Comparative Study on Null Boundary and Periodic Boundary

3-Neighborhood Multiple Attractor Cellular Automata for Classification

Anirban Kundu1,2

, Alok Ranjan Pal2, Tanay Sarkar

2, Moutan Banerjee

2, Sutirtha Kr. Guha

2, and

Debajyoti Mukhopadhyay2,3

1Netaji Subhash Engineering College

West Bengal University of Technology

Calcutta 700152, India 2Web Intelligence & Distributed Computing Research Lab (WIDiCoReL)

Green Tower C-9/1, Golf Green, Calcutta 700095, India 3Calcutta Business School

Diamond Harbour Road, Bishnupur, West Bengal 743503, India

{anik76in, chhaandasik, tanay.sarkar, moutanbanerjee, sutirthaguha, debajyoti.mukhopadhyay

}@gmail.com

Abstract

This paper reports a generic analysis on Null

Boundary and Periodic Boundary 3-neighborhood

Multiple Attractor Cellular Automata (MACA) for

showing the comparative study in classification

technique. Cellular Automata (CA) is now-a-days an

essential tool for researchers in the area of pattern

recognition, pattern generation, testing field, fault

diagnosis and so on. So, general knowledge on CA up

to some extent is a must for researchers in these areas.

A CA may be linear or non-linear in behavior. A

linear/additive CA employs XOR/XNOR logic, while a

non-linear CA employs AND/OR/NOT logic. This paper shows a graph analysis along with state

transition behavior of CA cells. A Rule Vector Graph

(RVG) is generated from the Rule Vector (RV) of a CA.

Linear time algorithms are reported for generation of

RVG. MACA provides an implicit memory to store the

patterns. Search operation to identify the class of a

pattern out of several classes boils down to running a

CA for one time step. This demands storage of RV and

seed values. MACA is based on sound theoretical

foundation of CA technology. This paper only

concentrates on MACA since it is responsible for

classifying the various types of patterns.

Keywords: Null Boundary Cellular Automata

(NBCA), Periodic Boundary Cellular Automata

(PBCA), State Transition, Rule Vector Graph (RVG).

1. Introduction

The researcher in artificial life inherits the

tradition of simulation. They model the world as a set

of dynamic systems. Difficulties in evaluating

behavior of a CA from the rules of CA cells or

designing a CA to realize the desired behavior have

restricted CA applications. In this background, an [3,

7] analytical formulation is reported in this paper to

analyze state transition behavior of a CA. Many

researchers in the field of artificial life have been

enamored of a mathematical formalism of computing

model known as CA. There are a lot of research works

available for designing MACA [6]. Synthesis of

MACA demands formation of a RV with group and

non-group rules in specific sequence. The method to

identify such a sequence is described in [1]. In CA

world, classification depends on the formation of

MACA. The RV of an n-cell CA is denoted as <R0,

R1, · · · , Ri, · · · , Rn-1>, where ith

cell is configured

with Ri. A linear or non-linear [4-5] MACA consists of

2n number of states where n is the size of MACA. The

structure of a MACA has attractors (self-loop or single

length cycle), non-reachable states, and transient

states. The attractors form unique classes (basins). All

other states reach the attractor basins after certain time

steps. To classify a set of k classes, (k-1) numbers of

attractors are used, each identifying a single class. [2,

8] reported illustrative ideas in this regard.

Rest of the paper is organized as follows: in

Section 2 CA Preliminaries are discussed; Section 3

978-1-4244-2917-2/08/$25.00 ©2008 IEEE 204

presents the Analysis of CA; and, Conclusion is

depicted in Section 4.

2. CA Preliminaries

A Cellular Automata (CA) is a model which can

be used to show how the elements of a system interact

with each other. Each element of the system is

assigned a cell. The cells are of various types like 1D,

2D, 3D, etc.. In this paper, we are only concerned

about 1D CA with 3-neiborhood configuration. To

form one-dimensional cellular automata, cells are

generally connected with its neighbors in both

directions. Cellular automata can serve as a source of

random numbers which are used for encrypting

messages, running simulations and other purposes.

