xt][T2]

Rule 4N : [Xt+1] = [T1][Xt][T2] Rule 8N : [Xt+1] = [T1][Xt]

Rule 16N : [Xt+1] = [T1][Xt][T1] Rule 32N : [Xt+1] = [Xt][T1]

Rule 64N : [Xt+1] = [T2][Xt][T1] Rule 128N : [Xt+1] = [T2][Xt]

Rule 256N : [Xt+1] = [T2][Xt][T2].

4.2. Rule matrices with primary rules under periodic boundary condition

The auxiliary matrices T1p and T2p for periodic boundary case are defined as follows:

T1p =

0 1 0 0 .

0 0 1 0 .

. . . . .

. . . . 11 0 0 0 0

and T2p =

0 0 . 0 11 0 . . .

0 1 0 . .

. . . . .

0 0 . 1 0

. (5)

Proposition 1. The fundamental periodic number rule matrices T1PR, T2PR, T4PR, T8PR, T16PR, T32PR ,T128PR , T256PR are respectively given by

T1PR =

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

. 0 0 0 In

T2PR =

T1p 0 0 0 .

0 T1p 0 0 .

. . . . .

. 0 0 T1p 0

. 0 0 0 T1p

T32PR =

T2p 0 0 0 .

0 T2p 0 0 .

. . . . .

. 0 0 T2p 0

. 0 0 0 T2p

T4PR =

0 T1p 0 0 .

0 0 T1p 0 .

. . . . .

0 0 0 0 T1p

T1p 0 0 0 .

T64PR =

0 0 0 0 T2p

T2p 0 0 0 .

0 T2p 0 0 .

. . . . .

. 0 0 T2p 0

T8PR =

0 In 0 0 .

0 0 In 0 .

. . . . .

0 0 0 0 In

In 0 0 0 .

T128PR =

0 0 0 0 In

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

T16PR =

0 T2p 0 0 .

0 0 T2p 0 .

. . . . .

0 0 0 0 T2p

T2p 0 0 0 .

T256PR =

0 0 0 0 T1p

T1p 0 0 0 .

0 T1p 0 0 .

. . . . .

. 0 0 T1p 0

where T1p and T2p are given as in (5). Also In is an identity matrix n × n type.

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Proof. In order to determine the rule matrix(TPR)mn×mn we need to determine the action of(TPR)mn×mn on the base vectors. First, we considerthe linear transformation (T) from m × n matrixspace to itself. Next, we relate the transformation(T) with (TPR). Let eij denote the matrix of sizem × n where the (i, j) position is equal to one andthe rest of the entries equal to zero. It is well knownthat these vectors give the standard basis for thisspace. Given eij , the image of eij which is T (eij)is related to the suitable nearest neighbors. ThenT (eij) equals to a linear combination of its suit-able neighbors corresponding to the rules. Hence,without getting into details, we obtain the rulematrices.

From Proposition 1, one can easily see the fol-lowing result which reduces the number of indepen-dent primary rules from eight to four. The followingis due to the matrix transpose equality (T1p)t = T2p.

Corollary 4.1. The following matrix transposeidentitites are satisfied:

(T2PR)t = T32PR, (T16PR)t = T256PR,

(T8PR)t = T128PR, (T4PR)t = T64PR.

Hence we get the following general rule matrixresult for the periodic case as a theorem.

Theorem 1 [Periodic Case]. The matrix for anyperiodic boundary CA rule (PB) can be representedas

[TPR]mn×mn

=

Ap Bp 0 0 . 0 Dp

Cp Ap Bp 0 . . 0

0 Cp Ap Bp 0 . .

. . . . . . .

. . 0 Cp Ap Bp 0

0 . . 0 Cp Ap Bp

Ep 0 . . 0 Cp Ap

where Ap, Bp, Cp, Dp, Ep are one of the follow-ing matrices of the order of n × n : 0, I, T1p, T2p,I + T1p, I + T2p, T1p + T2p and I + T1p + T2p.

Proof. By applying Proposition 1 and making useof linearity, the result is obtained. Note that inmatrix representation, T1p T2p, I = In×n = identityand 0 = 0n×n zero matrix are all of type n×n.

Lemma 2. The next state of all primary rules (1,2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ) of 2D periodic cellularautomaton over Z2 can be represented as follows:

Rule 1P : [Xt+1] = [Xt]

Rule 2P : [Xt+1] = [Xt][T2p]

Rule 4P : [Xt+1] = [T1p][Xt][T2p]

Rule 8P : [Xt+1] = [T1p][Xt]

Rule 16P : [Xt+1] = [T1p][Xt][T1p]

Rule 32P : [Xt+1] = [Xt][T1p]

Rule 64P : [Xt+1] = [T2p][Xt][T1p]

Rule 128P : [Xt+1] = [T2p][Xt]

Rule 256P : [Xt+1] = [T2p][Xt][T2p].

Proof. Let

[Xt] =

x11 x12 x13

x21 x22 x23

x31 x32 x33

(where x11, x12, . . . , x33 ∈ Z2) be the state of CA attime t. Let RP ([Xt]) denote the next state of ([Xt])with respect to the rule number. To obtain the stateof CA at time (t + 1) Rule1: [Xt+1] = [Xt], we willdo the following operations: we get the rule matri-ces TPR of designed orders. The periodic boundarycase of state [Xt] can be taken as

x33 x31 x32 x33 x31

x13 x11 x12 x13 x11

x23 x21 x22 x23 x21

x33 x31 x32 x33 x31

x13 x11 x12 x13 x11

.

We can show one of the states obtained explicitlyand the others follow in a similar way. Let us con-sider Rule 4P.

R4P ([Xt]) =

x22 x23 x21

x32 x33 x31

x12 x13 x11

=

0 1 00 0 11 0 0

x11 x12 x13

x21 x22 x23

x31 x32 x33

×

0 0 11 0 00 1 0

.

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Corollary 4.2. Rank of the periodic auxiliarymatrices [T1p]n×n and [T2p]n×n is n. In other words,the periodic auxiliary matrices are invertible.

