cellular automata evolution for pattern recognitionpmaji/index_files/pmajippt.pdf · cellular...

Department of Computer Science & Technology,

Bengal Engineering College (DU), Shibpur, Howrah, W. B. India

Cellular Automata Evolutionfor Pattern Recognition

Pradipta Maji Center for Soft Computing Research

Indian Statistical Institute, Kolkata, 700 108, INDIA

Under the supervision of

Prof. P Pal Chaudhuri Prof. Debesh K DasProfessor Emeritus Professor

Dept. of Comp. Sci. & Tech. Dept. of Comp. Sci. & Engg.Bengal Engineering College (DU), Shibpur, INDIA Jadavpur University, INDIA



Introduction

• Cellular Automata (CA) – promising research areaArtificial Intelligence (AI) Artificial Life (ALife)

• Considerable research in –modeling toolimage processinglanguage recognitionpattern recognitionVLSI testing

• Cellular Automata (CA) –learns association from a set of examples apply this knowledge-base to handle unseen cases such associations effective for classifying patterns



Contribution of the Thesis

• Analysis and synthesis of linear boolean CA (MACA) – CA with only XOR logicapplication of MACA in pattern recognition• data mining• image compression• fault diagnosis of electronic circuit

• Analysis and synthesis of non-linear boolean CA (GMACA) – CA with all possible logicapplication of non-linear CA in pattern recognition

• Analysis and synthesis offuzzy CA (FMACA) – CA with fuzzy logicapplication of fuzzy CA in pattern recognition



Cellular Automata (CA)

• A special type of computing model –50’s - J Von Neumann80’s - S. Wolfram

• A CA displays three basic characteristicsSimplicity: Basic unit of CA – cell – is simpleVast parallelism: CA achieves parallelism on a scale larger than massively parallel computersLocality: CA – characterized by local connectivity of its cell – all interactions take place on a purely local basis – a cell can only communicate with its neighboring cells – interconnection links usually carry only a small amount of information – no cell has a global view of the entire system



Cellular Automata (CA)• A computational model with discrete cells updated synchronously• Uniform CA, hybrid / non-uniform CA, null boundary CA, periodic boundary CA

outputInput

Combinational Logic

Clock

From left neighbor

From right neighbor

0/1

2 - state 3-neighborhood CA cell

………..

Each cell can have 256 different rules



CA – State Transition0 0 1 1

0 1 1 1

98 230 226 107

0 0 1 0

98 230 226 107

3

7

For 2nd Cell

Rule 230

PS NS111 1110 1101 1100 0011 0010 1001 1000 0 2



Different Types of CA

• Linear CA– Based on XOR logic– Total 7 rules (60, 90, 102, 150, 170, 204, 240) – Can be expressed through matrix (T), characteristic polynomial– Next state of the CA cell P(t+1) = T. P(t)

• Additive CA– Based on XOR and XNOR logic– Total 14 rules (linear rules + 195,165,153,105,85,51,15)– Can be expressed through matrix, inversion vector, characteristic polynomial– The next state of the CA cell P(t+1) = T. P(t) + F

60 102 150 2041 0 0 00 1 1 00 1 1 10 0 0 1

T =

60 153 105 204 0 1 1 0F =



Additive Cellular AutomataXOR Logic XNOR Logic

Rule 60 : qI(t+1) = qI-1(t) ⊕ qI(t) Rule 195 : qI(t+1) = qI-1(t) ⊕ qI(t)

Rule 90 : qI(t+1) = qI-1(t) ⊕ qI+1(t) Rule 165 : qI(t+1) = qI-1(t) ⊕ qI+1(t)

Rule 102 : qI(t+1) = qI(t) ⊕ qI+1(t) Rule 153 : qI(t+1) = qI(t) ⊕ qI+1(t)

Rule 150 : qI(t+1) = qI-1(t) ⊕ qI(t) ⊕ qI-1(t) Rule 105 : qI(t+1) = qI-1(t) ⊕ qI(t) ⊕ qI-1(t)

Rule 170 : qI(t+1) = qI-1(t) Rule 85 : qI(t+1) = qI-1(t)

Rule 204 : qI(t+1) = qI (t) Rule 51 : qI(t+1) = qI (t)

