Cellular Automata (CA)- Theory & Application

SUSMITA SUR KOLAY

P. PAL CHAUDHURI

I BACKGROUND

Time Frame Major Players Contribution

Early 50’s J. Von Neuman, E.F. Codd, Henrie & Moore , H Yamada & S. Amoroso

Modeling biological systems - cellular models

‘60s & ‘70s A. R. Smith , Hillis, Toffoli Language recognizer, Image Processing

‘80 s S. Wolfram ,Crisp,Vichniac Discrete Lattice,statistical systems, Physical systems

‘87 - ‘96 IIT KGP, Group Additive CA, characterization,applications

‘97 - ‘99 B.E.C Group GF (2p) CA

II CA PRELIMINARIES - GF(2) CA2.1 CA Structure and Rule

(a) 4-Cell Group CA Structure

Non-Group CA structure

0

0

0

0

D

QCell 0 Cell 1 Cell 2 Cell 3

0 1 2 3

(b) CA RULE :

qi t+1 = f (q i-1

t , q it , q i+1

t )

Neighborhood State

111 110 101 100 011 010 001 000

(90) 0 1 0 1 1 0 1 0

(150) 1 0 0 1 0 1 1 0

( c) XOR/ XNOR Rule ListWith XOR With XNOR

Rule 60: qi t+1 = q i-1 t (+) qit

Rule 90: qi t+1 = q i-1 t (+) q i+1 t

Rule 102: qi t+1 = qit (+) q i+1 t

Rule 150: qi t+1 = q i-1

t (+) qit (+) q i+1 t

Rule 170: qi t+1 = q i+1

t

Rule 204: qi t+1 = qit

Rule 240: qi t+1 = q i-1 t

195: qi t+1 = q i-1 t (+) qit

165: qi t+1 = q i-1 t (+) q i+1 t

153: qi t+1 = qit (+) q i+1

t

105: qi t+1 = q i-1

t (+) qit (+) q i+1 t

85: qi t+1 = q i+1

t

51: qi t+1 = qit

15: qi t+1 = q i-1 t

2.2 State Transition Behavior(a) Group CA structure state transition behavior

Non-Group CA structure & behavior

2

14

1211

133

705

8

410

16

9 15

3

10

13

4 11

14

1

5

12

2

7 8

6 9

15

0

T = 0 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0

Ch. Poly :

x4 + x2 + 1

Min Poly : (x2 +x+1)2

No. Of Pred = 2

Cycle Structure = 1 , 1(3) , 2(6)

T =

1 1 0 0

1 1 0 0

0 1 0 1

0 0 1 1Ch. Poly :x4 + x3 + x2

Min Poly : x2(x2+x+1)

No. Of Pred = 2 , Height = 2.

Cycle Structure = 1(3) , 1(1)

Complete Characterizations

• Th1 : A CA is a group CA iff the determinant det T = 1, where T is the characteristic matrix for the CA.

• Th2 : A group CA has cycle lengths of p or factors of p with a non-zero starting state iff det[Tp (+) I] = 0

• Th3 : If d is the dimension of the null space of the characteristic matrix of a non-group CA, then the total numbers of predecessors of the all-zero state(state 0) is 2d

• Th4 : If the number of predecessors of a reachable state and that of the state 0 in a non-complemented CA, are equal.

• Th5 : An exhaustive CA and exhaustive LFSR are isomorphic to each other.

A few important theorems for characterization on height ,cycle length & no. of components.

2.4 List Of Applications

• VLSI Testing

• Data Encryption

• Error Correcting Code Correction

• Testable Synthesis

• Generation of hashing Function

CA for generation of exhaustive PatternsA 4-cell CA : <90 150 90 150>

T = 0 1 0 0

1 1 1 0

0 1 0 1

0 0 1 1

Ch. Poly : x4 + x + 1Factors : (1 0 0 1 1)Minimal Polynomial : (1 0 0 1 1)

Rank of matrix = 4

Depth Of CA = 0

Null Space = ( 0 0 0 0 )

Cycle Structure : 1(1) , 1(15)

Determinant = 1

0

1

5

13

14

15

3

2

9

4

12

11

7

8

10

6

CA for Generation of exhaustive Two Patterns

T =

0 1 0 0 0 0

1 1 1 0 0 0

0 1 1 1 0 0

0 0 1 1 1 0

0 0 0 1 0 1

0 0 0 0 1 1

STG

0

1

6

3 45

50

11 59

………...

