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Nov 29th 2006 MS Thesis Defense 1
Minimizing Minimizing NN-Detect Tests for -Detect Tests for Combinational CircuitsCombinational Circuits
Master’s DefenseMaster’s DefenseKalyana R. KantipudiKalyana R. Kantipudi
Thesis Advisor: Dr. Vishwani D. Agrawal
Thesis Committee: Dr. Charles E. Stroud and Dr. Victor P. Nelson
Dept. of ECE, Auburn University
Nov 29th 2006 MS Thesis Defense 2
Outline
• Background
• Problem Statement
• ContributionsTheoretical Minimum for N-Detect TestsILP Based N-Detect Test MinimizationRelaxed LP based methods
The New Recursive Rounding Approach
• ConclusionsFuture work
Nov 29th 2006 MS Thesis Defense 3
Background• Defects are modeled as faults• Single stuck-at faults ease the test generation process• Bridging faults emulate the defects more accurately
• Test sets with greater than 95% fault coverage can produce only 33% coverage of node-to-node bridging faults (Krishnaswamy et al. ITC’01)
• About 80% of all bridges occur between a node and Vcc or Vss
W
K = 1
K = 0 K = 1K = 0
Dominate
OR
AND
1
W sa1
0
W sa0
Nov 29th 2006 MS Thesis Defense 4
N-Detect Tests
• Some applications need much lower DPM
• New test strategy which can be easily assimilated into the normal test generation process
• The problem with N-detect tests is their size
• There is no accurate way to achieve a minimal N-detect set
• There is no proven lower bound on the size of the N-detect vectors
Nov 29th 2006 MS Thesis Defense 5
Problem Statement
• To find a lower bound on the size of N-detect tests
• To find an exact method for minimizing a given N-detect test set
• To derive a polynomial time heuristic algorithm for the N-detect test minimization problem
Nov 29th 2006 MS Thesis Defense 6
The Independence Graph
• Independence graph: Nodes are faults and edges represent pair-wise independence relationships
• A clique is a fully connected sub-graph
Example: c17
1 2 3 4 5
6 7 8 9 10
11
1 2 3 4 5
6 7 8 9 10
11
A. S. Doshi, “Independence Fault Collapsing and Concurrent Test Generation,” Master’s thesis, Auburn University, May 2006.
Nov 29th 2006 MS Thesis Defense 7
Lower Bound on Single-Detection Tests
• The Independent Fault Set (IFS) is a maximal clique in the graph
• Theorem 1: The size of the IFS is a lower bound on the single detection test set size (Akers et al., ITC-87)
1
4 2
5
So, the lower bound for the single detection test set of c17 is ‘4’.
1 2 3 4 5
6 7 8 9 10
11
Nov 29th 2006 MS Thesis Defense 8
Theoretical Minimum of an N-Detect Test Set
• Theorem 2:Theorem 2: The lower bound on the size of the N-detect The lower bound on the size of the N-detect test set is N times the size of the largest clique in test set is N times the size of the largest clique in
the independence graph (Original Contribution) the independence graph (Original Contribution)
1
N test Vecs
5
N test Vecs
2
N test Vecs
4N test Vecs
So, at least 4N vectors are needed to detect each fault ‘N’ times.
1
4 2
5
Nov 29th 2006 MS Thesis Defense 9
Minimized N-Detect Vectors for 74181 ALU
N Lower Bound
(Theorem 2)(Theorem 2)
Minimized from Exhaustive set
1 12 12
10 120 120
20 240 240
30 360 360
40 480 480
50 600 607
60 720 742
70 840 877
80 960 1012
90 1080 1147
96 1152 1228
Nov 29th 2006 MS Thesis Defense 10
ILP Based N-Detect Test Minimization
• Use any N-detect test generation approach to obtain a set of k vectors which detect every fault at least N times.
• Use diagnostic fault simulation to get the vector subset Tj for each fault j.
• Assign integer variable ti to ith vector such that, ti = 1 if ith vector is included in the minimal set.
ti = 0 if ith vector is not included.
Nov 29th 2006 MS Thesis Defense 11
Objective and Constraints of ILP
Nj is the multiplicity of detection for the jth fault.
Nj can be selected for individual faults based on some criticality criteria or on the capability of the initial vector set.
Theorem 3: When the minimization is performed over an Theorem 3: When the minimization is performed over an exhaustive set of vectors, an ILP solution that satisfies exhaustive set of vectors, an ILP solution that satisfies the above expressions is a minimum N-detect test.the above expressions is a minimum N-detect test.
jfaults,Nt
t minimize
jT vectori
k
1ii
ji
:sConstraint
:Objective
Nov 29th 2006 MS Thesis Defense 12
Derivation of N-Detect Tests
• Generate an unoptimized M-detect test set (M N) using an ATPG (e.g., ATALANTA).