Rule 240 : qI(t+1) = qI+1(t) Rule 15 : qI(t+1) = qI+1(t)



CA - State Transition Diagram

9 15

613

7 12

3 1411

52 8

1 410

0

Group CA 5

15

10

0

4

14

11

1

2

7

13

8

3

6

12

9

Non-group CA

Associative Memory

Non-group Cellular Automata

Linear Non-linear Fuzzy

MACA GMACA FMACA Perform pattern recognition task



Pattern Recognition

• Pattern Recognition/Classificationmost important foundation stone of knowledge extraction methodologydemands automatic identification of patterns of interest (objects, images) from its background (shapes, forms, outlines, etc)conventional approach – machine compares given input pattern with each of stored patterns – identifies the closest matchtime to recognize the closest match – O(k) – recognition slow

• Associative Memory Entire state space - divided into some pivotal points

Transient

Transient

Transient

States close to pivot -associated with that pivot

Time to recognize a pattern -Independent of number of stored patterns1. MACA (linear) 2. GMACA (non-linear) 3. FMACA (fuzzy)



Multiple Attractor CA (MACA)Employs linear CA rules State with self loop – attractorTransient states and attractor – form attractor basinBehaves as an associative memory Forms natural clusters

10001 01001

1100010000 01000

10010 10011

1101101010 01011

00001 00000 11001 00010 00011 11010

10100 10101

1110001100 01101

10110 10111

1111101110 01111

00101 00100 11101 00110 00111 11110



Multiple Attractor CA (MACA)

• Next state: P(t+1) = T. P(t)

• Characteristic Polynomial: X (n-m) (1+X)m where m=log2(k)• n denotes number of CA cell• k denotes number of attractor basins

• Depth d of MACA –• number of edges between a non-reachable state and an attractor state

• Attractor of a basin: P(t+d) = Td P(t)

• m-bit positions pseudo-exhaustive: extract PEF (pseudo-exhaustive field) from attractor state

• Problem:

Complexity of identification of attractor basin is O(n3)Exponential search spaceRedundant solutions



Chromosome of Genetic Algorithm

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

T =Matrix

1 1 1 1 0 … … .. 1 0

102 60 204 204 240

Characteristic polynomial x3(1+x)2 Elementary divisorsx3(1+x)2 ……… x2 (1+x) (1+x) x

Rule vector

x2 (1+x) ( x1+x)



Dependency Vector/Dependency String

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

T =

Characteristic polynomial x3(1+x)2

Matrix T is obtained from T1 and T2 by Block Diagonal Method

1 1 01 1 00 0 1

T1 =

1 01 0

T2 =

Characteristic polynomial x2(1+x)

Characteristic polynomial x(1+x)




0 0 0 0 00 0 0 0 00 0 1 0 00 0 0 1 00 0 0 1 0

Td =

Dependency Vector DV1 = < 0 0 1 >

0 0 00 0 00 0 1

T1d =

1 01 0

T2d =

Dependency Vector DV1 = < 1 0 >

Dependency String DS = < 0 0 1 > < 1 0 >

Dependency String DS = < 0 0 1 2 0 >

0 0 1 2 0




0 00 11 01 1

Zero basin of T1 + Zero basin of T2 -----Zero basin of T1 + Non-zero basin of T2 ---

Non-zero basin of T1 + Zero basin of T2 ----Non-zero basin of T1 + Non-zero basin of T2 ---

PEF Bits

DV1 contributes 1st PEF Bits DV2 contributes 2nd PEF BitsPEF = [PEF1] [PEF2] = [DS.P] = [DV1.P1] [DV2.P2]

P = [ 1 1 1 1 1 ] DS = [ 0 0 1 2 0 ]

PEF = [PEF1][PEF2] = [<0 0 1><1 1 1>][<1 0><1 1>] = 1 1



Matrix/Rule from Dependency String

• ‘0’ in DV T1 = [ 0 ]

• ’11’ in DV T3 = [T1] 1

0 0

[T1] 1 0

0 1 1

0 1 1

• ‘1’ in DV T2 = [ 1 ]