The configurations of a succession of CA generations

can be used as a random sequence. One-dimensional

CA is normally used for this purpose. The simplest

nontrivial CA would be one-dimensional, with two

possible states per cell (1/0), and a cell’s neighbors

defined to be the adjacent cells on either side of it.

These neighbors are known as left-neighbor & right

neighbor. A cell and its two neighbors form a

neighborhood of 3 cells, so there are 23=8 possible

patterns for a neighborhood. There are then 28=256

possible rules.

An n cell CA consists of n cells (Figure 1(a)) with

local interactions [9]. It evolves in discrete time and

space. The next state function of three neighborhood

CA cell (Figure 1(b)) can be represented as a rule as

defined in Table 1 [7]. First row of Table 1 represents

23 = 8 possible present states of 3 neighbors of i

th cell -

(i-1), i, (i+1) cells. Each of the 8 entries (3 bit binary

string) represents a min term of a 3 variable Boolean

function for a 3 neighborhood CA cell. In subsequent

discussions, each of the 8 entries in Table 1 is referred

to as a Rule Min Term (RMT).

The decimal equivalent of 8 min terms are 0, 1, 2,

3, 4, 5, 6, 7 noted within ( ) below the three bit string.

Each of the next five rows of Table 1 shows the next

state (0 / 1) of ith

cell. Hence, there can be 28 = 256

possible bit strings. The decimal counterpart of such

an 8 bit combination is referred to as a CA rule [3, 7].

The rule of a CA cell can be derived from Table 1 of

the ith

cell.

2.1. Definitions

Definition 1: Group CA - Each state in the state

transition behavior of a group CA has only one

predecessor and consequently each state is reachable

from only one state. A group CA traverses all the

states in a cycle. A group CA is a reversible CA in the

sense that the CA will always return to its initial state.

Definition 2: Non-Group CA - A non-group CA has

states that have r number of predecessors, where r = 0,

1, 2, 3, · · ·

Definition 3: Attractor Cycle - The set of states in a

cycle is referred to as an attractor cycle.

Definition 4: Rule Vector (RV) - The sequence of

rules < R0R1 · · ·Ri · · ·Rn+1 >, where ith

cell is

configured with rule Ri.

Definition 5: Rule Vector Graph (RVG) - A graph,

derived from the RV (Rule Vector) of a CA is referred

to as RVG. A node in RVG represents a set of RMTs

(Rule Min Terms) while an edge between a pair of

nodes represents the next state value (0 / 1) of a cell

for specific RMTs. Derivation of the RVG from the

RV (Rule Vector) of a CA follows.

Figure 1 Local Interaction between Cellular

Automata Cells

Table 1 Truth Table of sample rules of a CA cell showing the next state logic for the Minterms of a 3 variable

Boolean function - The 8 minterms having decimal values 0, 1, 2, 3, 4, 5, 6, 7 are referred to as Rule Minterms

(RMTs)

205

3. Analysis of CA

One-dimensional (1D) Cellular Automata (CA)

over finite fields are studied in which each interior

(local) cell is updated to contain the sum of the

previous values of its two nearest (left & right)

neighbors along with its own cell value. Boundary

cells are updated according to Null Boundary

conditions. For a given initial configuration, the CA

evolves through state transitions to an attracting cycle

which is defined as attractor / basin (refer Definition

3). The number of cycles can be determined from the

minimal polynomial and characteristic polynomial of

the updated matrix which is formed by the linear CA.

For detailed theoretical study, follow [3]. But, in case

of non-linear CA, matrix can not be formed since it

does not follow any regular mathematics. So, as per

Definition 4, CA can be expressed as RV in both the

cases. Figure 1 shows an example of NBCA with RV

<0, 255, 2, 5>. Leftmost & rightmost cells are ground

as shown in the figure.

Figure 2 Example of Null Boundary CA

Figure 3 Example of Periodic Boundary CA

In case of periodic CA, we have considered the

rightmost cell as the left neighbor of leftmost cell.

Similarly, the leftmost cell is considered as the right

neighbor of rightmost cell. So, it is like a circular

linked list data structure. Figure 3 shows PBCA.

