Fault Equivalence

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 sa0 sa1 AND OR sa0 sa1 INVERTER sa0 sa1 sa0 sa1 NOT sa0 sa1 sa0 sa1 sa0 sa1 sa0 sa1 sa0 sa1 NAND NOR sa0 sa1 sa0 sa0 sa1 sa0 sa1 sa1 sa0 FANOUT sa1

5. 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 sa0 sa1 sa0 sa1 sa0 sa1 sa0 sa1 sa0 sa1 sa0 sa1 20 Collapse ratio = = 0.625 32 Equivalence Example

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. x y z Fault dominace • Combinational circuits • 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

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

9. All tests of F2 F1 s-a-1 001 110 010 000 101 100 F2 s-a-1 011 The only test of F1 s-a-1 s-a-1 s-a-1 A dominance collapsed fault set (after equivalence collapsing) s-a-0 Dominance Example

10. 0 0 1 1 1 0 1 0 0 0 1 1 MINIMAL SETS OF NON-DOMINATING FAULTS FOR TWO-INPUT GATES Or equivalently 0 0 1 1 0 0 1 1 1 1 0 0

11. Dominance Example 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. P M L J CIRCUIT WITH FANOUT-FREE SUBCIRCUITS SHOWN

16. b 0 e 1 0 J h c f a 0 0 0 1 Z 0 0 1 d g L 0 1 0 M 0 EXAMPLE OF AN FANOUT-FREE CIRCUIT WITH SET OF DOMINANCE-REDUCED FAULTS Equivalent to sa1 at the input indominance fault collapsing it is sufficient to consider only the input faults Equivalent to sa0 at the input

18. Total fault sites = 16 Checkpoints ( ) = 10 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.

20. b 0 1 0 e 0,1 c x1 0 1 0 h f 0 1 0 0 d L 1 1 0 M 1 g 1 0 1 P 0 DOMINANCE-REDUCED FAULT LIST

22. b e c b J f h J f h 1 d L L d M M g P Unused EXAMPLE OF PRUNING AND STRIPPING

23. The multiple stuck-fault model Definition : Let Tg be the set of all tests that detect a fault g, we say that a faultf 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. 0/1 1 1/0 1/0 0/1 a b c 1/1if double fault 0 if a single fault d 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

25. A B C D 1/0 0/1 0/1 1/0 1 1 0 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 0/1 1

26. A B C D 11,00 01,00,11 00,11 11,00,10 10,00,11 11,00,01 01,00,11 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 00,11

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. A B C D 1/0 0/1 0/1 1/0 1 1 0 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