• ’101’ in DV T4 =

DV = < 1 1 1 >

T2 =1 1 0

0 0 1

0 0 0

DS = < 1 0 1 1 2 2 2>

T =

1 1 0 0 0 0 0

0 1 1 0 0 0 0

0 1 1 1 0 0 0

0 0 0 0 0 0 0

0 0 0 0 1 1 0

0 0 0 0 0 0 1

0 0 0 0 0 0 0

CA Rule Vector

<102, 102, 150, 0, 102, 170, 0>

DV = < 1 0 1 1 >

T1 =

1 1 0 0

0 1 1 0

0 1 1 1

0 0 0 0




Characteristic polynomial x4(1+x)

0 1 0 0 00 0 0 0 00 1 1 1 00 0 0 0 00 0 0 1 0

T1 =

0 1 0 0 00 0 1 0 00 1 1 1 00 0 1 0 00 0 0 1 0

T2 =

0 1 0 0 00 0 1 0 00 1 1 1 00 0 1 0 00 0 0 0 0

T3 =

<170, 0, 150, 0, 240> <170, 0, 150, 0, 240> <170, 0, 150, 0, 240>

Dependency Vector <0 1 1 1 0>Identification of attractor basins in O(n)

Reduction of search space



Image CompressionBlock diagram of codebook generation scheme

Training Images

8 X 8 Set

4 X 4 Set

16 X 16 Set

TSVQ

8 X 8 Codebook

4 X 4 Codebook

16 X 16 Codebook

Spatial Domain

High Compression ratio

Acceptable image quality

Applications - Human Portraits



Tree-Structured Vector Quantization

N X N Set

Cluster 1 Cluster 2

Centroid 1 Centroid 2

S1, S2, S3, S4

S1, S2 S3, S4

S1 S2 S3 S4

Clusters and centroids generation using Tree-Structured Vector Quantization (TSVQ)

Logical structure of multi-class classifier equivalent to PTSVQ



MACA Based Two Stage Classifier

OutputLayer

Cla

ssif

ier

1

Cla

ssif

ier

2 Classifier 1: n-bit DS consists of m DVs

Classifier 2: m-bit DV

HiddenLayerInput

Layer

0.0280.00915

0.9790.9195300

0.0190.00615

0.9690.8845200

0.0100.00315

0.9420.7955100

Memory Size RatioSoftware Hardware

Value of PEF (m)

No of Bits (n)

MSR (software) = (n+m) / (n+2m)

MSR (hardware) = (3n+3m-4) / (3n-2+2m)



Image CompressionOriginal Decompressed

compression 96.43%, PSNR 32.81

compression 95.66%, PSNR 34.27

0.041020.131920.194116 X 16

0.013670.033120.04738 X 8

0.005620.008240.01214 X 4

CATSVQFull Search

Block Size

Execution Time (in milli seconds)

High compression

Acceptable image quality

Higher speed



MACA based Tree-Structured Classifier

I

MACA1

IV

IVII

III

MACA2 MACA3

I II III

MACA4

I II IV

Selection of MACA:

Diversity of ith attractor basin (node): Mi = max{Nij } / ∑j Nijwhere Nij - number of tuples of class j covered by ith attractor basin

Mi ≈ 1, ith attractor indicates class attractor indicates class j j for whichfor which Nij is maximumFigure of Merit: FM = 1/k ∑i Mi

where k denotes number of attractor basins



Fault Diagnosis of Digital CircuitFault

Injection Diagnosis of an example CUT ‘EC’

0.442695.1299.9410S6669(55,6358)

0.930879.5497.058S3271(14,2585)

0.750489.6392.4310S4863(16,4123)

0.223898.8198.966C7552(108,7053)

0.210499.3399.726C6288(32,7648)

0.910696.0398.836C1908(25,1801)

MemoryDictionaryMACA# PartitionCUT (n,p)

Module 1

Module 2

CUT ‘EC’Set of

Test Vectors

Fault Injection

S

ASignature

SetPattern

Classifier MACA



Fault Diagnosis of Analog Circuit

99.87756035610C2

99.83954215430C1

99.88654245430OTA2

99.91889628970OTA1

SRNot detectedDetected# SamplesComponent

OTA2OTA1

C2C1Vin

Vout

OTA2OTA1

Vin

X1

OTA3

X2

X1: Output of BPFX2: Output of LPF

97.6410242194321C2

97.3011641814297C1

99.88544364441OTA3

99.651543064321OTA2

99.89871937201OTA1

SRNot detectedDetected# SamplesComponent



Performance on STATLOG Dataset

Memory Overhead (Kbyte)Classification Accuracy (%)

