Sep. 30, 2003 Agrawal: ITC'03 1

Fault Collapsing Via Functional Dominance

Vishwani D. AgrawalRutgers University, ECE Dept., Piscataway, NJ 08854, USA

vishwani02@yahoo.com

http://cm.bell-labs.com/cm/cs/who/va

A. V. S. S. Prasad and Madhusudan V. AtreAgere Systems, Bangalore 560066, India

avssp@agere.com mvatre@agere.com

International Test Conference – ITC’03

Charlotte, NC, Sep. 30 -- Oct. 2, 2003

Sep. 30, 2003 Agrawal: ITC'03 2

Talk Outline

• Problem statement• Introduction to fault collapsing• Functional dominance• Hierarchical fault collapsing

– An example with functional dominance

• Larger examples– Fault collapsing– ATPG

• Conclusion

Sep. 30, 2003 Agrawal: ITC'03 3

Problem Statement

• Reduce the collapsed fault set below 40-60% level – to about 25%.

• Outline of method:– Use hierarchical fault collapsing (ITC’02)– Use functional dominance

Sep. 30, 2003 Agrawal: ITC'03 4

Role of Fault Collapsing

DUT

Generate fault list

Collapse fault list

Generate test vectors

Fault Model

Required fault coverage

Sep. 30, 2003 Agrawal: ITC'03 5

Definitions

• Given– T1 is set of all tests for fault F1– T2 is set of all tests for fault F2

• F1 dominates F2

• F1 and F2 are equivalent

T1 T2

T1=T2

Sep. 30, 2003 Agrawal: ITC'03 6

Structural Equ. and Dom.

• Structural Equivalence

• Structural Dominance

ab

ca0 a1

b0 b1

c0 c1

ab

ca0 a1

b0 b1

c0 c1

Sep. 30, 2003 Agrawal: ITC'03 7

ISCAS’85 Circuits

Circuit name

Total faults

Collapsed faults (collapse ratio)

Equivalence* Dominance**

C17 34 22 (0.65) 16 (0.47)

C432 864 524 (0.61) 449 (0.52)

C499 998 758 (0.76) 706 (0.71)

C1355 2710 1574 (0.58) 1210 (0.45)

C1908 3816 1879 (0.49) 1566 (0.41)

C2670 5276 2747 (0.52) 2318 (0.44)

C3540 7080 3428 (0.48) 2794 (0.39)

C5315 10630 5350 (0.50) 4500 (0.42)

C6288 12576 7744 (0.62) 5824 (0.46)

C7552 15012 7550 (0.50) 6134 (0.41)* Fastest, Gentest, Hitec, TetraMax **Fastest

Sep. 30, 2003 Agrawal: ITC'03 8

Functional Dominance

f1

f0

f2

Always 0

f1 f2 f0 + f1 f2 f0 = 0

T1 T2

Sep. 30, 2003 Agrawal: ITC'03 9

Hierarchical Collapsing

a

e c

a0 a1

b0 b1

c0 c1

d

fd0 d1

f0 f1e0 e1 b

Total faults = 12Structural Equivalence collapsed faults = 8Structural Dominance collapsed faults = 6