Example 4.1 [von Neumann Neighborhood]. SinceRule 170P = Rule 2P + Rule 8P + Rule 32P +Rule 128P, we have

[Xt+1] = [Xt][T2p] + [T1p][Xt] + [Xt][T1p]

+ [T2p][Xt].

Table 4. 2D finite CA2×2 withadiabatic boundary condition.

x11 x11 x12 x12

x11 x11 x12 x12

x21 x21 x22 x22

x21 x21 x22 x22

Then we obtain the following description:

T170P ([Xt]) = T170P

x11 x12 x13

x21 x22 x23

x31 x32 x33

=

x12 + x21 + x13 + x31 x11 + x13 + x22 + x32 x12 + x23 + x11 + x33

x11 + x22 + x31 + x23 x12 + x23 + x32 + dx21 x13 + x33 + x22 + x21

x21 + x32 + x11 + x33 x22 + x33 + x31 + x12 x23 + x32 + x31 + x13

=

x31 x32 x33

x11 x12 x13

x21 x22 x23

+

x12 x13 x11

x22 x23 x21

x32 x33 x31

+

x21 x22 x23

x31 x32 x33

x11 x12 x13

+

x13 x11 x12

x23 x21 x22

x33 x31 x32

= [Xt][T2p] + [T1p][Xt] + [Xt][T1p] + [T2p][Xt].

4.3. Rule matrices with primary rules under adiabatic boundary condition

The auxiliary matrices T1a and T2a for the adiabatic boundary case are defined as follows:

T1a =

0 1 0 0 .

0 0 1 0 .

. . . . .

. . . . 10 0 0 0 1

and T2a =

1 0 0 . 01 0 . . .

0 1 0 . .

. . . . .

0 0 . 1 0

. (6)

Proposition 2. The fundamental adiabatic boundary (AB), rule number (n), rule matrices TnAB : T1AB ,T2AB , T4AB , T8AB , T16AB , T32AB , T128AB , T256AB are

T1AB =

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

. 0 0 0 In

T2AB =

T1a 0 0 0 .

0 T1a 0 0 .

. . . . .

. 0 0 T1a 0

. 0 0 0 T1a

T32AB =

T2a 0 0 0 .

0 T2a 0 0 .

. . . . .

. 0 0 T2a 0

. 0 0 0 T2a

T4AB =

0 T1a 0 0 .

0 0 T1a 0 .

. . . . .

0 0 0 0 T1a

0 0 0 0 T1a

T64AB =

T2a 0 0 0 0

T2a 0 0 0 .

0 T2a 0 0 .

. . . . .

. 0 0 T2a 0

T8AB =

0 In 0 0 .

0 0 In 0 .

. . . . .

0 0 0 0 In

0 0 0 0 In

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T128AB =

In 0 0 0 0

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

T16AB =

0 T2a 0 0 .

0 0 T2a 0 .

. . . . .

0 0 0 0 T2a

0 0 0 0 T2a

T256AB =

T1a 0 0 0 0

T1a 0 0 0 .

0 T1a 0 0 .

. . . . .

. 0 0 T1a 0

where T1a and T2a are given as in (6). Also In is the identity matrix n × n type.

Proof. In order to determine the rule matrix (TAB)mn×mn we need to determine the action of (TAB)mn×mn

on the base vectors. The rule matrix related to these equations is obtained after similar calculations as inthe proof of Proposition 1.

Hence we get the following general rule matrix result for the adiabatic case as a theorem.

Theorem 2 [Adiabatic Case]. The rule matrix for any adiabatic boundary CA rule (AB) can be representedas

[TAB ]mn×mn =

Aa Ba 0 0 . 0 0

Ca Aa Ba 0 . . 0

0 Ca Aa Ba 0 . .

. . . . . . .

. . 0 Ca Aa Ba 0

0 . . 0 Ca Aa Ba

0 0 . . 0 Ca Aa

where Aa, Ba, Ca are one of the following matrices of the order of n × n : 0, I, T1a, T2a, I + T1a, I + T2a,T1a + T2a and I + T1a + T2a.

Proof. The desired result follows by applying Proposition 2.

Lemma 3. The next state of all primary rules (1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ) of 2D adiabatic cellularautomaton with Z2 can be represented as follows:

Rule 1AB: [Xt+1] = [Xt] Rule 2AB: [Xt+1] = [Xt][T1a]t

Rule 4AB: [Xt+1] = [T1a][Xt][T1a]t Rule 8AB: [Xt+1] = [T1a][Xt]

Rule 16AB: [Xt+1] = [T1a][Xt][T2a]t Rule 32AB: [Xt+1] = [Xt][T2a]t

Rule 64AB: [Xt+1] = [T2a][Xt][T2a]t Rule 128AB: [Xt+1] = [T2a][Xt]

Rule 256AB: [Xt+1] = [T2a][Xt][T1a]t.

It is easily seen that the adiabatic auxiliary matrices are not invertible.

Corollary 4.3. The rank of the adiabatic auxiliary matrices [T1a]n×n and [T2a]n×n is (n − 1).

4.4. Rule matrices with primary rules under reflexive boundary condition

The auxiliary matrices T1r and T2r for the reflexive boundary case are defined as follows:

T1r =

0 1 0 0 .

0 0 1 0 .

. . . . .

. . . . 10 0 0 1 0

and T2r =

0 1 0 . 01 0 . . .

0 1 0 . .

. . . . .

0 0 . 1 0

. (7)

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Proposition 3. The fundamental reflexive boundary (RB), rule number (n), rule matrices TnRB : T1RB ,T2RB , T4RB , T8RB , T16RB , T32RB , T128RB , T256RB are

T1RB =

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

. 0 0 0 In

T2RB =

T1r 0 0 0 .

0 T1r 0 0 .

. . . . .

. 0 0 T1r 0

. 0 0 0 T1r

T32AB =

T2r 0 0 0 .

0 T2r 0 0 .

. . . . .

. 0 0 T2r 0

. 0 0 0 T2r

T4RB =

0 T1r 0 0 .

0 0 T1r 0 .

