1. A Primal-Dual Solution to Minimal Test Generation Problem
Mohammed Ashfaq Shukoor
Vishwani D. Agrawal
Auburn University, Department of Electrical and Computer Engineering Auburn, AL 36849, USA
12th IEEE VLSI Design and Test Symposium, 2008, Bangalore
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2. Problem Statement
To find a minimal set of vectors to cover all stuck-at faults in a combinational circuit
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3. A Known Method: Test Minimization ILP[1]
vj is a variable assigned to each of the J vectors with the following meaning:
If vj = 1, then vector j is included in the minimized vector set
If vj = 0, then vector j is not included in the minimized vector set
Objective: minimize
constant akj is 1 only if the fault k is detected by vector j, else it is 0
(1)
Subject to conditions:
Fault
number ( k)
Vector number ( j ) J 1 . 2 3 4 K
(2)
k = 1, 2, , K vj
integer [0, 1], j = 1, 2, , J
(3)
K is the number of faults in a combinational circuit
J is the number of vectors in the unoptimized vector set
[1] P. Drineas and Y. Makris, “Independent Test Sequence Compaction through Integer Programming,” Proc. International Conf. Computer Design, 2003, pp. 380–386.
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4. Motivation
When test minimization is performed over an exhaustive set of vectors, the ILP solution is the smallest possible test set.
For most circuits exhaustive vector sets are impractical.
We need a method to find a non-exhaustive vector set for which the test minimization ILP will give a minimal test set.
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5. Definitions Independent Faults [2]: Independent Fault Set (IFS) [2]:
Two faults are independent if and only if they cannot be detected by the same test vector.
T(f1)
T(f2)
T(f1)
T(f2)
f1 and f2 are independent
f1 and f2 are not independent
Independent Fault Set (IFS) [2]:
An IFS contains faults that are pair-wise independent.
[2] S. B. Akers, C. Joseph, and B. Krishnamurthy, “On the Role of Independent Fault Sets in the Generation of Minimal Test Sets,” Proc. International Test Conf., 1987, pp. 1100–1107.
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6. Independence Graph
Independence graph: Nodes are faults and and an edge between two nodes means that the corresponding faults are independent. Example: c17[3].
An Independent Fault Set (IFS) is a maximum clique in the graph.
Size of IFS is a lower bound on test set size (Akers et al., ITC-87) 1 2 4 5 1 2 4 5 3 10 11 6 7 8 9 6 7 8 9
[3] A. S. Doshi and V. D. Agrawal, “Independence Fault Collapsing,” Proc. 9th VLSI Design and Test Symp., Aug. 2005, pp
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7. New Definitions Conditionally Independent Faults:
Two faults, detectable by vector set V, are conditionally independent with respect to the vector set V if no vector in the set detects both faults.
Conditionally Independent Fault Set (CIFS):
For a given vector set, a subset of all detectable faults in which no pair of faults can be detected by the same vector, is called a conditionally independent fault set (CIFS).
Conditional Independence Graph:
An independence graph in which the independence relations between faults are relative to a vector set is called a conditional independence graph
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8. Primal and Dual Problems[4]
An optimization problem in an application may be viewed from either of two perspectives, the primal problem or the dual problem
These two problems share a common set of coefficients and constants.
If the primal minimizes one objective function of one set of variables then its dual maximizes another objective function of the other set of variables
Duality theorem states that if the primal problem has an optimal solution, then the dual also has an optimal solution, and the optimized values of the two objective functions are equal.
[4] G. Strang, Linear Algebra and Its Applications, Fort Worth: Harcourt Brace Javanovich College Publishers, third edition, 1988.
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9. Dual ILP Formulation maximize Subject to,
integer [0, 1], k = 1, 2, , K fk
j = 1, 2, , J
(5)
(6)
(4)
fk is a variable assigned to each of the K faults with the following meaning,
If fk = 1, then fault k is included in the fault set
If fk = 0, then fault k is not included in the fault set
Theorem 1: A solution of the dual ILP of 4, 5 and 6 provides a largest conditionally independent fault set (CIFS) with respect to the vector set V.