Three tests, {00,01,10}, cover all faults

Sep. 30, 2003 Agrawal: ITC'03 10

AND Gate

a

bc

a0 a1

b0 b1

c0 c1

a0 b0

c0

a1 b1

c1

a0

a1

b0

b1

c0

c1

a0

1

1

1

a1

1

b0

1

1

1

b1

1

c0

1

1

1

c1

1

1

1

Dominancegraph

Dominancematrix

Sep. 30, 2003 Agrawal: ITC'03 11

OR Gate

c0 d0

f0

c1 d1

f1

c0

c1

d0

d1

f0

f1

c0

1

c1

1

1

1

d0

1

d1

1

1

1

f0

1

1

1

f1

1

1

1

Dominancegraph

Dominancematrix

c

df

c0 c1

d0 d1

f0 f1

Sep. 30, 2003 Agrawal: ITC'03 12

Fanout

a

e c

a0 a1

b0 b1

c0 c1

d

fd0 d1

f0 f1e0 e1 b

b0 d0

e0

b1 d1

e1

e0

e1

b0

b1

d0

d1

e0

1

e1

1

b0

1

b1

1

d0

1

d1

1

Dominancegraph

Dominancematrix

Sep. 30, 2003 Agrawal: ITC'03 13

Functional Dominances

e0

e1

b0

b1

c0

c1

d0

d1

f0f1

e0

1

1

e1

1

1

1

1

f1

1

1

f01

d1

1

1

fi fk f + fi fk f = 0

a0 a1

b0 b1

c0 c1

d0 d1

f0 f1e0 e1

c1

1

Sep. 30, 2003 Agrawal: ITC'03 14

Dominance matrix of Circuit

a0

a1

b0

b1

c0

c1

a0

1

1

1

a1

1

b0

1

1

1

b1

1

c0

1

1

1

c1

1

1

1d0

d1

f0

f1

1

1

1

d0

1

d1

1

1

1

1

1

f0

1

1

1

1

f1

1

1

1

1

1

e0

e1

e0

1

1

1

e1

1

1

1

1

1

Entries in purple obtained from functional dominance expression.

Sep. 30, 2003 Agrawal: ITC'03 15

Transitive Closure of Dominance matrix

a0

a1

b0

b1

c0

c1

a0

1

1

1

a1

1

b0

1

1

1

b1

1

c0

1

1

1

c1

1

1

1d0

d1

f0

f1

1

1

1

d0

1

d1

1

1

1

1

1

1

f0

1

1

1

1

1

1

f1

1

1

1

1

1

1

e0

e1

e0

1

1

1

1

1

1

e1

1

1

1

1

1

1Entries in orange are added in transitive closure.

Sep. 30, 2003 Agrawal: ITC'03 16

Dominance Fault Collapsing

a0

a1

b0

b1

c0

c1

a0

1

1

1

a1

1

b0

1

1

1

b1

1

c0

1

1

1

c1

1

1

1d0

d1

f0

f1

1

1

1

d0

1

d1

1

1

1

1

1

1

f0

1

1

1

1

1

1

f1

1

1

1

1

1

1

e0

e1

e0

1

1

1

1

1

1

e1

1

1

1

1

1

1

Sep. 30, 2003 Agrawal: ITC'03 17

Dominance Collapsed Set

a

e c

a0 a1

b0 b1

c0 c1

d

fd0 d1

f0 f1e0 e1 b

Total faults = 12Structural Equivalence collapsed faults = 8Structural Dominance collapsed faults = 6Functional dominance collapsed faults = 4

Two tests, {01,10}, cover all faults

Sep. 30, 2003 Agrawal: ITC'03 18

8-bit Ripple Carry Adder (RCA)

Sep. 30, 2003 Agrawal: ITC'03 19

XOR Cell

a

b

c

d

e

f

g

h

i

j

k

m

c0 c1

d0

d1

Functional Dom. examples: d0 j0, k1 g0

Sep. 30, 2003 Agrawal: ITC'03 20

Collapsed Dominance Matrixof XOR Cell

a0

a1

b0

b1

c0

c1

d0

d1

m0

m1

a0

1

1

1

a1

1

1

1

b0

1

11

b1

111

c0

1

c1

1

d0

1

d1

1

m0

1

11

m1

11

1

24x24 matrix is reduced to a 10x10 matrix.

Inputs

Output

Collapsedfaults

Sep. 30, 2003 Agrawal: ITC'03 21

Fault Collapsing Using Functional Dominance

Circuit name

All faultsNumber of collapsed faults

Structural

equivalence

Functional

dominance

xor cell 24 16 (0.67) 4 (0.17)

Full-adder 60 38 (0.63) 14 (0.23)

8-bit adder 466 290 (0.62) 112 (0.24)

C499exp* 2710 1574 (0.58) 586 (0.22)

* C499exp implements C1355 with XOR cells.

Sep. 30, 2003 Agrawal: ITC'03 22

ATPG for 8-Bit Adder

Fillmode

100% coverage ATPG* vectors

290-fault struc. equ. set

112-fault func. dom. set

Don’t care 65 49

0s 35 31

1s 32 27

Random 16 13

* Gentest

Sep. 30, 2003 Agrawal: ITC'03 23

Conclusion• Functional dominances can be found for

small cells and then applied via hierarchical collapsing to large circuits.

• With functional dominances, the number of faults for ATPG reduces to about 25%; usually gives smaller test set.

• Caution: fault coverage may not be correct when the collapsed fault set contains redundant faults; coverage may be evaluated for equivalence collapsed set.

• Reference: Prasad et al., Proc. ITC’02, pp. 391-397.