In this paper, we have mainly concentrated on the

variation of state transition of Null Boundary &

Periodic Boundary MACA for the sake of

classification techniques in various fields. The

attractors of MACA can be treated as the classifier

basins for separating different type of data or topic. It

has been found that there is a tendency for achieving

more number of attractors in a PBCA compared to

NBCA. In this paper, the overall analysis is done

based on state transition behavior and RVG of CA

cells.

3.1. State Transition based Observation

Observation 1: State Transition behaviors of Rule

Vector (RV) <2, 44, 4, 21> in Null Boundary &

Periodic Boundary mode have been shown in Figure 4.

In Null Boundary case, only one attractor (state “1”)

has been found whereas in Periodic Boundary case,

two attractors (states “1” & “5”) have been found. It is

found through exhaustive experimentation that there is

a tendency to have more number of attractors in case

of Periodic Boundary MACA compared to Null

Boundary MACA in maximum cases.

Null Boundary Periodic Boundary

Attractor - 1

Attractor - 1, 5

Figure 4 State Transition behaviors of Rule Vector (RV) <2, 44, 4, 21>

0 255 2 5

0 255 2 5

206

Null Boundary Periodic Boundary

Attractor - 0

Attractor - 0

Figure 5 State Transition behaviors of Rule Vector (RV) <90, 90, 90, 90>

Observation 2: State Transition behaviors of Rule

Vector (RV) <90, 90, 90, 90> in Null Boundary &

Periodic Boundary mode have been shown in Figure 5.

In both Null Boundary & Periodic Boundary case,

only one attractor (state “0”) has been found. Here,

Null Boundary case is basically a Group CA

(Definition 1) having all the states within some cycles

of state transition. After applying the Periodic

Boundary concept on this RV, a MACA is found. So,

there is a tendency to generate a Non-group CA

(Definition 2) in case of PBCA compared to Null

Boundary CA in maximum cases. In this particular

case, a MACA has been formed in Periodic Boundary

by applying the periodic neighborhood concept upon

the initial Group CA of Null Boundary. So, it is

possible to get a MACA structure from Group CA for

formation of a classifier.

Algorithm 1 depicts the generic state transition

technique of any NBCA or PBCA.

Algorithm 1: Null / Periodic Boundary State

Transition

Input: CA Size and Rule Vector

Output: State transition diagram

Step 1: Convert decimal rule values into binary

Step 2: Create RMT Table (refer Table 1)

Step 3: Loop (start)

Step 4: Pick seedi and apply corresponding rule on it

Step 5: Store next state value

Step 6: Loop (Stop)

Step 7: Stop

Firstly, CA Size and Rule Vector are taken as

inputs. After that, conversion of decimal rule into

binary for creating RMT table (8 X CA size) has been

done. Since, 3-neighborhood Cellular Automata (CA)

has been used in our approach, that’s why it should be

23 = 8 combinations. It means {0 to 7} columns should

be there in the RMT table and the number of rows will

depend on CA size at runtime. Then, pick a seed

arbitrarily and apply corresponding rule on it. Store

the next state value within a temporary variable. The

state diagrams of NBCA and PBCA have been created

using Algorithm 1. The only difference of these two

boundary concepts is whether leftmost [0th

] &

rightmost [(n-1)st] cells are ground or interconnected.

Generally, we have to consider (i-1, i, i+1) cells for

calculating the next state value of the ith

cell of a

particular CA in 3-neighborhood condition. In Null

Boundary case, for 0th

cell, the left neighbor [(i-1)th

cell] should be ground. Similarly, for (n-1)th

cell, the

right neighbor [(i+1)th

cell] should be ground. On the

other hand, in case of Periodic Boundary, leftmost &

rightmost cells are connected to each other. Through

our experimentation, it is clear that we can get greater

than or equal number of attractors in Periodic

Boundary case as compared to Null Boundary 3-

neighborhood CA in most of the cases. So, more

number of classes can be achieved for classification of

patterns by increasing the number of attractors within

the structure of a CA. Most benefited advantage is that

we can search more number of pages in a short time

using less memory [1, 10].

3.2. RVG based Observation

An arbitrary CA is shown in Figure 6 for

representing it in RVG form. RVG consists of several

nodes and each node consists of maximum two links.