. . . . .

0 0 0 0 T1r

0 0 0 T1r 0

T64RB =

0 T2r 0 0 0

T2r 0 0 0 .

0 T2r 0 0 .

. . . . .

. 0 0 T2r 0

T8RB =

0 In 0 0 .

0 0 In 0 .

. . . . .

0 0 0 0 In

0 0 0 In 0

T128RB =

0 In 0 0 0

In 0 0 0 .

0 In 0 0 .

. . . . .

. 0 0 In 0

T16RB =

0 T2r 0 0 .

0 0 T2r 0 .

. . . . .

0 0 0 0 T2r

0 0 0 T2r 0

T256RB =

0 T1r 0 0 0

T1r 0 0 0 .

0 T1r 0 0 .

. . . . .

. 0 0 T1r 0

where T1r and T2r are given as in (7).

Proof. The rule matrix related to these equationscan be found by using the base vectors.

Hence we get the following general rule matrixresult for the reflexive case as a theorem.

Theorem 3 [Reflexive Case]. The rule matrix for areflexive boundary CA rule (RB) is represented as

[TRB ]mn×mn

=

Ar Br 0 0 . 0 0

Cr Ar Br 0 . . 0

0 Cr Ar Br 0 . .

. . . . . . .

. . 0 Cr Ar Br 0

0 . . 0 Cr Ar Br

0 0 . . 0 Cr Ar

where Ar, Br, Cr are one of the following matricesof the order of n×n: 0, I, T1r, T2r, I +T1r, I +T2r,T1r + T2r and I + T1r + T2r.

Proof. From Proposition 3 and linearity, the proofis obtained.

Lemma 4. The next state of all primary rules (1 ,2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 ) of 2D reflexive cellularautomaton over Z2 can be represented as follows:

Rule 1RB: [Xt+1] = [Xt]

Rule 2RB: [Xt+1] = [Xt][T1r]t

Rule 4RB: [Xt+1] = [T1r][Xt][T1r]t

Rule 8RB: [Xt+1] = [T1r][Xt]


Rule 32RB: [Xt+1] = [Xt][T2r]t


Rule 128RB: [Xt+1] = [T2r][Xt]

Rule 256RB: [Xt+1] = [T2r][Xt][T1r]t.

It is easily seen that the reflexive auxiliarymatrices are not of full rank, i.e. these are nonin-vertible matrices.

Corollary 4.4. The rank of the reflexive auxiliarymatrices [T1r]n×n and [T2r]n×n is (n − 1).

Here we briefly emphasize on the importanceof the reversibility property of linear CA. Thedimension of the kernel of the transition matrixof CA gives a clue to draw the state transition

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Fig. 2. An application of Rule 8 with null (NB), periodic (PB), adiabatic (AB) and reflexive (RB) boundary respectivelyafter 16 iterations of the first image.

Fig. 3. An application of Rule 72 after 16 iterations of the first image.




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Fig. 13. An application of Rule 315 after 5, 12, 15 and 16 iterations of the first image.

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diagram [Chattopadhyay et al., 1999] to determineits reversibility. Thus, in order to determine thedimension of the kernel of a 2D CA, one can studythe rank of (Trules)mn×mn. For finite CA, in orderto obtain the inverse of a 2D finite linear CA manyauthors [Khan et al., 1997, 1999; Siap et al., 2011b]have made use of the rule matrices. Since we havealready found the rule matrix Trules correspond-ing to 2D finite CA, we can state the followingrelation between the column vectors X(t) and therule matrix Trules: X(t+1) = TrulesX

(t) (mod 2). Ifthe rule matrix Trules is nonsingular, then we have

X(t) = (Trules)−1X(t+1) (mod 2).

Thus a main problem will be whether the rulematrix Trules is invertible or not. If the rule matrixTrules has full rank, then it is invertible, so the 2Dfinite CA is reversible, otherwise it is irreversible.Reversibility investigation on the rule matricesgiven in Theorems 1–3 is left for another work orfor the interested readers as open problems.

In the next section, by applying the rulematrices to the patterns, we generate the new pat-terns and classify CA linear rules based on thebehavior of the rule in the nth iteration and theirboundaries.

5. Application of 2D CA in ImageProcessing: Self-ReplicatingPatterns

Self-replicating pattern generation is one of themost interesting topics and research areas in non-linear science. A motif is considered as a basicsubpattern. Pattern generation is the process oftransforming copies of the motif about the array(1D), plane (2D) or space (3D) in order to cre-ate the whole repeating pattern with no overlapsand blanks [Gravner & Griffeath, 2011; Packard &Wolfram, 1985; von Neumann, 1966; Wolfram,1983]. These patterns have some mathematicalproperties which make generating algorithm possi-ble. A cellular automaton is a good candidate for

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an algorithmic approach used for pattern gener-ation. The history of self-replicating pattern pro-grams starts with John von Neumann who contriveda cellular automaton that took some input, and pro-duced as output that input [von Neumann, 1966].However if the automaton itself was given as input,the automaton will reproduce itself as output.von Neumann’s automaton program is an exam-ple of what is known as trivial self-replicating pro-gram because the structure to reproduce is encodeddirectly within the program or the input. Hence

this kind of trivial self-replicating procedure is eas-ily implemented in any programming language.

Creating an algorithmic approach for generat-ing self-replicating patterns of digital images (motifas in first image) is important and sometimesa difficult task. Meanwhile many researchers arefaced with many challenges in building and devel-oping tiling algorithms such as providing simpleand applicable algorithm to describe high com-plex patterns models. Growth from simple motifin 2D CAs can produce self-replicating patterns

Fig. 18. An application of nonsymmetric pattern for Rule 312 after 16 iterations of the first image.