State Pattern

(tapping 0,2,4)

000001 000

000011 001

000110 001

…….. …..

S0 0 1 0 0 0 0 S0

S2 = 0 1 1 1 0 0 S1

S4 0 0 0 1 0 1 S2

S3

S4

S5

000 S0 100 S1

= 010 S2 + 110 S3

000 S4 011 S5

Sv = Tv X Sv + T0 X S0

Th. For the given characteristic matrix of a 2n-cell CA and a set of n visible bits , an exhaustive 2-pattern generation is ensured if the rank of the corresponding obscurity matrix is n.

Proof : An arbitrary two-pattern Sv->Sv is obtained iff the following equation is satisfied.

Sv = Tv X Sv + T0 X S0

=> T0 X S0 = Sv - Tv X Sv

= X

where X is the n-dimensional vector (Sv - Tv X Sv). A solution for S0 exists iff

rank [T0] = rank [T0 X]

and is ensured if rank [T0] = n. Hence the theorem.

2.7 GeneralizationLemma : A 2n-cell null boundary CA with any combination of Rule 90 & 150 over the cell is capable of generating exhaustive two-patterns at the cells 0,2,4,…,(2n-1)

Proof : The characteristic matrix of a 2n-cell CA with an arbitrary combination of Rule 90 (I.e. gi = 0) & Rule 150 (gi = 1) is given by :

T = g0 1 0 …. 0

1 g1 1 …. 0

0 1 g2 ….0

……………

0 0 0 ..1..g2n-1

Here, T0 =

1 0 …. 0

1 1 …. 0

0 1 1 .. 0

………..

0 0 …. 1

Obviously, Rank(T0) = n. Hence, the lemma is proved by the prev. th.

2.8 ILLUSTRATIONConsider a six-cell null boundary CA with rule vector (90, 150, 150, 150, 90, 150). The next state function of this is represented by

S0

S1

S2

S3

S4

S5

=

0 1 0 0 0 0

1 1 1 0 0 0

0 1 1 1 0 0

0 0 1 1 1 0

0 0 0 1 0 1

0 0 0 0 1 1

S0

S1

S2

S3

S4

S5

Or, in general, S = T X S

Since, T0 has rank n=3, the CA generates exhaustive two patterns on 0,2,4 bit positions.

VISIBLE BITS : <0,2,4>

0 0 0

Tv = 0 1 0

0 0 0

1 0 0

T0 = 1 1 0

0 1 1

CA Based Test Pattern generator

• Exhaustive two patterns - All possible transitions on Primary inputs (PI’s).

• For circuits with large number of PI’s, tune CATPG for the CUT– For a CUT with n PI’s , use k-cell GF(2p) CA where n <= kp

– Fix the value of p

– Define the CA structure that suits best for testing the CUT.

C U T

…..q1

2 i k

.. 1 2 p

K-cell GF(2p)

GF(2p) CA based CATPG and Experimental results on synchronous circuit.

Circuit Name Number of Input / Output

Fault efficency(%) P Value Test Vector HITECH/PROOFSEfficiency(%)

C880 60/26 100 4 2.4 100 C6288 32/32 99.99 4 .08 99.99 C1908 33/5 99.58 2 4.2 a C499 41/32 98.96 4 .70 a C3540 50/22 98.77 4 3.5 98.59 C1355 41/32 99.49 4 1.8 99.89 C499m 41/32 99.89 4 1.8 99.89 C1355m 41/32 95.77 8 11 96.05 C432 36/7 99.24 4 .40 a C2670 233/140 90.58 8 03 98.76 C7552 207/108 95.73 8 12 98.86 C432m 36/7 98.23 4 4 92.06

S344 9/11 97.87 4 .30 98.4 S349 9/11 97.33 4 .30 98 S713 35/23 100 2 3.6 100 S641 35/24 100 2 4 100 S953 16/23 100 2 .04 100 S1196 14/14 98.47 4 13 100 S35932 35/320 95.40 4 11 99.38 S5378 35/49 87.94 2 8 71.61 S1238 14/14 98.21 4 11.50 100

S1423 17/5 64.00 4 16 38.15

Ic1 19/24 78.14 8 15 100

Ic2 56/50 99.34 8 10 100

* Total no of est vectors = testvectorr x 1000, m= mutant circuit,, a = aborted

TEST RESULTS WITH UNIFORM CA :

