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April 3, 2003 Agrawal: Fault Collapsing 1
Hierarchical Fault Collapsing; Functional Equivalences
and Dominances
Vishwani D. AgrawalRutgers University, Dept. of ECE
[email protected]://cm.bell-labs.com/cm/cs/who/va
April 3, 2003 Agrawal: Fault Collapsing 2
Test Vector Generation Flow
DUT
Generate fault list
Collapse fault list
Generate test vectors
Fault Model
Required fault coverage
April 3, 2003 Agrawal: Fault Collapsing 3
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 (f1f2).
a0 a1
b0 b1
c0 c1
a1 c1: Dominanceb1 c1: Dominance
a0 = b0 = c0 : Equivalence
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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)
April 3, 2003 Agrawal: 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)
April 3, 2003 Agrawal: Fault Collapsing 6
• 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
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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., a0,b0,c0)
• #faults = 6 (dimension of dominance matrix) 2-input AND gate
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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.
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Transitive Closure
• (F1 F2) and (F2 F3) => (F1 F3)
F1 F2 F3
F1 F2 F3
F1 1 1
F2 1 1
F3 1
Graph
F1 F2 F3
F1 F2 F3
F1 1 1 1
F2 1 1
F3 1
Transitive Closure
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ExampleA
B
C
D
E
A0 B0
D0
E0
C0
A1 B1
D1
E1
C1
Dominance Graph
Transitiveclosureedges
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Finding Functional Equivalences
f0
f2
Always 0
Always 0f1
f2
f1
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XOR Circuit
Functional Equivalences : (c1,f1), (g1,h1,i1), (g0,m0)
c1
f1
g1
h1
i1
g0m0
Also (d1,f0) and (e1,c0) not used here
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Dominance matrix (XOR)(2
4x24)
Functional equivalences shown as boxed entries
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Transitive Closure (XOR)
j0 k0 m1 f1 f0…c1 a0
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Results for XOR Circuit
#faults #Eq. Faults #Dom. faults
24 16 13
With functional equivalence
101224
#Dom. faults#Eq. Faults#faults
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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 B1B0C0 C0
C1
C0 C0
C1
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XOR Library Cell
• Useful for hierarchical fault collapsing• Dimension of the matrix = 14
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8-bit Ripple Carry Adder (RCA)
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Fault Collapsing in 8-bit RCAUsing Functional Equivalences
Number of collapsed faults
Flat
structural only
Hierarchical
with functional
Equ. Dom. Equ. Dom.
xor cell 24 16(0.63) 13(0.54) 12(0.50) 10(0.42)
Full-adder 60 38(0.63) 30(0.50) 30(0.50) 24(0.40)
8-bit adder 466 290(0.62) 226(0.49) 226(0.49) 178(0.38)
Circuit name
All faults
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ISCAS’85 CircuitsCircuit name
Total faults
Equivalence fault set size Dominance fault set size
Graph method Other programs* Graph method Fastest
C17 34 22 22 16 16
C432 864 524 524 449 449
C432exp 1044 560560 632632 449449 503503
C499 998 758 758 706 706
C499exp 2710 11581158 15741574 898898 12101210
C1355 2710 1574 1574 1210 1210
C1908 3816 1879 1879 1566 1566
C2670 5276 2747 2747 23172317 23182318
C3540 7080 3428 3428 27862786 27942794
C5315 10630 5350 5350 44924492 45004500
C6288 12576 7744 7744 5824 5824
C7552 15012 7550 7550 61326132 61346134* Fastest, Gentest, Hitec, TetraMax
April 3, 2003 Agrawal: Fault Collapsing 21
Finding Dominances
f1
f0
f2
Always 0
April 3, 2003 Agrawal: Fault Collapsing 22
Fault Collapsing in 8-bit RCAUsing Functional Dominances
Number of collapsed faults
Flat
structural only
Hierarchical
with functional
Equ. Dom. 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)
Circuit name
All faults
April 3, 2003 Agrawal: Fault Collapsing 23
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