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with complicated boundaries (null, periodic, adia-batic and reflexive), characterized by a variety ofgrowth dimensions. The approach given here leadsto an interesting algorithm for generating differentpatterns. In this paper we use the CAs with allthe nearest neighborhoods to generate self-replicatepatterns of digital images. For applying 2D null,periodic, adiabatic and reflexive CA linear rules inimage processing, we take a binary matrix of size(100 × 100) due to computational limitations. Wemap each element of the matrix to a unique pixelon the screen (writing new MATLAB codes) and wecolor a pixel white for 0, black for 1 for the matrixelements. Then we take another image (as a motif)whose size is less than (30 × 30) for which patternsare to be generated and put it in the center of thebinary matrix. This way the image is drawn withinan area of (100×100) pixels. To classify with respectto their boundary conditions, we add null, periodic,adiabatic and reflexive boundary conditions to theinput image. Then we apply 2D CA linear rules onthat (100 × 100) matrix and each time the rule isapplied using the changed matrix and a new imageis redrawn (Figs. 2–12). In our model, two states, i.e.black and white, are used to represent the states ofcells. Therefore the pattern is treated as the devel-oping and redrawing of black and white patterns.Then one can give different colors to these patternsusing different types of graphics packages. The rulesother than the fundamental rules generate differentpatterns of the given image. It is observed from thefigures that the self-replicating patterns can be gen-erated only when the number of repetitions is 2n

where (n = 4).

5.1. Self-replicating patterns: Typen and multiple copies of anyarbitrary image on null,periodic, adiabatic andreflexive 2D linear CA

It is observed from the corresponding figures thatthe rules other than the fundamental ones createmultiple copies of the given image, the number ofcopies being the same as the type number to whichthe applied rule belongs. Hence we obtain the max-imum number of copies an image can have on theapplication of such rules which is 9 for null andperiodic boundary case, because the maximum typenumber is 9 as found before in null case [Choudhuryet al., 2005, 2010].

Remark 5.1. In Figs. 13 and 14, one can see that theevolutions from steps 15 to 16 change drasticallyon the number of live cells. For visual inspection,the density population always seems to increasegradually but in step 16 many cells die quickly[Chopard & Droz, 1998; Reggia et al., 1998; Mar-tinez et al., 2011, 2012; von Neumann, 1966]. Suchevolutions become more important for some appli-cations areas of CA.

We present an illustrative example for the casen = 4, i.e. the iteration number 24 = 16. It may alsobe observed that the rules belong to the same typethough they create equal number of copies, the dis-tributions of these copies differ (see Figs. 6 and 7).For example (see Fig. 10), the Rule 415 is of Type 7for null and periodic boundary case but Type 6 foradiabatic and reflexive boundary case, that is, theseventh and sixth copies of that image are found inthe display matrix corresponding to the boundarycases. Such copies of the iterating image, may how-ever be formed within the display matrix providedthat the maximum length of the image consideredin all directions, does not exceed 30% of the lengthof row or column (since rule matrices are a squarematrix) of the display matrix (here it is 100). Moreresearch effort is required to explain why only t = 24

times repetition of the rules causes the appearanceof self-replicating patterns of any arbitrary image.

5.2. From chaotic behavior to orderor vice versa

Figures 13 and 14 present the case that help us todistinguish between the types and have this typedefined as the case from chaos to order. This can beseen very easily from these figures. When the iter-ation numbers increase from t = 1 to t = 15, thereis chaotic behavior of the images. When t = 16, akind of miracle happens, and then self-replicatingpatterns appear in the space diagrams. This repli-cation is valid when the first seed image decorateddoes not exceed 30% of the length of row or columnof the display matrix.

5.3. From order behavior to order

If we consider Figs. 15–17, the symmetric beautyorder is preserved when increasing the iterationnumber t. This is also valid for the nonsymmet-ric figure as given in Fig. 18. Moreover, there is noimportance for order behaviors of the seed image

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whether its size is larger than 30% of the length ofrow or column of the display matrix. Our examplesof programmatic CAs and repetition of simple rulescan create very nice complex patterns of archety-pal beauty (see Fig. 16). There is also an importantobservational result in Fig. 17. If the first imageexceeds 30% of the length of row or column of thedisplay matrix, the self-replication pattern when theiteration number t reaches 16 does not occur. Alsobehaviors for different boundaries produce differentshapes when t = 16. Hence we have a classifica-tion device and tables up to self-replicating patternnumber and for the case the seed image is less than30% of the display matrix. These will be presentedin the next section.

6. The Classification Types of 2DCA Linear Rules

In this section our aim is to classify 512 linear rulesinto different types which we called Type m (seeTables 5 and 6). The natural number m is the num-ber of self-replicating patterns after some iterationsof the first image (i.e. see Figs. 2–12).

First we studied the patterns being generatedfor a given first image (or seed image) for all linearrules iterating it a fixed number of times (we getreplicating images for t = 16). Hence we classifiedthe rules on the basis of these generated patterns.

The entire classification for four different type ofboundaries is shown in Tables 5 and 6.

6.1. Null and periodic boundaryrules

Remark 6.1. The problem of deciding self-replicating patterns (i.e. m) for Rule n with NB,PB, AB, RB algebraically is interesting. This isequivalent to determining which Rule n correspondsto Type m classification scheme given in Tables 5and 6. Also there is a need for more research effortto explain why only t = 24 = 16 times iteration ofthe rules causes the appearance of self-replicatingpattern phenomena.

6.2. Adiabatic and reflexiveboundary rules

Here the linear CA rules are classified on the basis oftheir capacities in producing multiple similar (self-replicating) figures. The relation between Tables 5and 6 is summarized in the following remark.

Remark 6.2. In Tables 5 and 6, each pattern is gen-erated for a fixed seed by the application of theCA rules on each cell in the matrix for tth itera-tions. The differences in Tables 5 and 6 are observed

Table 5. Null and periodic boundary linear 2D CA rules can be classified based on the natural number of the patterngenerated. Type m means that m copies of the images appear after the first image iterates.