Table 2 : Test Results with Hybrid CA

Circuit Name Fault efficency(%) P Value Test Vectors FinalEfficiency(%)

Obspt/No of FF cut

Hitech / ProofEfficiency(%)

C880 100 4 2500 -------- ----- 100 C6288 99.99 4 60 -------- ---- 99.99 C1908 99.58 2 4000 --------- ------ a C499 98.96 4 600 --------- ------ a C3540 98.59 4 3500 -------- ------ 98.59 C1355 99.49 4 1800 -------- ----- 99.89 C499m 99.89 4 2000 -------- ----- 99.89 C1355m 96.05 8 12000 99.24 4/0 96.05 C432 99.24 4 400 ------ ------ a C2670 92.31 16 3500 99.67 14/0 98.76 C7552 96.98 8 12000 98.69 11/0 98.86 C432m 98.55 4 4000 ------- -------- 92.06 S344 100 4 300 -------- ------ 98.40 S349 100 4 300 -------- ------ 98.00 S713 100 4 2000 ------- ---- -- 100 S641 100 4 2000 --------- ------ 100 S953 100 8 20 ------- ------ 100 S1196 98.44 4 12000 -------- ------ 100 S35932 97.20 4 14000 98.00 0/17 99.38 S5378 98.69 2 8000 96.24 0/17 71.61 S1238 98.21 4 10000 ------- ------ 100

S1423 64.35 4 16000 72.68 0/8 38.15

Ic1 78.14 8 14000 86.12 6/0 100

Ic2 100 8 10000 ------ ------- 100

CA based Response Evaluator

C U T

C A R E

CUT OUTPUT :

7-value logic

<000> , <111>, <0^1> , <1V0>,

<0X0>, <1X1>, <0X1>For n PO CUT, CARE : n cell GF(23) CA

- Maximum Length GF(23) CA - minimum aliasing.

- Compare with Golden signature

- Diagnosis ?

DIAGNOSIS• CUT is divided into k number of blocks

B1 B2

B3 B4

…..

…...

C A R E

------------------

CA Classifier

Faulty Block

DIAGNOSIS (Contd.)

• Introduce a fault on jth component /gate of the ith block Bi and generate the signature Bij.

• Design the CA-based classifier based on the given classes.

{{B1},{B2},……….{Bi},…….}

• With Bij as the input, the classifier identifies the faulty block Bi.

IllustrationG6

G4

G8

G11

G10

G1

G2

G3

G7

G4

G9

G5

G1

G2

Fig : t4.v

Test Vectors :

G1 <111> G2 <1V0> G3 <1V0> ;

G1 <111> G2 <000> G3 <0^1> ;

G1 <1V0> G2 <0^1> G3 <111> ;

Wire instance classG3,G7 AND2_0 1

G2 NAND2_2

G9 NAND2_2

G5 NAND2_3 2

G1 NAND2_3

G1 base & NAND2_0

G4 base & NAND2_4

G2 base & NAND2_0, 3

INV1_0

G5 base & NAND2_5

G4 NAND2_5

G1 NAND2_5 4

G6 BUF1_0

4

3

1 2

AND2_0

NAND2_0

NAND2_2

NAND2_4

BUF1_0

INV1_0

NAND2_3

NAND2_5

Faults - detected and diagnosed

GOLDEN SIGNATURE : 10011

Faulty Signature Faults

00100 BLOCK 3 [G1(base),G2(base)]

10000 BLOCK 3 [G1(NAND2_0)]

00011 BLOCK 3 [G2(INV1_0),G5(base)]

01100 BLOCK 3 [G5(NAND2_4)]

CA CLASSIFIER

1

2

3

4

1

23

4

Why Cellular Automata ?• Regular, Modular, Cascadable structure.• Use of GF(2p) CATPG with p and CA structure

tuned to match the test requirement of the circuit. (Implemented for synchronous circuits with promising results)

• CA Toolkit has been developed based on the Theory of Extension Field and analytical study of GF (2p) CA state transition behavior.

• The Toolkit enables– identification of cycle structure of the CA.

– Design a CA, to realize a given state transition behavior

• Diagnosis tool based on Multiple Attractor CA to diagnose the faulty block within the CUT.

THANK YOU ALL !!