Jan. 6, 2006VLSI Design '061 On the Size and Generation of Minimal N-Detection Tests Kalyana R....

Jan. 6, 2006 VLSI Design '06 1

On the Size and Generation of Minimal On the Size and Generation of Minimal N-Detection TestsN-Detection Tests

Kalyana R. KantipudiVishwani D. Agrawal

Department of Electrical and Computer EngineeringAuburn University, AL 36849 USA

19th International Conference on VLSI Design, Hyderabad, January 3-7, 2006


Motivation for This Work

• N-detection vectors are known to have a higher coverage of “real defects”.

• Increased test length for N-detection tests directly impacts testing cost.

• Efficient ways to find minimal N-detection test set are still evolving.

• There is no proven lower bound on the size of the N-detection vectors.


Outline

• Introduction

• Theoretical minimum for an N-detection test set

• ILP based N-detection approach

• Results

• Conclusions


Minimal Single-Detection Tests• Independence graph: Nodes are faults and edges represent pair-

wise independence relationships. Example: c17[1].• An Independent Fault Set (IFS) is a maximal clique in the graph.• Size of IFS is a lower bound on test set size (Akers et al., ITC-87)

1 2 3 4 5

6 7 8 9 1 0

11

1 2 3 4 5

6 7 8 9 1 0

11

1

4 2

5So, the minimal test set for these 11 faults is ‘4’.

[1]A. S. Doshi and V. D. Agrawal, “Independence Fault Collapsing,” Proc. 9th VLSI Design and Test Symp., Aug. 2005, pp. 357-364.


Theoretical Minimum of an N-Detection Test Set

• Theorem 1: The size of the largest clique in the independence graph is a lower bound on single detection test set size [Akers et al. ITC-87].

• Theorem 2: A lower bound on the size of the N-detection test set is N times the size of the largest clique in

the independence graph.

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.


4-Bit ALU: N-Detection Vectors

0

100

200

300

400

500

600

700

Vec

tors

1 2 3 4 5 6 7 8 9 10 20 40 50

N

Atalanta vectors Min. ILP vectors Lower bound


ILP Based N-Detection Approach

• Use any ATPG to obtain a set of k vectors that detects every fault at least M times, where M N.

• 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 included in the minimal set ti=0 if ith vector not included


Objective and Constraints of ILP

Objective:

k

1iitimize min

Constraints:

jfaults,Nt jTvectori

ji

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 exhaustive set of vectors, an ILP solution that satisfies the above expressions is a minimum N-detection test set.


Minimal 3-Detection Test Set for c17

• ATALANTA generates 4 test sets (M = 4); repeated vectors are removed.

• HOPE is used to perform diagnostic fault simulation on the vector set.

• Simulation information is used to create constraints for the ILP.

sa1

x

sa1

x

sa1

x

sa1

x

sa1

x

sa1

x

sa1

x

sa1

xsa1

x

sa1

xsa1

xsa1

x

sa1

x

sa1

x

sa1

x

X

sa0

X

sa0

X

sa0

X

sa0

X

sa0sa1

x

sa1

x1511

8

21

12

97

6

54

314

13 12

10 1617

18

19

20

22


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

1iitimize min

jfaults,3t

ji Tvectori

and the constraints are:

Constraint_‘F1’: t1+t2+t15+t16+t22+t24 ≥ 3

Constraint_‘F21’: t13+t15+t16+t19+t23+t24 ≥ 3


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_‘F21’: 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 the change in test size is negligible.For large circuits the change in test size is negligible.

35


15-Detection Tests for Benchmarks

0

500

1000

1500

2000

2500

Te

st

se

t s

ize

Lower Bound

Present Work

Previous Result

Hamzaoglu and Patel, IEEE-TCAD, Aug. 2000Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002


CPU Time for 15-Detection Tests

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

CP

U s

ec

on

ds

Present Work

Previous Result

Ultra-5, * Ultra-10, ** Sun Fire 280R

Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002


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 less probability of detecting any other fault.

`

Primary Inputs

PO1

PO2

PO3

F1X

X F2

XF3

F4X


Example: Ripple Carry Adders

1-b

1-b

1-b

1-b

a

b c_in

s_out

c_out


Conclusion

• The theoretical lower bound on the N-detection tests is useful in assessing the minimality of such tests.

• ILP is an effective method for minimizing tests, though further improvement is possible.

• The formulation of ILP allows custom selection of multiplicity of detection for individual faults.

• Contributions of this work:– Theoretical lower bound on N-detection tests.– ILP method for deriving minimal N-detection tests.


Thank You . . .

Jan. 6, 2006VLSI Design '061 On the Size and Generation of Minimal N-Detection Tests Kalyana R....

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