2. Fault Equivalence
Number of fault sites in a Boolean gate circuit is = #PI + #gates + # (fanout branches)
Fault equivalence: Two faults f1 and f2 are equivalent if all tests that detect f1 also detect f2.
If faults f1 and f2 are equivalent then the corresponding faulty functions are identical.
Fault collapsing: All single faults of a logic circuit can be divided into disjoint equivalence subsets, where all faults in a subset are mutually equivalent.
A collapsed fault set contains one fault from each equivalence subset.

3. Fault equivalence & collapsing
Combinational circuits
faults f and g are equivalent iff Zf(x) = Zg(x)
equivalent faults are not distinguishable
For gate with controlling value c and inversion i :
all input sac faults and output sa(c  i) faults are equivalent

4. Equivalence Rules WIRE/BUFFER sa0 sa0 sa0 sa1 sa0 sa1 sa1 sa1 sa0 sa1
AND OR
sa0 sa1
sa0 sa1
INVERTER
sa0
sa1
NOT
sa1
sa0
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
NAND
NOR
sa0
sa1
sa0 sa1
sa0 sa1
sa0
sa1
sa0
FANOUT
sa1

5. Equivalence Example Faults in red removed by sa0 sa1 equivalence
collapsing 20
Collapse ratio = = 0.625 32

6. Fault Dominance
If all tests of some fault F1 detect another fault F2, then F2 is said to dominate F1.
Dominance fault collapsing: If fault F2 dominates F1, then F2 is removed from the fault list.
Any set that detects F1 also detects F2 (that dominates F1)
When dominance fault collapsing is used, it is sufficient to consider only the input faults of Boolean gates.
In a tree circuit (without fanouts) PI faults form a dominance collapsed fault set.

7. Fault dominace Combinational circuits Example :
If f dominates g => any test that detects g will also detect f .
Therefore, only dominating faults must be detected
Example :
[x, y]=[1 0] is the only test to detect
f1 = y sa1,
Since it also detects
f2 = z sa0 => f2 dominates x y z

8. Fault dominance & collapsing
For gate with controlling value c & inversion i, the output sa(c’i) dominates any input sac’
sequential circuits dominance fault collapsing is not useful

9. A dominance collapsed fault set (after equivalence collapsing)
Dominance Example
s-a-1 F1 F2
001
000
101
100
011
All tests of F2
The only test of F1
s-a-0
A dominance collapsed fault set
(after equivalence collapsing)

10. MINIMAL SETS OF NON-DOMINATING FAULTS FOR TWO-INPUT GATES1
Or equivalently 1 1 1 1 1 1

11. Dominance Example Faults in red removed by equivalence collapsing
sa0 sa1
Faults in red
removed by
equivalence
collapsing
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
Faults in green
removed by
dominance
collapsing
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1
sa0 sa1 15
Collapse ratio = ── = 0.47 32

13. CIRCUIT WITH FANOUT-FREE SUBCIRCUITS SHOWNM P L J

16. EXAMPLE OF AN FANOUT-FREE CIRCUIT WITH SET OF DOMINANCE-REDUCED FAULTS
Equivalent to sa1 at the input b c d e f g h M L Z J 1 a
in dominance fault collapsing it is sufficient to consider only the input faults
Equivalent to sa0 at the input

18. Checkpoint Theorem
Primary inputs and fanout branches of a combinational circuit are called checkpoints.
Checkpoint theorem: A test set that detects all single (multiple) stuck-at faults on all checkpoints of a combinational circuit, also detects all single (multiple) stuck-at faults in that circuit.
Total fault sites = 16
Checkpoints ( ) = 10

20. DOMINANCE-REDUCED FAULT LISTx1 b c e f d h g P M L 1
0,1

22. EXAMPLE OF PRUNING AND STRIPPING1 d f e b c g J h M P L
Unused b f d h J M L

23. The multiple stuck-fault model
Definition :
Let Tg be the set of all tests that detect a fault g, we say that a fault f functionally masks the fault g iff the multiple fault {f, g} is not detected by any test in Tg
If f masks g then {f, g} is not detected by t Tg
but it may be detected by other tests

24. The multiple stuck-fault model
Example :
Consider the faults c sa0, a sa1
t= 011 is the only test that detects fault c sa0
but t does not detect {c sa0, a sa1} =>
a sa1 masks c sa0 a b c d
0/1 1
1/0
1/1 if double fault
0 if a single fault

25. The multiple stuck-fault model
Example :
Test set T= {1111, 0111, 1110, 1001, 1010, 0101}
detects all SSF in following circuit, but the only
test which detects B sa1 and C sa1 is 1001 1
0/1 A B C D
1/0
0/1 1

26. The multiple stuck-fault model
Example :
Using Rajski’s method
B sa1 detectable if no A sa0
while C sa1 detected with no conditions
11,00
00,11
01,00,11 A B C D
11,00
01,00,11
00,11
11,00,10
10,00,11
11,00,01

27. The multiple stuck-fault model
Properties of MSF(Multiple Stuck Faults)
in a irredundant two_level circuit , any complete test set for SSF detects all MSF
in a fanout-free circuit , any complete test set for SSF detects all double and triple faults & there is a complete test set for SSF that detects all MSF

28. The multiple stuck-fault model
in an internal fanout-free circuit, any complete test for SSF detects at least 98% of MSF with K < 6 and detects all MSF unless C contains a subcircuit with interconnection A B C D
1/0
0/1 1