Types Null and Periodic Boundary Rules

Type 1 1, 2, 4, 8, 16, 32, 64, 128, 256

Type 2 18, 20, 34, 66, 68, 72, 80, 132, 136, 144, 192, 257, 260, 264, 272, 288, 320, 3, 5, 6, 9, 10, 12, 17, 24, 33, 36, 40,48, 65, 129, 257, 258, 384

Type 3 21, 22, 28, 35, 38, 42, 50, 52, 69, 76, 84, 88, 100, 104, 112, 137, 140, 148, 152, 162, 196, 200, 208, 232, 262, 268,273, 276, 280, 290, 292, 296, 304, 322, 324, 328, 336, 352, 388, 392, 400, 448, 7, 11, 13, 14, 19, 25, 26, 37, 41, 44,49, 56, 67, 73, 74, 81, 82, 97, 98, 104, 131, 133, 134, 138, 145, 146, 161, 164, 168, 176, 193, 194, 224, 259, 265,266, 289, 292, 321, 385, 386, 416

Type 4 29, 30, 39, 43, 46, 51, 53, 54, 58, 60, 77, 83, 85, 87, 90, 92, 99, 101, 106, 108, 116, 120, 141, 149, 153, 154, 156,163, 166, 172, 178, 180, 184, 197, 201, 204, 212, 216, 226, 228, 240, 263, 269, 270, 277, 278, 281, 284, 291, 293,294, 298, 300, 305, 308, 312, 323, 325, 326, 329, 330, 332, 353, 354, 356, 360, 368, 389, 390, 393, 396, 401, 404,408, 418, 420, 424, 432, 449, 450, 452, 456, 464, 480, 15, 23, 27, 45, 57, 71, 75, 78, 86, 89, 102, 105, 114, 135,139, 142, 147, 150, 165, 170, 177, 195, 202, 209, 210, 225, 267, 279, 297, 306, 387, 394, 402, 417

Type 5 31, 47, 55, 59, 61, 62, 79, 91, 103, 108, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 171, 173,174, 181, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217, 218, 220, 227, 229, 230, 233, 234, 236, 241, 242,244, 248, 271, 279, 283, 285, 286, 295, 301, 302, 303, 307, 309, 310, 313, 314, 316, 327, 331, 333, 334, 339, 341,342, 345, 346, 348, 355, 357, 358, 361, 362, 364, 369, 370, 372, 376, 391, 395, 397, 398, 403, 405, 406, 409, 410,412, 419, 421, 422, 425, 426, 428, 433, 434, 436, 440, 453, 454, 457, 458, 460, 465, 466, 468, 472, 481, 482, 484,488, 496, 107, 167, 179, 451

(Continued)

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Table 5. (Continued)

Types Null and Periodic Boundary Rules

Type 6 63, 95, 119, 125, 126, 159, 182, 183, 187, 190, 207, 215, 221, 222, 231, 237, 245, 246, 249, 252, 287, 311, 317,318, 335, 343, 347, 349, 350, 359, 365, 366, 373, 374, 377, 378, 380, 399, 407, 413, 414, 423, 429, 430, 435, 437,438, 444, 455, 459, 461, 462, 467, 469, 470, 473, 474, 476, 483, 485, 486, 489, 490, 492, 497, 498, 500, 504, 111,123, 175, 187, 219, 231, 235, 238, 243, 250, 315, 365, 371, 411, 442, 427, 435, 441

Type 7 127, 191, 223, 247, 253, 254, 319, 351, 367, 375, 379, 381, 382, 415, 431, 439, 445, 446, 463, 471, 475, 477, 478,487, 491, 493, 494, 499, 501, 502, 505, 506, 508, 239, 251, 443

Type 8 255, 383, 447, 479, 495, 503, 507, 509, 510

Type 9 511

when the different boundaries are applied to thefixed seed image. Let us consider for example howthe pattern for Rule 149 is generated. For Rule149, the cells under consideration are highlightedas shown in Fig. 5. Applying this rule on an ini-tial seed produces Fig. 5 at the 16th iteration. Rule149 is of Type 4 for Null and Periodic boundaries(see Table 5), whereas Rule 149 is Type 3 for Adia-batic and Reflexive boundaries (see Table 6). Thisis easily observed from Fig. 5.

Since rule matrices are necessarily squarematrices (i.e. mn×mn), the nonsingular rule matrixcalled reversible CA deserves to be studied closelybecause a nonsingular matrix must produce cyclicstate transition diagram having enumerable appli-cations in different areas such as Pattern Classifica-tion, Clustering, Encryption and Decryption, DataCompression etc. [Martinez et al., 2011, 2012; Stin-son, 2005; Wuensche & Adamatzky, 2006; Wuen-sche, 2009].

Table 6. Adiabatic and reflexive boundary linear 2D CA rules can be classified based on the natural number of thepattern generated.

Types Adiabatic and Reflexive Boundary Rules

Type 1 1, 2, 3, 4, 5, 8, 9, 16, 17, 32, 33, 64, 65, 128, 129, 256, 257

Type 2 6, 7, 10, 11, 12, 13, 18, 19, 20, 21, 24, 25, 34, 35, 36, 37, 40, 41, 48, 49, 66, 67, 68, 69, 72, 73, 80, 81, 96, 97,130, 131, 132, 133, 136, 137, 144, 145, 160, 161, 192, 193, 258, 259, 260, 261, 264, 265, 272, 273, 288, 289, 320,321, 384, 385

Type 3 112, 113, 134, 135, 138, 139, 140, 141, 146, 147, 148, 149, 152, 153, 162, 163, 164, 165, 168, 169, 176, 177, 194,195, 196, 197, 200, 201, 208, 209, 224, 225, 262, 263, 266, 267, 268, 269, 274, 275, 276, 277, 280, 281, 290, 291,292, 293, 296, 297, 304, 305, 322, 323, 324, 325, 328, 329, 336, 337, 352, 353, 386, 387, 388, 389, 392, 393, 400,401, 416, 417, 448, 449