These links are known as edges. Node values are RMT

values of specific rules. A RVG always starts with a

single node containing {0, 1, 2, 3} as node value in

Null Boundary case. In 3-neighbourhood NBCA, R0

(Leftmost cell) and Rn-1 (Rightmost cell) have four

207

don’t care positions in its concerned RMTs. So, in case

of R0, the effective RMT positions are 0, 1, 2 & 3.

Similarly, for Rn-1, the effective RMT positions are 0,

2, 4 & 6 as discussed in [10]. In 3-neighbourhood

NBCA, RMT values range from (0-7). So, maximum

eight values are there within the RVG. Any

combination of (0-7) can be a node value depending

on the applied rule on the particular level of RVG.

Each node can have maximum of two edges (0-edge &

1-edge). The reason behind this is each cell’s next state

value may be either ‘0’ or ‘1’. So, depending on the

probability to appear ‘0’ or ‘1’ as the next state value,

RMTs of each node is divided into two categories. So,

at any level of graph, maximum of two nodes can be

formed from each node of previous level. Figure 6

shows the RVG and corresponding RMT table of a 4-

cell CA with RV <2, 44, 4, 21> in a Null Boundary

condition.

Figure 6 Null Boundary RVG & related RMT of a 4-cell CA with RV <2, 44, 4, 21>

Figure 7 Periodic Boundary RVG & related RMT of a 4-cell CA with RV <2, 44, 4, 21>

In case of PBCA, the leftmost & rightmost cells

are interconnected. So, there is no such don’t care

RMT positions for R0 & Rn-1. The RVG of RV <2, 44,

4, 21> is shown in Figure 7 with Periodic Boundary

condition. Here, root (starting) node as well as leaf

(ending) node consists of all RMTs {0, 1, 2, 3, 4, 5, 6,

7}. So, there are some changes over the edges of the

RVG.

Observation 3: At any intermediate level of RVG, if

there is any node containing {01234567} RMTs, the

CA tends to be a MACA.

Observation 4: Number of nodes containing

{01234567} RMTs increases when Periodic Boundary

concept is imposed on the Null Boundary RVG.

Observation 5: Load balancing over the edges of

RVG is equally distributed in case of Group CA.

Observation 6: Load balancing over the edges of

RVG is not equally distributed in case of Non-group

CA like MACA.

Algorithm 2 and Algorithm 3 show how to

generate RVG for NBCA and PBCA respectively.

Rule Min Term (RMT)

111 110 101 100 011 010 001 000

0 0 0 0 0 0 1 0

0 0 1 0 1 1 0 0

0 0 0 0 0 1 0 0

0 0 0 1 0 1 0 1

R0 = 2

R1 = 44

R2 = 4

R3 = 21

01234567

01234567

01234567

01234567

01234567

0, 2, 3, 4, 5, 6, 7 / 0

0, 1, 4, 6, 7 / 0

0, 1, 3, 4, 5, 6, 7 / 0

1, 3, 5, 6, 7 / 0

1 / 1

2, 3, 5 / 1

2 / 1

0, 2, 4 / 1

2

44

4

21

208

Algorithm 2: Generating RVG for NBCA

Input: RV < R0 R1 · · ·Ri · · ·Rn-1 > of ‘n’ cell CA

Output: RVG

Step 1: For a NBCA fix the Root Node as T(0, 1, 2, 3)

which represents the 0th

cell.

Step 2: For ith

level (i = 0, 1, 2, ...., (n-2)) execute Step

3 and Step 4

Step 3: For ith

cell, draw the outgoing edges from each

of input nodes generated in the previous level. Both 0-

edge and 1-edge are derived from RMT table of Rule

Ri. Two edges are marked with weight {Ti}/bi, where

{Ti}={subset of RMTs for which the next state value

of ith

cell is bi (bi € {0, 1}) as per Rule Ri}. Put a tag

on the node if 0-edge or 1-edge is missing.

Step 4: Generate next level nodes out of the subset of

RMTs {Ti} marked as weight on the edges derived in

Step 3. A node having the RMTs covered by another

node of ith

level is merged with it. The output nodes of

ith

level serve as input nodes for (i+1)th

level.

Step 5: The edges from the output nodes of (n-2)th

level (i.e., input nodes of (n-1)th

level) to the leaf node

T(0, 1, 4, 5) are derived as per the rule Rn-1.