• Remove repeated vectors.
• Perform diagnostic fault simulation of the remaining vectors using a fault simulator (e.g., HOPE).
• If |Tj | < N for any fault, obtain additional vectors for that fault.
• Generate ILP constraints and use an ILP solver to determine the values of the variables ti that minimize the number of vectors = Σti .
Nov 29th 2006 MS Thesis Defense 13
Minimal 3-Detect Test Set for c17
• ATALANTA is used to generate 4 test sets (M = 4 iterations) and the repeated vectors are removed.
• HOPE is used to perform diagnostic fault simulation on the remaining vectors.
• The simulation information is used to create constraints for the ILP
Fault NumbersFault Numbers
sa1x
sa1x
sa1x
sa1x
sa1x
sa1x
sa1x
sa1x sa1
x
sa1xsa1
xsa1
x
sa1x
sa1x
sa1x
Xsa0
Xsa0
Xsa0
Xsa0
Xsa0
sa1x
sa1x 1511
8
21
12
97
6
54
14
13
10 1617
18
312
19
20
22
Nov 29th 2006 MS Thesis Defense 14
Constraint Generation
• Fault 1 is detected by the vectors 1, 2, 15, 16, 22, 24.• Fault 2 is detected by the vectors 1, 2, 3, 4, 5, 6, 7, 8, 9,
15, 16, 22, 24, 28, 29..... so on ....
Now the Objective is:
29
1iit minimize
jfaults,t
ji T vectori
3and the constraints are:
Constraint for fault 1: t1+t2+t15+t16+t22+t24 ≥ 3
Constraint for fault 21: t13+t15+t16+t19+t23+t24 ≥ 3
Nov 29th 2006 MS Thesis Defense 15
Minimum Test Sets from ILP
• The minimum 3-detect test set size is 13 (lower bound = 12). Vectors are: 2, 6, 7, 11, 14, 15, 16, 17, 18, 21, 23, 24, 28.
Suppose ‘fault 21’ is a critical fault to be detected 5 times:
Constraint for fault 21: t13+t15+t16+t19+t23+t24
• The minimum test set given by ILP has 14 vectors. Vectors are: 2, 6, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 28.
For large circuits this change in test size can be quite small.For large circuits this change in test size can be quite small.
35
Nov 29th 2006 MS Thesis Defense 16
Results
CircuitCircuitNameName
No. ofNo. ofUn Opt.Un Opt.
VecsVecs
Single DetectionSingle Detection 2-Detect2-Detect 3-Detect3-Detect 5-Detect5-Detect
ILP TimeILP Time(sec.s) (sec.s)
LowerLowerboundbound
SetSetSizeSize
LowerLowerboundbound
SetSetSizeSize
LowerLowerBoundBound
SetSetSizeSize
LowerLowerBoundBound
SetSetSizeSize
c432c432 14882 82.3 27 27 54 55 81 83 135 140
c499c499 397 5.34 52 52 104 104 156 156 260 260
c880c880 3042 306.81 13 25 26 44 39 63 65 105
c1355c1355 755 16.71 84 84 168 168 252 252 420 420
c1908c1908 2088 97 106 106 212 212 318 318 530 530
c2670c2670 8767 *1568.62 44 71 88 145 132 224 220 391
c6288c6288 243 519.67 6 18 12 27 18 37 30 57
c7552c7552 2156 *1530 65 148 130 298 195 468 325 841
Results on Ultra-5 * Ultra-10
Nov 29th 2006 MS Thesis Defense 17
Results for 15-Detect Tests
CircuitCircuit
ILPILP Prev. Result [1]Prev. Result [1]Lower Lower BoundBound15 x [2]15 x [2]CPU sCPU s No. of No. of
vectorsvectors CPU sCPU s No. of No. of vectorsvectors
c432 444.8 430 292.1 505 405
c499 24.9 780 153.2 793 780
c880 521.4 321 229.6 338 195
c1355 52.0 1260 5674.6 1274 1260
c1908 191.0 1590 1563.9 1648 1590
c2670 *607.8 1248 9357.6 962 660
c3540 1223.7 1411 - - 1200
c5315 *1368.4 924 - - 555
c6288 1206.3 134 1813.8 144 90
c7552 **346.1 2370 - 975
c499, c1355, c1908 - Type – I
C880,c2670,c7552 - Type – II
Results on Ultra-5
* Ultra-10
** Sun Fire 280R
[1] Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002
[2] Hamzaoglu and Patel, IEEECAD, 2000.
