cellular automata (ca) - theory & application susmita sur kolay p. pal chaudhuri
TRANSCRIPT
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.