1. Cellular Automata (CA) - Theory & Application
SUSMITA SUR KOLAY
P. PAL CHAUDHURI

2. I BACKGROUND Contribution Time Frame Major Players Early 50’s
J. Von Neuman, E.F. Codd, Henrie & Moore , H Yamada & S. Amoroso
Early 50’s
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
IIT KGP, Group
Additive CA, characterization,applications
‘87 - ‘96
‘97 - ‘99
B.E.C Group
GF (2p) CA

3. II CA PRELIMINARIES - GF(2) CA
2.1 CA Structure and Rule
(b) CA RULE :
(a) 4-Cell Group CA Structure
qi t+1 = f (q i-1t , q it , q i+1 t ) Q
Cell 0
Cell 1
Cell 2
Cell 3 D
Non-Group CA structure
Neighborhood State 2 1 3
(90)
(150)

4. ( c) XOR/ XNOR Rule List With 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

5. 2.2 State Transition Behavior
Non-Group CA structure & behavior
(a) Group CA structure state transition behavior 5 10 4 11 6 9 7 2 13 14 15 3 13 5 1 1 3 6 11 12 8 2 10 9 15 14 4 12 7 8
T =
T =
Ch. Poly :
x4 + x2 + 1
Ch. Poly :x4 + x3 + x2
Min Poly : x2(x2+x+1)
Min Poly : (x2 +x+1)2
No. Of Pred = 2 , Height = 2.
No. Of Pred = 2
Cycle Structure = 1 , 1(3) , 2(6)
Cycle Structure = 1(3) , 1(1)

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

7. 2.4 List Of Applications VLSI Testing Data Encryption
Error Correcting Code Correction
Testable Synthesis
Generation of hashing Function

8. CA for generation of exhaustive Patterns
A 4-cell CA : < >
T =
Ch. Poly : x4 + x + 1
Factors : ( )
Minimal Polynomial : ( )
Rank of matrix = 4 1 3 6
Depth Of CA = 0 5 2 11
Null Space = ( ) 13 9 7
Cycle Structure : 1(1) , 1(15) 14 4 8
Determinant = 1 15 12 10

9. CA for Generation of exhaustive Two Patterns
STG
S S0
S2 = S1
S S2 S3 S4 S5
T = 1 3 45 50 6
State Pattern
(tapping 0,2,4)
…… ….. 59 11
000 S S1
= S S3
000 S S5
………...
Sv = Tv X Sv + T0 X S0

10. 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.

11. Generalization
Lemma : 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 :
g0 1 0 …. 0
1 g1 1 …. 0
0 1 g2 ….0
……………
g2n-1
1 0 …. 0
1 1 …. 0
………..
0 0 …. 1
T =
Here, T0 =
Obviously, Rank(T0) = n. Hence, the lemma is proved by the prev. th.

12. 2.8 ILLUSTRATION
Consider a six-cell null boundary CA with rule vector (90, 150, 150, 150, 90, 150). The next state function of this is represented by
VISIBLE BITS : <0,2,4> S0 S1 S2 S3 S4 S5 S0 S1 S2 S3 S4 S5
Tv = =
T0 =
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.

13. 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. p 1 2 .. 2 i k
K-cell GF(2p)
….. 1 q
C U T

14. GF(2p) CA based CATPG and Experimental results on synchronous circuit.
TEST RESULTS WITH UNIFORM CA :

15. Table 2 : Test Results with Hybrid CA

16. CA based Response Evaluator
CUT OUTPUT :
7-value logic
<000> , <111>, <0^1> , <1V0>,
<0X0>, <1X1>, <0X1>
C U T
C A R E
For n PO CUT, CARE : n cell GF(23) CA
- Maximum Length GF(23) CA - minimum aliasing.
- Compare with Golden signature
- Diagnosis ?

17. DIAGNOSIS CUT is divided into k number of blocks ….. …... C A R E
C A R E
CA Classifier
Faulty Block

18. 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.

19. Illustration 4 3 2 1 Wire instance class G3,G7 AND2_0 1 G2 NAND2_2
G1 base & NAND2_0
G4 base & NAND2_4
G2 base & NAND2_0, 3
INV1_0
G5 base & NAND2_5
G4 NAND2_5
G1 NAND2_
G6 BUF1_0 G1 G4 G6 G2
INV1_0
NAND2_4 G4
NAND2_0 G5
NAND2_5 G8 3
BUF1_0 G1
G11 G7
NAND2_3 2 1 G2
G10 G9 G3
NAND2_2
AND2_0
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> ;

20. Faults - detected and diagnosed
GOLDEN SIGNATURE : 10011
Faulty Signature Faults
BLOCK 3 [G1(base),G2(base)]
BLOCK 3 [G1(NAND2_0)]
BLOCK 3 [G2(INV1_0),G5(base)]
BLOCK 3 [G5(NAND2_4)] 1 1 2 2
CA CLASSIFIER 3 3 4 4

21. 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.

22. THANK YOU ALL !!