89.594.494.696.7Segment

78.779.368.572.9Vehicle

84.467.286.687.4Letter

94.199.999.999.9Shuttle

77.586.285.285.4Satimage

86.680.779.380.1Heart

74.667.167.466.8German

87.991.493.390.3DNA

75.975.374.272.9Diabetes

86.584.785.883.4Australian

MACAMLPC4.5BayesianDataset

24.512.74370.42121.89

29.831.3272.1447.02

354.112.571299.28766

55.170.711513.571500

222.7411.33709.72669

3.171.5219.469.8

17.4213.6499.3149

50.8637.221067.961000

9.960.4427.1514

8.041.9337.8519

MACAMLPC4.5Bayesian



MACATree VS C4.5 on Statlog Dataset

24.51370.4289.594.6Segment

29.8372.1478.768.5Vehicle

354.111299.2884.486.6Letter

55.171513.5794.199.9Shuttle

222.74709.7277.585.2Satimage

3.1719.4686.679.3Heart

17.4299.3174.667.4German

50.861067.9687.993.3DNA

9.9627.1575.974.2Diabetes

8.0437.8586.585.8Australian

Memory OverheadC4.5 MACA

Classifn. AccuracyC4.5 MACA

STATLOGDataset

58250839.782

28196350.8139

563314352524.62107

6633398547.249

49432255199.2433

1718198.633

19196749.8134

494655124.3127

10173934.239

446127.335

Retrieval Time(ms)C4.5 MACA

No of NodesC4.5 MACA

Comparable classification accuracy

Low memory overhead

Lesser number of intermediate nodes

Lesser retrieval time



Conclusion

• Advantages:– Explore computational capability of MACA– Introduction of Dependency Vector (DV)/String (DS) to characterize MACA– Reduction of complexity to identify attractor basins from O(n3) to O(n)– Elegant evolutionary algorithm – combination of DV/DS and GA– MACA based tree-structured pattern classifier– Application of MACA in

• Classification• image compression• fault diagnosis of electronic circuits• Codon to amino acid mapping, S-box of AES

• Problems:– Linear MACA – employs only XOR logic, functionally incomplete– Distribution of each attractor basin is even – Can handle only binary patterns

• Solutions:– Nonlinear MACA (GMACA) – Fuzzy MACA (FMACA)



Generalized MACA (GMACA)

• Employs non-linear hybrid rules with all possible logic• Cycle or attractor length greater than 1• Can perform pattern recognition task• Behaves as an associative memory

0100 1000

1010 0001

0101

0011

0010

0000

11010111

1100 1001

10110110

1110

1111

P1 attractor-1

P2 attractor-2

Rule vector:<202,168,218,42>



Basins of Attraction (Theoretical)

n = 50

k = 10Error correcting capability

at multiple bit noise

Error correcting capability at single bit noise



Distribution of CA Rule (Theoretical)

More homogeneous – less probability of occurrence

Degree of Homogeneity

DH = | 1- r/4 | where r = number of 1’s of a rule



Synthesis of GMACA

0100 1000

0001

0010 0000

Basin 1

Basin

1110 11011011

0111

1111

Basin 2

Next State

01110111011111111111

Present State Next StatePhase I: Random Generation of a directed sub-graph

0100 00011 1000 0001Phase II: State transition table

from sub-graph 0001 00000000 00000010 0000

Phase III: GMACA rule vector from State transition table

g1For 2nd Cell:-

111 1110 1101 1100 0011 1 010 0001 0000 0

Rule 232

Basin Present State

1110

2 1011110101111111

g2



Genetic AlgorithmResolution of Collision:

if n0 = n1; next state is either `0’ or `1’

if n0 > n1; next state is `0’

if n0 < n1; next state is `1’

where n0 = Occurrence of state `0’ for a configuration

n1 = Occurrence of state `1’

g1 g2 ….. gkExample chromosome format – each gx a basin of a pattern Px

k numbers of genes in a chromosome

Each gene - a single cycle directed sub-graph with p number of nodes, where p = 1 + n