Type 4 30, 31, 46, 47, 54, 55, 58, 59, 60, 61, 78, 79, 86, 87, 90, 91, 92, 93, 102, 103, 106, 107, 108, 109, 114, 115, 116,117, 120, 121, 142, 143, 150, 151, 154, 155, 156, 157, 166, 167, 170, 171, 172, 173, 178, 179, 180, 181, 184, 185,198, 199, 202, 203, 204, 205, 210, 211, 212, 213, 216, 217, 226, 227, 228, 229, 232, 233, 240, 241, 270, 271, 278,279, 282, 283, 284, 285, 294, 295, 298, 299, 300, 301, 306, 307, 308, 309, 312, 313, 326, 327, 330, 331, 332, 333,338, 339, 340, 341, 344, 345, 354, 355, 356, 357, 360, 361, 368, 369, 390, 391, 394, 395, 396, 397, 402, 403, 404,405, 408, 409, 418, 419, 420, 421, 424, 425, 432, 433, 450, 451, 452, 453, 456, 457, 464, 465, 480, 481

Type 5 62, 63, 94, 95, 110, 111, 118, 119, 122, 123, 124, 125, 158, 159, 174, 175, 182, 183, 186, 187, 188, 189, 190, 191,206, 207, 214, 215, 218, 219, 220, 221, 230, 231, 234, 235, 236, 237, 242, 243, 244, 245, 248, 249, 286, 287, 302,303, 310, 311, 314, 315, 316, 317, 334, 335, 342, 343, 346, 347, 348, 349, 358, 359, 362, 363, 364, 365, 370, 371,372, 373, 376, 377, 398, 399, 406, 407, 410, 411, 412, 413, 422, 423, 426, 427, 428, 429, 434, 435, 436, 437, 440,441, 454, 455, 458, 459, 460, 461, 466, 467, 468, 469, 472, 473, 482, 483, 484, 485, 488, 489, 496, 497

Type 6 126, 127, 222, 223, 238, 239, 246, 247, 250, 251, 252, 253, 318, 319, 350, 351, 366, 367, 374, 375, 378, 379, 380,381, 414, 415, 430, 431, 438, 439, 442, 443, 444, 445, 462, 463, 470, 471, 474, 475, 476, 477, 486, 487, 490, 491,492, 493, 498, 499, 500, 501, 504, 505

Type 7 254, 255, 382, 383, 446, 447, 478, 479, 494, 495, 502, 503, 506, 507, 508, 509

Type 8 510, 511

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7. An Application in Cryptography

An important need in applications of cryptographyis to generate a random number of binary bits ofparticular length. This is a very difficult problemsince the generation of such sequence of bits relieson algorithms and programs that are built system-atically. Further, a method of deciding the random-ness is an important and challenging issue. In orderto solve this problem a method of randomizing thesequences is needed. In this section we present sucha method by utilizing the hybrid structure of the 2DCA studied in the previous sections. Though this isnot a new method, the success of CA on generatingpseudo numbers is not always possible. For instance,if the hybridization is not used, in general, such aprocess fails. Further, the success even in the caseof hybridization is still not straight forward. How-ever, in this section we prove statistically that asuggested specific hybridization is very successful.In order to test such a sequence for randomness wehave applied five basic tests [Menezes et al., 1996].In a sequel we present these five basic tests brieflywith the application under the consideration.

The following basic tests are applied tothe sequences obtained herein for testing theirrandomness:

I. Frequency Test: The aim of this test is todetermine whether the number of 0’s and 1’s in asequence of bits are approximately the same, as wewould expect for a random sequence.

II. Serial Test: The aim of this test is to deter-mine if the number of occurrences of 00, 01, 10 and11 are approximately the same or not.

III. Poker Test: The poker test aims to deter-mine if the number of the distinct subsequences oflength m are approximately the same or not suchthat n/m ≥ 5 · 2m.

IV. Run Test: The run test aims to see if the devi-ation of the number of runs for several lengths fromthe expected values is negligible or not.

V. Autocorrelation Test: This test determinesthe correlation between the sequence and noncyclicshifted version of it.

Pseudo random numbers or bits are used forseveral purposes in cryptography but one of themain applications is the generation of a key for acryptosystem [Stinson, 2005]. The algorithm used ina cryptosystem may be very efficient, however if thekey is vulnerable to some attacks, then the cipherwill be insecure. Therefore, generating pseudo ran-dom numbers or bits is vital.

CAs are used for pseudo random bit genera-tors as besides some other methods [Martin & Sole,2008; Schiff, 2010; Guan & Tan, 2004]. In [Rubioet al., 2004] the use of one-dimensional linear hybridCAs as pseudo random bit generators is investi-gated. Here in this work, we also show that, two-dimensional linear hybrid CAs can be used as apseudo random bit generator.

To this end, we generated 100 sequences of bitsof length 1024 in a such a way that we evolved a5 × 5 lattice of cells which are chosen randomly1024 times and we concatenate a fixed cell of eachtime step. The cells were evolved by hybridizationof the rule numbers 109 and 308 respectively. Inthe sequel, 99 of the sequences that we generatedpassed the frequency test, all of them passed theserial test, 95 of them passed the poker test, 98 ofthem passed the run test and all of them passed theautocorrelation test.

Now we present an example to illustrate theapplication among the tested sequences that passedall the tests. We use the hexadecimal notation inorder to save space of a sequence of bits that wegenerated and that is as follows:

3C56E72D 5EECC297 286D52A9 6A0D841F CE45E198 EBCE81BB 114109A8 75D9F85F 5561261752CB4749 517C5AA2 85C440B0 BC4E6C7 C6BB427A EF17A787 6C89279F E7D812C7 2E1CBD0283E694F D5DFDA8B 18371F59 248E8DFE B9DBA5DA A212A52F A51EB3BC D90D137EE93573CA 3AB069E9 A150F4EF B1D04A13 2D9AB64B 3DEF449C.

There are 504 zeros and 520 ones in the sequence and the value of the statistic X1 is 0.25.The number of 00, 01, 10 and 11’s are 236, 267, 267 and 253 respectively and the value of the statistic

X2 is 2.2944.The number of nonoverlapping subsequences 000, 001, 010, 011, 100, 101, 110 and 111 are 44, 42,

47, 46, 33, 50, 31 and 48 respectively and the value of the statistic X3 is 8.0674.