Nov 29th 2006 MS Thesis Defense 18
Minimized Vectors for 15-Detect Tests
0
500
1000
1500
2000
2500
Te
st
Se
t S
ize
(V
ec
tors
)
c432
c499
c880
c135
5
c190
8
c267
0
c354
0
c531
5
c628
8
c755
2
Lower Bound
Present (ILP)
Previous
Nov 29th 2006 MS Thesis Defense 19
CPU Time for Minimizing 15-Detect Tests
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Tim
e T
ak
en
(S
ec
on
ds
)
c432
c499
c880
c135
5
c190
8
*c26
70
c354
0
*c53
15
c628
8
**c75
52
Present (ILP)
Previous
Nov 29th 2006 MS Thesis Defense 20
Classifying Combinational Circuits
F1 X
F3 X
F2X
F4 X
PRIMARY INPUTS
PO1
PO3
PO4
PO2
TYPE - I: TYPE – II:
c499, c1355, c1908 c880, c2670, c7552
Output cones have large overlap.
Any vector detecting a fault F2 will have high probability of
detecting other faults, say fault F3 or F1.
Non-overlapping output cones.
Any vector detecting a particular fault, will have very low probability
of detecting any other fault.
`
PRIMARY INPUTS
PO1
PO2
PO3
F1X
X F2
XF3
F4X
Nov 29th 2006 MS Thesis Defense 21
Ripple Carry Adders
1-b
1-b
1-b
1-b
Ai
Bi
Ci
Ci+1
Si
Iterations: Number of times test sets are taken from Atalanta ATPG
Nov 29th 2006 MS Thesis Defense 22
Relaxed-LP Approach• Though ILP guarantees an optimal solution, it takes
exponential time to generate the solution.
• Time bounded ILP solutions deviate from optimality.
• LP takes polynomial time (sometimes in linear time) to generate a solution.
• Redefining the variables tis as real variables in the range [0.0,1.0] converts the ILP problem into a linear one.
• The problem now remains to convert it into an ILP solution.
The optimal value of the relaxed-LP of the ILP minimization The optimal value of the relaxed-LP of the ILP minimization problem is a lower bound on the value of the optimal integer problem is a lower bound on the value of the optimal integer solution to the problem.solution to the problem.
Nov 29th 2006 MS Thesis Defense 23
Previous Solutions (Randomized Rounding)
• The real variables are treated as probabilities.
• A random number xi uniformly distributed over the range [0.0,1.0] is generated for each variable ti.
• If ti ≥ xi then ti is rounded to 1, otherwise rounded to 0.
• If the rounded variables satisfy the constraints, then the rounded solution is accepted.
• Otherwise, rounding is again performed starting from the original LP solution.
Nov 29th 2006 MS Thesis Defense 24
Limitations of Randomized Rounding• Consider three faults f1,f2 and f3, and three vectors.
• We assign a real variable ti to vector i.
• Now the single detection problem is specified as:Minimize t1 + t2 + t3
Subject to constraints, f1 : t1 + t2 ≥ 1
f2 : t2 + t3 ≥ 1
f3 : t3 + t1 ≥ 1
• The number of tests is much larger
than the size of the minimal test set.
• The randomized rounding becomes a random search.
t3
LP Solution (0.5,0.5,0.5)
t1
t20
1
1
1
Nov 29th 2006 MS Thesis Defense 25
Recursive Rounding (New Method)
• Step 1: Obtain an LP solution. Stop if each ti is either 0.0 or 1.0
• Step 2: Round the largest ti and fix its value to 1.0 If several ti’s have the largest value, arbitrarily
set only one to 1.0. Go to Step 1.
Maximum number of LP runs is bounded by the final Maximum number of LP runs is bounded by the final minimized test set size.minimized test set size.
Final set is guaranteed to cover all faults. This method takes polynomial time even in the worst case.This method takes polynomial time even in the worst case. LP provides a lower bound on solution.
Lower Bound ≤ exact ILP solution ≤ recursive LP solution
Absolute optimality is not guaranteed.
Nov 29th 2006 MS Thesis Defense 26
The 3V3F Example• Step 1:
LP gives t1 = t2 = t3 = 0.5
• Step 2:
We arbitrarily set t1 = 1.0
• Step 1:
Gives t2 = 1, t3 = 0 ■
or t2 = 0, t3 = 1 ■
or t2 = t3 = 0.5
• Step 2: (last case)
We arbitrarily set t2 = 1.0
• Step 1: Gives t3 = 0
t3
t2
t1
LP Solution (0.5,0.5,0.5)
0
1
1
1
Non-optimum solution
ILP solutions (optimum)
Step 1
Step 2
Nov 29th 2006 MS Thesis Defense 27
Minimal Tests for Array Multipliers
• There exists a huge difference between its theoretical lower bound of six and its practically achieved test set of size 12.