Maximum Permissible Noise/Height

Minimum value of maximum permissible height hmax = 2

Minimum value of maximum permissible noise rmax = 1



Performance Analysis of GMACA

Higher memorizing capacity than Hopfield network

Cost of computation is constant –depends on transient length of CA



Basins of Attraction (Experimental)

n = 50 k = 10



Distribution of CA Rule (Experimental)



Conclusion

• Advantages:– Explore computational capability of non-linear MACA– Characterization of basins of attraction of GMACA– Fundamental results to characterize GMACA rules– Reverse engineering method to synthesize GMACA– Combination of reverse engineering method and GA– Higher memorizing capacity than Hopfield network

• Problems:– Can handle only binary patterns

• Solutions:– Fuzzy MACA (FMACA)



Fuzzy Cellular Automata (FCA)

• A linear array of cells• Each cell assumes a state - a rational value in [0, 1]• Combines both fuzzy logic and Cellular Automata• Out of 256 rules, 16 rules are OR and NOR rules (including 0 and 255)

Boolean Function Opeartion FCA Operation

OR (a + b) min{1, (a + b)}

AND (a.b) (a.b)

NOT (~a) (1 – a)



Fuzzy Cellular Automata (FCA)OR Logic NOR Logic

Rule 170 : qI(t+1) = qI-1(t) Rule 85 : qI(t+1) = qI-1(t)

Rule 204 : qI(t+1) = qI (t) Rule 51 : qI(t+1) = qI (t)

Rule 238 : qI(t+1) = qI (t) + qI+1(t) Rule 17 : qI(t+1) = qI (t) + qI+1(t)

Rule 240 : qI(t+1) = qI+1(t) Rule 15 : qI(t+1) = qI+1(t)

Rule 250 : qI(t+1) = qI-1(t) + qI+1(t) Rule 5 : qI(t+1) = qI-1(t) + qI+1(t)

Rule 252 : qI(t+1) = qI-1(t) + qI (t) Rule 3 : qI(t+1) = qI-1(t) + qI (t)

Rule 254 : qI(t+1) = qI-1(t) + qI (t) + qI+1(t) Rule 1 : qI(t+1) = qI-1(t) + qI (t) + qI+1(t)



Fuzzy Cellular Automata (FCA)

• 16 OR and NOR rules can be represented by n x n matrix T and an ndimensional binary vector F

• Si(t) represents the state of ith cell at tth time instantSi(t+1) = | Fi - min{1, Σj Tij.Sj(t)}|

where Tij = 1 if next state of ith cell dependents on jth cell

0 otherwise F = Inversion vector, contains 1 where NOR rule is applied

4-cell null boundary hybrid FCA

<238,1,238,3>T =

1 1 0 0

1 1 1 0

0 0 1 1

0 0 1 1

F =

0

1

0

1



Fuzzy Multiple Attractor CA (FMACA)




