1. Fault Collapsing via Functional Dominance
Vishwani D. Agrawal
Rutgers University, Dept. of ECE, Piscataway,
New Jersey, USA
A. V. S. S. Prasad and M. V. Atre
Agere Systems, Bangalore, India
May 15, 2003
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
May 15, 2003
Agrawal et al.: Fault Collapsing

3. Agrawal et al.: Fault Collapsing
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 = b0 = c0 : Equivalence
a1  c1 : Dominance
b1  c1 : Dominance
a0 a1
c0 c1
b0 b1
May 15, 2003
Agrawal et al.: Fault Collapsing

4. Agrawal et al.: Fault Collapsing
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 = #equ. = 7744 (0.62), #dom. = 5824 (0.46)
May 15, 2003
Agrawal et al.: Fault Collapsing

5. Agrawal et al.: Fault Collapsing
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)
May 15, 2003
Agrawal et al.: Fault Collapsing

6. Agrawal et al.: Fault Collapsing
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.
May 15, 2003
Agrawal et al.: Fault Collapsing

7. Agrawal et al.: Fault Collapsing
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
May 15, 2003
Agrawal et al.: Fault Collapsing

8. Agrawal et al.: Fault Collapsing
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.
May 15, 2003
Agrawal et al.: Fault Collapsing

9. Agrawal et al.: Fault Collapsing
Transitive Closure
(F1  F2) and (F2  F3) => (F1  F3) F1 F2 F3 1
Transitive Closure F1 F2 F3 1
Graph
May 15, 2003
Agrawal et al.: Fault Collapsing

10. Agrawal et al.: Fault Collapsing
Example A D E B C
Dominance Graph E0 E1
Transitive
closure
edges D0 C0 D1 C1 B0 A0 B1 A1
May 15, 2003
Agrawal et al.: Fault Collapsing

11. Agrawal et al.: Fault Collapsing
Functional Dominance f1
Always 0 f0 f2
f1 dominates f2
May 15, 2003
Agrawal et al.: Fault Collapsing

12. Functional Equivalence
Always 0 f0 f2
f1 dominates f2 and f2 dominates f1
May 15, 2003
Agrawal et al.: Fault Collapsing

13. Functional Equivalence
Always 0 f2 f1
Always 0 f2
May 15, 2003
Agrawal et al.: Fault Collapsing

14. Agrawal et al.: Fault Collapsing
XOR Circuit c1 h1 g1 g0 m0 i1 f1
Functional Equivalences : (c1,f1), (g1,h1,i1), (g0,m0),
(d1,f0) and (e1,c0); additional dominances not shown
May 15, 2003
Agrawal et al.: Fault Collapsing

15. Agrawal et al.: Fault Collapsing
XOR Circuit
Structural equivalence collapsing
16 faults
May 15, 2003
Agrawal et al.: Fault Collapsing

16. Agrawal et al.: Fault Collapsing
XOR Circuit
Functional equivalence collapsing
10 faults
May 15, 2003
Agrawal et al.: Fault Collapsing

17. Agrawal et al.: Fault Collapsing
XOR Circuit
Functional dominance collapsing
4 faults
May 15, 2003
Agrawal et al.: Fault Collapsing

18. Agrawal et al.: Fault Collapsing
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
May 15, 2003
Agrawal et al.: Fault Collapsing

19. 8-bit Ripple Carry Adder
May 15, 2003
Agrawal et al.: Fault Collapsing

20. Fault Collapsing Using Functional Dominances of xor
Number of collapsed faults
Flat
structural only
Hierarchical
with functional
Equ.
Dom.
xor cell 24
16(0.63)
13(0.54)
10(0.41)
4(0.17)
Full-adder 60
38(0.63)
30(0.50)
26(0.43)
14(0.23)
8-bit adder
466
290(0.62)
226(0.49)
194(0.42)
112(0.24)
c499exp
2710
1574(0.58)
1210(0.45)
950(0.35)
586(0.22)
Circuit
name
All
faults
May 15, 2003
Agrawal et al.: Fault Collapsing

21. Agrawal et al.: Fault Collapsing
References
A. Lioy, “Looking for Functional Equivalence,” Proc. ITC, 1991, pp
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
H. Al-Asaad and R. Lee, “Simulation-Based Approximate Global Fault Collapsing,” Proc. Int. Conf. VLSI, 2002, pp
V. D. Agrawal, A. V. S. S. Prasad and M. V. Atre, “Fault Collapsing via Functional Dominance,” Proc. ITC, 2003 (accepted).
May 15, 2003
Agrawal et al.: Fault Collapsing

22. Agrawal et al.: Fault Collapsing
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
May 15, 2003
Agrawal et al.: Fault Collapsing