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10. Theorem 2: For a combinational circuit, suppose V1 and V2 are two vector sets such that and V1 detects all detectable faults of the circuit. If CIFS(V1) and CIFS(V2) are the largest CIFS with respect to V1 and V2, respectively, then |CIFS(V1)| ≥ |CIFS(V2)|. 1 2 4 5 6 7 8 9 10 11 3
|CIFS(V1)| = 5
Conditional Independence Graph for vector set V1 1 2 3 4 5 6 7 8 9 10 11
|CIFS(V2)| = 4
Conditional Independence Graph for vector set V2
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11. Primal Dual ILP Algorithm for Test Minimization
Generate an initial vector set to detect all (or most) faults
Obtain diagnostic matrix (conditional independence graph) by fault simulation without fault dropping
Solve dual ILP to determine CIFS. Go to 6 if CIFS has converged
Augment vector set by additional tests for CIFS
Go to step 2
Solve primal ILP for final compacted vector set
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12. Example 1: c1355 Problem type Iteration number No. of vectors ATPG
CPU s
Fault sim.
CIFS
size
No. of min. vectors
ILP
Dual 1 2 3
114
507
903
0.033
0.085
0.333
1.517
2.683 85 84
0.24
0.97
1.91
Primal
3.38
SUN Fire 280R, 900 MHz Dual Core machine
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13. Example 2: c2670 Problem type Iteration number No. of vectors ATPG
CPU s
Fault sim.
CIFS
size
No. of min. vectors
ILP
Dual 1 2 3 4 5 6 7 8 9 10 11 12
194
684
1039
1424
1738
2111
2479
2836
3192
3537
3870
4200
2.167
1.258
1.176
1.168
1.136
1.128
1.112
1.086
1.073
1.033
1.048
3.670
5.690
6.895
8.683
10.467
12.333
14.183
15.933
17.717
19.267
20.983
22.600
102 82 79 78 76 74 73 72 70
1.99
3.22
7.90
3.69
5.89
7.43
7.16
8.45
9.81
10.90
12.02
13.44
Primal
316.52
SUN Fire 280R, 900 MHz Dual Core machine
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14. Comparing primal_LP–dual_ILP solution with LP-alone solution
Circuit Name
Lower bound on vectors
LP-alone minimization [5]
Primal-dual minimization [this paper]
Unopt. vectors
LP CPU s
Minimized vectors
Total CPU s
c432 27
608
2.00 36
983
5.52 31
c499 52
379
1.00
221
1.35
c880 13
1023
31.00 28
1008
227.21 25
c1355 84
755
5.00
507
1.95
c1908
106
1055
8.00
107
728
2.50
c2670 44
959
9.00
1039
17.41 79
c3540 78
1971
197.00
105
2042
276.91 95
c5315 37
1079
464.00 72
1117
524.53 67
c6288 6
243
78.00 18
258
218.9 17
c7552 65
2165
151.00
145
2016
71.21
139
SUN Fire 280R, 900 MHz Dual Core machine
[5] K. R. Kantipudi and V. D. Agrawal, “A Reduced Complexity Algorithm for Minimizing N-Detect Tests,” Proc. 20th International Conf. VLSI Design, Jan. 2007, pp. 492–497.
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15. Conclusion Future Work
A new algorithm based on primal dual ILP is introduced for test optimization.
The dual ILP helps in obtaining proper vectors, which then can be optimized by the primal ILP.
Future Work
According to Theorem 2, CIFS must converge to IFS as the vector set approaches the exhaustive set. We should explore strategies for generating vectors for the dual problem in order to have the CIFS quickly converge to IFS before vector set becomes exhaustive.
A useful application of the dual ILP and the conditionally independent fault set (CIFS), we believe, is in fault diagnosis. We hope to explore that in the future.
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16. Thank you …
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