• Dependency Vector (DV) – corresponding to matrix

• Derived Complement Vector (DCV) – corresponding to inversion vector

• Pivot cell (PC) – represents an attractor basin uniquely

• State of Pivot Cell (PC) of attractor of the basin where a statebelongs

qm = min {1, Σj |DCVj - DVj.Sj(t)|}

• Size of attractor basins – equal as well as unequal

• Matrix, inversion vector from DV/DCV




1.0 0.0 1.0

1.0 0.5 1.00.5 0.5 1.0

0.5 1.0 1.0

1.0 1.0 1.0

0.0 1.0 1.0

1.0 1.0 0.0

0.0 0.5 0.50.5 0.0 0.5

0.5 0.5 0.0

0.5 0.0 1.0

0.0 0.5 1.0 0.5 1.0 0.0 0.0 1.0 0.0

0.5 0.0 0.0

0.0 0.5 0.0

1.0 0.0 0.0

0.0 0.0 0.5

0.0 0.0 0.0

0.0 0.0 1.0

1.0 0.5 0.0

1.0 1.0 0.50.5 0.5 0.5

0.0 1.0 0.5

0.5 1.0 0.5

1.0 0.5 0.5

1.0 0.0 0.5

T = 0 0 1

1 1 0

0 0 0

0

0

0

1

1

1

0

0

0 F = DV = DCV =




1.0 0.0 0.0

1.0 0.5 0.00.5 0.5 0.0

0.5 1.0 0.0

1.0 1.0 0.0

0.0 1.0 0.0

1.0 1.0 1.0

0.0 0.5 0.50.5 0.0 0.5

0.5 0.5 1.0

0.5 0.0 0.0

0.0 0.5 0.0 0.5 1.0 1.0 0.0 1.0 1.0

0.5 0.0 1.0

0.0 0.5 1.0

1.0 0.0 1.0

0.0 0.0 0.5

0.0 0.0 1.0

0.0 0.0 0.0

1.0 0.5 1.0

1.0 1.0 0.50.5 0.5 0.5

0.0 1.0 0.5

0.5 1.0 0.5

1.0 0.5 0.5

1.0 0.0 0.5

T = 0 0 1

1 1 0

0 0 0

0

1

1

1

1

1

0

0

1 F = DV = DCV =



FMACA based Tree-Structured Classifier

I

FMACA1

IV

IVII

III

FMACA2 FMACA3

I II III

FMACA4

I II IV

• FMACA based tree-structured pattern classifier

• Can handle binary as well as real valued datasets

• Provides equal and unequal size of attractor basins

• Combination of GA and DV/DCV



Experimental Setup

• Randomly generate K number of centroids• Around each centroid, generate t number of tuples• 50 % patterns are taken for training• 50 % patterns are taken for testing

A2A1

AK

dmax

Dmin



Performance Analysis of FMACA

Dataset Depth TrainingAccuracy

TestingAccuracy

Breadth

(n=5,K=2,t=4000) 12

16.981.8

15.680.9

121

34

97.798.4

91.692.4

359Can generalize dataset irrespective of classes,

tuples, attributes

• Generalization of FMACA tree

– Depth: Number of layers from root to leaf

– Breadth: Number of intermediate nodes

Attributes (n) Size (t) No of Classes FMACA C4.5

6 6000 48

96.6889.26

93.1081.60

8 10000 610

85.9185.61

79.6073.93

10 10000 48

83.8274.01

77.1066.90

Higher classification accuracy compared to C4.5



Performance Analysis of FMACA

High generation time - but, one time cost

Lower retrieval time compared to C4.5

Generation Time (ms) Retrieval Time (ms)Dataset

FMACA C4.5 FMACA C4.5(n=5,k=2,t=2000) 14215 273 3 306

(n=5,k=2,t=20000) 52557 756 80 812

(n=6,k=2,t=2000) 722725 162 4 259

(n=6,k=2,t=20000) 252458 791 35 874

Attributes (n) Size (t) No of Classes FMACA C4.5

6 6000 48

13621331

51096932

8 8000 68

12611532

95198943

10 10000 610

13641481

77277984

Lower memory overhead compared to C4.5



Performance on STATLOG Dataset

Classification Accuracy (%)

85.193.181.988. 1

FMACA

84.4Letter94.1Shuttle77.5Satimage87.9DNA

MACADataset

16

936180

FMACA

1008

567540180

MACANumber of CA cells

661.8

51.8161.6122.1

FMACA

524.6

47.2199.2124.3MACA

No of Nodes of Tree

261.0818.29189.6251.22

FMACA

354.11Letter55.17Shuttle222.74Satimage50.86DNAMACADataset

Memory Overhead

1382

19822419491

FMACA

5633

66334943494

MACARetrieval Time (ms)

Comparable accuracy

Lesser CA cells

Lesser memory overhead

Lesser retrieval time



Conclusion

Introduction of fuzzy CA in pattern recognition

New mathematical tools –• Dependency matrix, Dependency vector• Complement vector, Derived complement vector

Reduction of complexity to identify attractors from O(n3) to O(n)

Both equal and unequal size of attractor basins

Movement of patterns from one to another basin

Reduction of search space

Elegant evolutionary algorithm – combination of DV/DCV and GA

FMACA based tree-structured pattern classifier



Future Extensions

Applications in pattern clustering, mix-mode learning

Theoretical analysis of memorizing capacity of non-linear CA

Combination of fuzzy set and fuzzy CA 1-D CA to 2-D CA

Development of hybrid systems using CA• CA + neural network + fuzzy set• CA + fuzzy set + rough set

Boolean CA to multi-valued / hierarchical CA

Application of CA in• Bioinformatics• Medical Image Analysis• Image Compression• Data Mining



Thank You

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