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Fig. 19. Lena’s original image.

The value of the statistic X4 is 6.5407.Choosing d = 8 the value of the statistic X5

is 0.3764.For a significance level of α = 0.05, the thresh-

old values for X1,X2,X3,X4 and X5 are 3.8415,5.9915, 14.0671, 9.4877 and 1.96 [Menezes et al.,1996]. Hence, this sequence will pass all the tests.This shows statistically that the sequence is apseudo number.

Now, in order to see concretely that thesequence is pseudo random, we will encrypt animage by using the sequence as a key in such a waythat we will consider the matrix of the pixels of theimage and we consider our key a matrix of the samesize with the matrix of the image and we will bitwiseXOR. Afterwards we will consider the result matrixand the transpose of the matrix which we obtained

Fig. 20. Lena’s encrypted image.

from the sequence and add them up. If we applythis random number obtained via this method tothe picture of Lena 19, we get the encrypted imageof Lena 20 and by XORing again we get the originalimage 19 back.

8. Conclusion

In this paper, we discuss in theory two-dimensional,uniform periodic, adiabatic and reflexive boundaryCAs of linear rules and applications of image pro-cessing. It is seen that CAs theory can be appliedsuccessfully in self-replicating patterns of imageprocessing. Just a nontrivial self-replicating pat-tern is shown in the paper (see Fig. 18). We inves-tigate 2D CA transition matrix rules with theseboundaries over the binary field Z2. We also studythe applications of image processing correspondingto the linear rules of 2D uniform CA with theseboundary conditions over Z2. Properties of the 2Dfinite CA over other fields (see [Akın & Siap, 2007])remain to be of great research interest. Some char-acterization and applications on a 2D finite CA byusing matrix algebra built on Z3 are planned infuture studies. Also, some results will be analyzedon the rule numbers 2460N and 2460P [Siap et al.,2010]. However after making use of the matrix rep-resentation of 2D CA, an algorithm will be providedto obtain the number of Garden of Eden configura-tions for the 2D CA defined by some rules.

In future studies the application of two-dimensional periodic, adiabatic and reflexive CAsto the problems of noise removal, border detectionin digital images, also CA with extended neigh-borhood for epidemic propagation are planned tobe explored with imaging science. Although onlysome important primary image transformations arebeing investigated for symmetric binary images, onecan consider that the work can extend further forany other complex image transformations as well asmany colored images (i.e. colors are chosen in Zp forp > 2 prime number). Our forthcoming research iscontinuing in that direction. Finally, there are sev-eral open questions raised by the authors that mayattract many researchers.

Acknowledgments

The first author thanks Prof. Dr. Ferat Sahin andespecially Dr. Ugur Sahin for the kind hospitalityduring the visit at the Rochester Institute of Tech-nology (RIT). S. Uguz would also like to thank

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The Council of Higher Education (YOK) for thesupport of his visit in RIT. The authors wish tothank TUBITAK (Project Number: 110T713) aswell for their support. We would also like to thankthe anonymous referees for their valuable sugges-tions that has lead to an improved version.

References

Adamatzky, A., Martinez, G. J. & Mora, J. C. [2006]“Phenomenology of reaction–diffusion binary-statecellular automata,” Int. J. Bifurcation and Chaos 16,2985–3005.

Akın, H. [2005] “On the directional entropy of Z2-actions

generated by additive cellular automata,” Appl. Math.Comput. 170, 339–346.

Akın, H. & Siap, I. [2007] “On cellular automata overGalois rings,” Inform. Process. Lett. 103, 24.

Alvarez, G., Encinas, L. H. & Rey, A. M. [2008] “A mul-tisecret sharing scheme for color images based on cel-lular automata,” Inform. Sci. 178, 4382.

Bilotta, E., Pantano, P. & Lohn, J. D. [2005] “Emer-gent patterning phenomena in 2D cellular automata,”Artif. Life 11, 339–362.

Blackburn, S. R., Murphy, S. & Peterson, K. G. [1997]“Comments on theory and applications of cellularautomata in cryptography,” IEEE Trans. Comput.46, 637.

Chattopadhyay, P., Choudhury, P. P. & Dihidar, S. K.[1999] “Characterisation of a particular hybrid trans-formation of two-dimensional cellular automata,”Comput. Math. Appl. 38, 207–216.

Chopard, B. & Droz, M. [1998] Cellular AutomataModelling of Physical Systems (Cambridge UniversityPress).

Chou, H. H. & Reggia, J. A. [1997] “Emergence of self-replicating structures in a cellular automata space,”Physica D 110, 252–276.

Choudhury, P. P., Nayak, B. K., Sahoo, S. & Rath, S. P.[2005] “Theory and applications of two-dimensional,null-boundary, nine-neighborhood, cellular automatalinear rules,” Tech. Report No. ASD/2005/4, 13arXiv: 0804.2346.

Choudhury, P. P., Sahoo, S., Hassan, S. S., Basu, S.,Ghosh, D., Kar, D., Ghosh, Ab., Ghosh, Av. & Ghosh,A. K. [2010] “Classification of cellular automata rulesbased on their properties,” Int. J. Comp. Cogn. 8,50–54.

Chua, L. O. & Yang, L. [1988] “Cellular neural networks:Theory,” IEEE Trans. Circuits Syst. 35, 1257–1272.

Chua, L. O. & Mainzer, K. [2011] The Universe asAutomaton: From Simplicity and Symmetry to Com-plexity (Springer).

Das, A. K. [1990] “Additive cellular automata: Theoryand application as a built-in self-test structure,” PhDthesis, I. I. T. Kharagpur, India.

Dihidar, S. K. & Choudhury, P. P. [2004] “Matrix alge-braic formulae concerning some exceptional rules oftwo dimensional cellular automata,” Inform. Sci. 165,91.

Dogaru, R. & Chua, L. O. [1999] “Emergence of uni-cellular organisms from a simple generalized cellularautomata,” Int. J. Bifurcation and Chaos 9, 1219–1236.

