Fault Collapsing via Functional Dominance

1. Fault Collapsing via Functional Dominance • Vishwani D. Agrawal • Rutgers University, Dept. of ECE, Piscataway, • New Jersey, USA • vishwani02@yahoo.com • http://cm.bell-labs.com/cm/cs/who/va • A. V. S. S. Prasad and M. V. Atre • Agere Systems, Bangalore, India Agrawal et al.: Fault Collapsing

2. Test Vector Generation Flow DUT Generate fault list Collapse fault list Generate test vectors Fault Model Required fault coverage Agrawal et al.: Fault Collapsing

3. a0 = b0 = c0 : Equivalence a1 c1 : Dominance b1 c1 : Dominance Background • Single stuck-at fault model is the most popularly used model. • Two faults f1 and f2 are equivalent if the same tests detect f1 and f2 (f1=f2) • If all tests of fault f2 also detect fault f1, then f1 is said to dominate f2 (f2f1). a0 a1 c0 c1 b0 b1 Agrawal et al.: Fault Collapsing

4. Background • Both equivalence and dominance relations are transitive in nature. [ (f1  f2) and (f2  f3) => (f1  f3) ] • If f1 dominates f2 and f2 dominates f1 then f1 and f2 are equivalent. [ (f1  f2) and (f2  f1) => (f1 = f2) ] • Number of faults in a 2-input AND gate reduces from 6 to 4 (by equivalence) and to 3 (by dominance) collapsing. Example: c6288, #faults =12576 #equ. = 7744 (0.62), #dom. = 5824 (0.46) Agrawal et al.: Fault Collapsing

5. Problem Statement • To devise a new method for fault collapsing with following attributes: • A single procedure for equivalence and dominance • Global analysis (independence from direction, and other choices, in collapsing) • Use functional equivalences and dominances • Hierarchical fault collapsing (collapsing in large circuits using pre-collapsed sub networks) Agrawal et al.: Fault Collapsing

6. Dominance Graph • A fault in the circuit is represented by a node in the graph. • A directed edge from f2 to f1 indicates that f1 dominates f2 (f2 f1). • Edges can represent either structural or functional relations. Agrawal et al.: Fault Collapsing

7. Dominance Matrix • Graph is represented as a connectivity matrix • Each fault is assumed to be equivalent to itself • Treats functional and structural relations identically • (f1  f2) and (f2 f1) => f2 = f1. Appear as symmetrical components in the matrix (e.g., a0,b0,c0) • #faults = 6 (dimension of dominance matrix) 2-input AND gate Agrawal et al.: Fault Collapsing

8. Transitive Closure • Transitive closure (TC) of the dominance matrix gives all dominance relations between faults. • TC is computed by the O(n3) Floyd-Warshall algorithm, where n is the dimension of the dominance matrix. Agrawal et al.: Fault Collapsing

9. F1 F1 F2 F2 F3 F3 F1 F1 1 1 1 1 1 F2 F2 1 1 1 1 F3 F3 1 1 F1 F2 F3 F1 F2 F3 Graph Transitive Closure Transitive Closure • (F1  F2) and (F2  F3) => (F1  F3) Agrawal et al.: Fault Collapsing

10. Transitive closure edges C1 E1 D1 B1 C0 E0 B0 D0 Example A D E B C Dominance Graph A0 A1 Agrawal et al.: Fault Collapsing

11. Functional Dominance f1 Always 0 f0 f2 f1 dominates f2 Agrawal et al.: Fault Collapsing

12. Functional Equivalence f1 Always 0 f0 f2 f1 dominates f2 and f2 dominates f1 Agrawal et al.: Fault Collapsing

13. Functional Equivalence f1 f0 Always 0 f2 f1 Always 0 f2 Agrawal et al.: Fault Collapsing

14. XOR Circuit c1 h1 g1 m0 g0 i1 f1 Functional Equivalences : (c1,f1), (g1,h1,i1), (g0,m0), (d1,f0) and (e1,c0); additional dominances not shown Agrawal et al.: Fault Collapsing

15. XOR Circuit Structural equivalence collapsing 16 faults Agrawal et al.: Fault Collapsing

16. XOR Circuit Functional equivalence collapsing 10 faults Agrawal et al.: Fault Collapsing

17. XOR Circuit Functional dominance collapsing 4 faults Agrawal et al.: Fault Collapsing

18. Design Hierarchy • Large designs are modular and hierarchical. • Advantageous to store the fault information of repeated blocks in a library. • When configured as a library cell the fault list includes cell PI & PO faults for transitivity. Top module B1 B1 B0 C0 C0 C0 C0 C1 C1 Agrawal et al.: Fault Collapsing

19. 8-bit Ripple Carry Adder Agrawal et al.: Fault Collapsing

20. Fault Collapsing Using Functional Dominances of xor Circuit name All faults Agrawal et al.: Fault Collapsing

21. References • A. Lioy, “Looking for Functional Equivalence,” Proc. ITC, 1991, pp. 858-863. • A. V. S. S. Prasad, V. D. Agrawal and M. V. Atre, “A New Algorithm for Global Fault Collapsing into Equivalence and Dominance Sets,” Proc. ITC, 2002, pp. 391-397. • H. Al-Asaad and R. Lee, “Simulation-Based Approximate Global Fault Collapsing,” Proc. Int. Conf. VLSI, 2002, pp. 72-77. • V. D. Agrawal, A. V. S. S. Prasad and M. V. Atre, “Fault Collapsing via Functional Dominance,” Proc. ITC, 2003 (accepted). Agrawal et al.: Fault Collapsing

22. Conclusion • A new algorithm for global fault collapsing • With functional equivalence number of faults for ATPG reduces • Fault set reduced below 25% with functional dominances (Caution: fault coverage not correct when redundant faults are present) • Library based hierarchical fault collapsing is a useful concept • Further studies are being carried out on independent fault sets Agrawal et al.: Fault Collapsing