Step 6: Stop.

Algorithm 3: Generating RVG for PBCA

Input: RV < R0 R1 · · ·Ri · · ·Rn-1 > of ‘n’ cell CA

Output: RVG

Step 1: For a PBCA fix the Root Node as T(0, 1, 2, 3,

4, 5, 6, 7) which represents the 0th

cell.

Step 2: For ith

level (i = 0, 1, 2, ...., (n-2)) execute Step

3 and Step 4

Step 3: For ith

cell, draw the outgoing edges from each

of input nodes generated in the previous level. Both 0-

edge and 1-edge are derived from RMT table of Rule

Ri. Two edges are marked with weight {Ti}/bi, where

{Ti}={subset of RMTs for which the next state value

of ith

cell is bi (bi € {0, 1}) as per Rule Ri}. Put a tag

on the node if 0-edge or 1-edge is missing.

Step 4: Generate next level nodes out of the subset of

RMTs {Ti} marked as the weight on the edges derived

in Step 3. A node having the RMTs covered by

another node of ith

level is merged with it. The output

nodes of ith

level serve as input nodes for (i+1)th

level.

Step 5: The edges from the output nodes of (n-2)th

level (i.e., input nodes of (n-1)th

level) to the leaf node

T(0, 1, 2, 3, 4, 5, 6, 7) are derived as per the rule Rn-1.

Step 6: Stop.

4. Conclusion

This paper reports analysis of MACA behavior for

classification purpose. The enhancement of pattern

classification can be performed using this analysis.

Both the state transition and graph analysis have been

discussed separately for finding out the distinguished

characteristic changes. In terms of pattern classifier,

the superiority of periodic boundary MACA with

reference to null boundary MACA is achieved with

respect to the generation of number of classes. Group

CA can also be modified into MACA structure

introducing Periodic Boundary.

References

[1] Anirban Kundu, Ruma Dutta, Debajyoti Mukhopadhyay,

“Generation of SMACA and its Application in Web

Services,” 9th International Conference on Parallel

Computing Technologies, PaCT 2007 Proceedings, Russia,

September 3-7, 2007.

[2] S. Chattopadhyay, S. Adhikari, S. Sengupta, M. Pal,

“Highly regular, modular, and cascadable design of cellular

automata-based pattern classifier,” IEEE Trans. on VLSI

Systems, Volume 8, No. 6, December 2000.

[3] P. P. Chaudhuri, D. R. Chowdhury, S. Nandi, S.

Chatterjee, “Additive Cellular Automata, Theory and

Applications, Vol.1,” IEEE Computer Society Press, Los

Alamitos, California, vol. ISBN-0-8186-7717-1, 1997.

[4] S. Das, A. Kundu, S. Sen, B. K. Sikdar, P. P. Chaudhuri,

“Non-Linear Celluar Automata Based PRPG Design

(Without Prohibited Pattern Set) In Linear Time

Complexity,” Asian Test Symposium, 78-83, 2003.

[5] S. Das, A. Kundu, B. K. Sikdar, “Nonlinear CA Based

Design of Test Set Generator Targeting Pseudo-Random

Pattern Resistant Faults,” Asian Test Symposium, 196-201,

2004.

[6] P. Maji, C. Shaw, N. Ganguly, B. K. Sikdar, P. P.

Chaudhuri, “Theory and Application of Cellular Automata

For Pattern Classification,” Fundamenta Informaticae, vol.

58, pp. 321-354, December 2003.

[7] S. Wolfram, “Theory and Application of Cellular

Automata,” World Scientific, 1986.

[8] S. Nandi, P. Chaudhuri, “Analysis of Periodic and

Intermediate Boundary 90/150 Cellular Automata,” IEEE

Transactions on Computers, 45(1), 1-11.

[9] J. V. Neumann, “The Theory of Self-Reproducing

Automata,” A. W. Burks, Ed. University of Illinois Press,

Urbana and London, 1966.

[10] Anirban Kundu, Ruma Dutta, Debajyoti

Mukhopadhyay, “Design of SMACA: Synthesis & its

Analysis through Rule Vector Graph for Web based

Application,” International Journal of Intelligent

Information and Database Systems, Vol. 2, No. 4, 2008.

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