Dogaru, R. & Chua, L. O. [2000] “Mutations of the gameof life: A generalized cellular automata perspective ofcomplex adaptive systems,” Int. J. Bifurcation andChaos 10, 1821–1866.

Gravner, J. & Griffeath, D. [2011] “The one-dimensionalexactly 1 cellular automaton: Replication, periodicity,and chaos from finite seeds,” J. Statist. Phys. 142,168–200.

Gravner, J., Gliner, G. & Pelfrey, M. [2011] “Replica-tion in one-dimensional cellular automata,” Physica D240, 1460–1474.

Guan, S. U. & Tan, S. K. [2004] “Pseudorandomnumber generation with self-programmable cellularautomata,” IEEE Trans. Computer-Aided Design ofIntegrated Circuits and Systems 23, 1095–1101.

Hedlund, G. A. [1969] “Endomorphisms and automor-phisms of full shift dynamical system,” Math. Syst.Th. 3, 320.

Inokuchi, S. & Sato, T. [2000] “On limit cycles andtransient lengths of cellular automata with thresholdrules,” Bull. Inform. Cybernet. 32, 2360.

Julian, P. & Chua, L. O. [2002] “Replication propertiesof parity cellular automata,” Int. J. Bifurcation andChaos 12, 477–494.

Khan, A. R., Choudhury, P. P., Dihidar, K., Mitra,S. & Sarkar, P. [1997] “VLSI architecture of a cel-lular automata machine,” Comput. Math. Appl. 33,79.

Khan, A. R., Choudhury, P. P., Dihidar, K. & Verma,R. [1999] “Text compression using two dimensionalcellular automata,” Comput. Math. Appl. 37, 115.

Martin, B. & Sole, P. [2008] Pseudo-Random SequencesGenerated by Cellular Automata, Int. Conf. Relations,Orders and Graphs : Interactions with Computer Sci-ence (Mahdia, Tunisie).

Martinez, G. J., Adamatzky, A., Stephens, C. R. & Hoe-flich, A. F. [2011] “Cellular automaton supercollid-ers,” Int. J. Mod. Phys. C 22, 419–439.

Martinez, G. J., Adamatzky, A. & Sanz, R. A. [2012]“Complex dynamics of elementary cellular automataemerging from chaotic rules,” Int. J. Bifurcation andChaos 22, 1250023.

1430002-23

Int.

J. B

ifur

catio

n C

haos

201

4.24

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

SHE

FFIE

LD

on

07/0

8/14

. For

per

sona

l use

onl

y.

January 29, 2014 6:59 WSPC/S0218-1274 1430002

S. Uguz et al.

Menezes, A. J., Oorschot, P. C., Menezes, A. & Van-stone, S. A. [1996] Handbook of Applied Cryptography(CRC Press).

Mihaljevic, M., Zheng, Y. & Imai, H. [1999] “A family offast dedicated one-way hash functions based on linearcellular automata over GF(q),” IEICE Trans. Fund.E82-A, 40–47.

Mitra, S. & Kumar, S. [2005] “Fractal replication in time-manipulated one-dimensional cellular automata,”Compl. Syst. 16, 191–208.

Nishinari, K. & Takahashi, D. [2000] “Multi-value cel-lular automaton models and metastable states in acongested phase,” J. Phys. A 33, 7709.

Packard, N. H. & Wolfram, S. [1985] “Two-dimensionalcellular automata,” J. Stat. Phys. 38, 901.

Reggia, J. A., Chou, H. H. & Lohn, J. D. [1998]“Emergence of self-replicating structures in a cellularautomata space,” Adv. Comput. 47, 141–183.

Rubio, C. F., Encinas, L. H., White, S. H., Rey, A. M. &Sanchez, G. R. [2004] “The use of linear hybrid cel-lular automata as pseudorandom bit generators incryptography,” Neural Parall. Scient. Comput. 12,175.

Schadschneider, A. & Schreckenberg, M. [1993] “Cellularautomaton models and traffic flow,” J. Phys. A 26,679–683.

Schiff, J. L. [2010] Cellular Automata: A Discrete Viewof the World (Wiley & Sons).

Siap, I., Akın, H. & Sah, F. [2010] “Garden of edenconfigurations for 2-D cellular automaton with rule2460N,” Inform. Sci. 180, 3562.

Siap, I., Akın, H. & Sah, F. [2011a] “Characterizationof two dimensional cellular automata over ternaryfields,” J. Franklin Instit. 348, 1258.

Siap, I., Akın, H. & Uguz, S. [2011b] “Structure andreversibility of 2D hexagonal cellular automata,”Comput. Math. Appl. 62, 4161.

Smith, A., Turney, P. & Ewaschuk, R. [2003] “Self-replicating machines in continuous space with virtualphysics,” Artif. Life 9, 21–40.

Stinson, D. R. [2005] Cryptography: Theory and Practice(Chapman and Hall/CRC Press, Ontario).

Uguz, S., Siap, I. & Akın, H. [2013a] “2-dimensionalreversible hexagonal cellular automata with periodicboundary,” Acta Physica Polonica A 123, 480–483.

Uguz, S., Akın, H. & Siap, I. [2013b] “Reversibility algo-rithms for 3-state hexagonal cellular automata withperiodic boundaries,” Int. J. Bifurcation and Chaos23, 1350101.

von Neumann, J. [1966] The Theory of Self-ReproducingAutomata (Univ. of Illinois Press, Urbana, USA).

Wolfram, S. [1983] “Statistical mechanics of cellularautomata,” Rev. Mod. Phys. 55, 601.

Wuensche, A. & Adamatzky, A. [2006] “On spiralglider-guns in hexagonal cellular automata: Activator-inhibitor paradigm,” Int. J. Mod. Phys. C 17, 1009.

Wuensche, A. [2009] “Cellular automata encryption: Thereverse algorithm, z-parameter and chain-rules,” Par-all. Process. Lett. 19, 283.

Ying, Z., Zhong, Y. & Pei-Min, D. [2009] “On behaviorof two-dimensional cellular automata with an excep-tional rule,” Inform. Sci. 179, 613.

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