1. April 3, 2003Agrawal: Fault Collapsing1 Hierarchical Fault Collapsing; Functional Equivalences and Dominances Vishwani D. Agrawal Rutgers University, Dept. of ECE vishwani02@yahoo.com http://cm.bell-labs.com/cm/cs/who/va va@agere.com mvatre@agere.com

2. April 3, 2003Agrawal: Fault Collapsing2 Test Vector Generation Flow DUT Generate fault list Collapse fault list Generate test vectors Fault Model Required fault coverage

3. April 3, 2003Agrawal: Fault Collapsing3 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 f1 also detect fault f2, then f2 is said to dominate f1 (f1  f2). a 0 a 1 b 0 b 1 c 0 c 1 a 1  c 1 : Dominance b 1  c 1 : Dominance a 0 = b 0 = c 0 : Equivalence

4. April 3, 2003Agrawal: Fault Collapsing4 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)

5. April 3, 2003Agrawal: Fault Collapsing5 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)

6. April 3, 2003Agrawal: Fault Collapsing6 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. A New Dominance Graph Model

7. April 3, 2003Agrawal: Fault Collapsing7 Computational Model 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., a 0,b 0,c 0 ) #faults = 6 (dimension of dominance matrix) 2-input AND gate

8. April 3, 2003Agrawal: Fault Collapsing8 Transitive Closure Transitive closure (TC) of the dominance matrix gives all dominance relations between faults. TC is computed by the O(n 3 ) Floyd- Warshall algorithm, where n is the dimension of the dominance matrix.

9. April 3, 2003Agrawal: Fault Collapsing9 Transitive Closure (F1  F2) and (F2  F3) => (F1  F3) F1F1 F2F2 F3F3 F1F2F3 F1 11 F2 11 F3 1 Graph F1F1 F2F2 F3F3 F1F2F3 F1 111 F2 11 F3 1 Transitive Closure

10. April 3, 2003Agrawal: Fault Collapsing10 Example A B C D E A0A0 B0B0 D0D0 E0E0 C0C0 A1A1 B1B1 D1D1 E1E1 C1C1 Dominance Graph Transitive closure edges

11. April 3, 2003Agrawal: Fault Collapsing11 Finding Functional Equivalences f0f0 f2f2 Always 0 f1f1 f2f2 f1f1

12. April 3, 2003Agrawal: Fault Collapsing12 XOR Circuit Functional Equivalences : (c 1,f 1 ), (g 1,h 1,i 1 ), (g 0,m 0 ) c1c1 f1f1 g1g1 h1h1 i1i1 g0g0 m0m0 Also (d 1,f 0 ) and (e 1,c 0 ) not used here

13. April 3, 2003Agrawal: Fault Collapsing13 Dominance matrix (XOR) (24x24) Functional equivalences shown as boxed entries

14. April 3, 2003Agrawal: Fault Collapsing14 Transitive Closure (XOR) j 0 k 0 m 1 f 1 f 0 …c 1 a 0

15. April 3, 2003Agrawal: Fault Collapsing15 Results for XOR Circuit #faults#Eq. Faults#Dom. faults 241613 With functional equivalence 101224 #Dom. faults#Eq. Faults#faults

16. April 3, 2003Agrawal: Fault Collapsing16 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 B0 C0 C1 C0 C1

17. April 3, 2003Agrawal: Fault Collapsing17 XOR Library Cell Useful for hierarchical fault collapsing Dimension of the matrix = 14

18. April 3, 2003Agrawal: Fault Collapsing18 8-bit Ripple Carry Adder (RCA)

19. April 3, 2003Agrawal: Fault Collapsing19 Fault Collapsing in 8-bit RCA Using Functional Equivalences Number of collapsed faults Flat structural only Hierarchical with functional Equ.Dom.Equ.Dom. xor cell2416(0.63)13(0.54)12(0.50)10(0.42) Full-adder6038(0.63)30(0.50) 24(0.40) 8-bit adder466290(0.62)226(0.49) 178(0.38) Circuit name All faults

20. April 3, 2003Agrawal: Fault Collapsing20 ISCAS’85 Circuits Circuit name Total faults Equivalence fault set sizeDominance fault set size Graph methodOther programs*Graph methodFastest C173422 16 C432864524 449 C432exp1044560632449503 C499998758 706 C499exp2710115815748981210 C135527101574 1210 C190838161879 1566 C267052762747 23172318 C354070803428 27862794 C5315106305350 44924500 C6288125767744 5824 C7552150127550 61326134 * Fastest, Gentest, Hitec, TetraMax

21. April 3, 2003Agrawal: Fault Collapsing21 Finding Dominances f1f1 f0f0 f2f2 Always 0

22. April 3, 2003Agrawal: Fault Collapsing22 Fault Collapsing in 8-bit RCA Using Functional Dominances Number of collapsed faults Flat structural only Hierarchical with functional Equ.Dom.Equ.Dom. xor cell2416(0.63)13(0.54)10(0.41)4(0.17) Full-adder6038(0.63)30(0.50)26(0.43)14(0.23) 8-bit adder466290(0.62)226(0.49)194(0.42)112(0.24) Circuit name All faults

23. April 3, 2003Agrawal: Fault Collapsing23 Conclusion A new algorithm for global fault collapsing With functional equivalence number of faults for ATPG reduces considerably Further reduction with functional dominances (Caution: fault coverage not correct when redundant faults are present) Library based hierarchical fault collapsing is a new concept Further studies are being carried out on independent fault sets Reference: Prasad et al., ITC-02, pp. 391-397