Cellular Automata CA Theory

1
Cellular Automata (CA)- Theory Application

• SUSMITA SUR KOLAY
• P. PAL CHAUDHURI

2
I BACKGROUND
Contribution
Time Frame
Major Players
J. Von Neuman, E.F. Codd, Henrie Moore , H
Yamada S. Amoroso
Early 50s
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 t1 f (q i-1t , q it , q i1 t )
Q
Cell 0
Cell 1
Cell 2
Cell 3
D
0
0
Non-Group CA structure
Neighborhood State
2
0
1
3
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
0
0
4
( c) XOR/ XNOR Rule List
With XOR
With XNOR
Rule 60 qi t1 q i-1 t () qit Rule 90 qi t1
q i-1 t () q i1 t Rule 102 qi t1 qit ()
q i1 t Rule 150 qi t1 q i-1 t () qit () q
i1 t Rule 170 qi t1 q i1 t Rule 204 qi t1
qit Rule 240 qi t1 q i-1 t
195 qi t1 q i-1 t () qit 165 qi t1 q i-1
t () q i1 t 153 qi t1 qit () q i1 t 105
qi t1 q i-1 t () qit () q i1 t 85 qi t1
q i1 t 51 qi t1 qit 15 qi t1 q i-1 t
5
2.2 State Transition Behavior
Non-Group CA structure behavior
(a) Group CA structure state transition behavior
10
4
11
5
6
9
7
2
13
14
0
15
13
3
5
1
3
1
0
6
12
11
8
2
10
14
9
15
4
12
1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1
0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0
7
8
T
T
Ch. Poly x4 x2 1
Ch. Poly x4 x3 x2
Min Poly x2(x2x1)
Min Poly (x2 x1)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
  detTp () 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
0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1
A 4-cell CA lt90 150 90 150gt
T
Ch. Poly x4 x 1
Factors (1 0 0 1 1)
Minimal Polynomial (1 0 0 1 1)
0
Rank of matrix 4
1
3

6
Depth Of CA 0
5
2
11
Null Space ( 0 0 0 0 )
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
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 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
T
0
1
3
45
6
50
State Pattern
(tapping 0,2,4) 000001 000 000011
001 000110 001 .. ..
11
59
000 S0 100 S1 010 S2
110 S3 000 S4 011 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-gtSv is
obtained iff the following equation is
satisfied. Sv Tv X Sv T0 X S0 gt 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
2.7 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 0 0
0 ..1..g2n-1
1 0 . 0 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 lt0,2,4gt
S0 S1 S2 S3 S4 S5
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
0 0 0 Tv 0
1 0 0 0 0

1 0 0 T0
1 1 0 0 1 1
Or, in general, S T X S
Since, T0 has rank n3, 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 (PIs).
• For circuits with large number of PIs, tune
  CATPG for the CUT
• For a CUT with n PIs , use k-cell GF(2p) CA
  where n lt 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 lt000gt , lt111gt, lt01gt
, lt1V0gt, lt0X0gt, lt1X1gt, lt0X1gt
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

B1
B2
..
B4
...
B3
------------------
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
Wire instance class
4
G3,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 NAND
2_5 4 G6 BUF1_0
G1
G6
G4
G4
G2
NAND2_4
INV1_0
NAND2_0
G5
NAND2_5
G8
3
BUF1_0
G11
G1
G7
2
NAND2_3
G2
1
G10
G9
G3
NAND2_2
AND2_0
Fig t4.v
Test Vectors
G1 lt111gt G2 lt1V0gt G3 lt1V0gt G1 lt111gt G2 lt000gt G3
lt01gt G1 lt1V0gt G2 lt01gt G3 lt111gt
20
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)
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 !!