• A 15 x 16 matrix of full-adders (FA) and half-adders (HA).
• To make use of its recursive
structure and apply
linear programming
techniques.
HA
HA
HA
FA
FA
FA
FA
FA
FA
FA
HA
FA
A0B0A1B0A2B0B0An-1
An-1B1
B2An-1
Bn-1An-1
P2n-1P2n-2 Pn
P0
P1
P2
P3
Pn+1
A0B1
B2
A0
A0B3
11
n-21
21n-2
2
n-2 133
Nov 29th 2006 MS Thesis Defense 28
Tests for c6288: 16-Bit Multiplier
• Known results (Hamzaoglu and Patel, IEEE-TCAD, 2000):
• Theoretical lower bound = 6 vectors
• Smallest known set = 12 vectors, 306 CPU s
• Our results:• Up to four-bit multipliers need six vectors
• Five-bit multiplier requires seven vectorsFive-bit multiplier requires seven vectors
• c6288900 vectors constructed from optimized vector sets of smaller
multipliers ILP, 10 vectors in two days of CPU timeRecursive LP, lower bound = 7, optimized set = 12, in 301 CPU s
Nov 29th 2006 MS Thesis Defense 29
Comparison of ILP and Recursive LP method
0
20
40
60
80
100
120
140
160
180
3 4 5 6 7 8 9 10 11 12
Bit Multiplier (Bits)
CP
U S
econ
ds
3
6
9
12
15
18
21
24
Test Set S
ize (Vectors)
LP CPUSecs
ILP CPUSecs
LP Set size
ILP Setsize
Timebound ILPSet size1000 sec
Nov 29th 2006 MS Thesis Defense 32
Optimized 15-Detect Tests
Circuit Circuit NameName
Unopti.Unopti.VecsVecs
LP/recursive LP/recursive RoundingRounding ILPILP Previous Previous
Result [1]Result [1]L.B.L.B.
Vect.Vect. CPU sCPU s Vect.Vect. CPU sCPU s Vect.Vect. CPU sCPU s
c432 14882 430 83.5 430 444.8 505 292.1 405
c499 1850 780 17.8 780 24.9 793 153.2 780
c880 4976 322 94.5 321 521.4 338 229.6 195
c1355 2341 1260 41.2 1260 52.1 1274 5674.6 1260
c1908 6609 1590 150.4 1590 191 1648 1563.9 1590
c2670 8767 1248 380.6 1248 607.8* 962 9357.6 660
c3540 4782 1407 239.6 1411 1223.7 - - 1200
c5315 4318 924 494.3 924 1368.4* - - 555
c6288 731 134 250.5 134 1206.3 144 1813.8 90
c7552 6995 2371 359.1 2370 346.1** - - 975[1] Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002
Nov 29th 2006 MS Thesis Defense 33
Conclusion
• A Lower Bound for N-Detect tests is derived.
• An N-Detect test minimization method based on ILP is formulated which always guarantees optimality.
• A polynomial time consuming recursive rounding LP, which can give close to optimal solutions for single and N-detect tests is presented.
• A smallest ever, 10 vector set derived for c6288 signifies the shortcomings of present test minimization techniques.
• The new recursive rounding LP method has numerous other applications where ILP is traditionally used and is found to be expensive.
Nov 29th 2006 MS Thesis Defense 34
Future Work• The dual problem of the test minimization problem
looks promising.
• The dual problem:
1 ,f
jvectors ,f
:sConstraint
f maximize :function bjectiveO
i
F fi
p
1ii
ji
0
1
• The Duality Theorem: If m is the minimum value of the primal problem and M is the maximum
value of the dual problem, then m = M.
Nov 29th 2006 MS Thesis Defense 35
The Previous c17 Example
• The primal problem gave a solution of 4 vectors.• The dual problem also gave a solution of 4, selecting
faults 1, 10, 16 and 18.• It is observed that these four faults are independent
of each other. • So the dual problem yielded an IFS of the circuit.• In cases where relaxed-LP gives non-integer
solutions for the dual problem, rounding techniques can be used.
This new approach has the potential of generating This new approach has the potential of generating much tighter lower bound compared to themuch tighter lower bound compared to the IFSIFS..