1. 1 Fault Nodes in Implication Graph for Equivalence/Dominance Collapsing, and Identifying Untestable and Independent Faults R. Sethuram rms@qualcomm.com M. L. Bushnell bushnell@caip.rutgers.edu V. D. Agrawal vagrawal@eng.auburn.edu

2. 2 Outline  Purpose  Introduction Terms and definitions Implication Graph  The New Functional Fault graph (FFG)  Identify Equivalence/Dominance/Independence and redundant faults using FFG  Results and analysis  Conclusion

3. 3 Purpose  Fault collapsing Reducing ATPG Time Test data volume  Identifying independent faults Compaction  Extension for other fault models  Polynomial-time complexity

4. 4 Definitions  Fault dominance f1 dominates f2 if every test of f2 detects f1  Fault equivalence If f1 and f2 dominate each other  Fault independence If every test of f1 does not detect f2

5. 5 Implication Graph (IG) example  Graph represents Boolean expressions  Nodes represent literals, edges represent implications 1 2 3 1 2 3 1 2 3

6. 6 Operations on IG  Transitive Closure: adds an edge For every path For every common ancestor  Graph condensation 4 2 3 1 5 b d e a f g c C1C1 C2C2 C3C3 C4C4 Condense Strongly Connected Component (SCC)

7. 7 Functional Fault Graphs (FFG)  New fault node to represent the fault detectability status a b c a sa0 a a OaOa a0a0 Complete FFG of a 2-input AND gate

8. 8 Deriving Information  Perform transitive closure and graph condensation. Then, identify:  Equivalence: Fault nodes f 1, f 2, …, f k are in one SCC  Dominance:  Independence:  Untestable faults: f1f1 f2f2 f1f1 f2f2 f1f1 f1f1

9. 9 IGs are incomplete  Observability relationship between stem and its branches cannot be represented  It can be partially overcome If p and q are two signals such that q is the dominator of p then O p  O q pq

10. 10 An example – i 0 dominates d 1 a b c d e f g h i k j m 1. d 1  O d  O g  O m and d 1  O d  b 2. d 1  a  j and d 1  a  I 3. O m  j  O k and O k  b  O i. Hence, d 1  O i 4. d 1  i and d 1  O i. Hence, d 1  i 0

11. 11 Extending for other Fault Models  Adding special nodes and edges in IG enables extending them for other fault models Time frame edges for implication across time frames Edges annotated with multiple bits l SR OlOl lOlOl l +1 01 l SR = Slow to raise fault

12. 12 Results – Fault Collapsing c6288 has several stems and our technique could not identify additional observability related implications. # Dominant Faults

13. 13 Results – Comparison using the circuit c1355 # Dominant Faults

14. 14 Results - Independent fault pairs for c432

15. 15 Results – Untestable delay faults * Only the first K levels of the netlist were analyzed # Transition Delay Faults

16. 16 Conclusion  Proposed and implemented a functional fault graph for equivalence/dominance fault collapsing and identifying independent fault pairs  Requires only polynomial-time algorithms  Can be easily enhanced for other fault models  Reduces fault set size by up to 66%