CSE 498M598M, Fall 2002 Digital Systems Testing

1
CSE 498M/598M, Fall 2002 Digital Systems Testing

• Instructor Maria K. Michael
• CSE Dept., University of Notre Dame
• LECTURE 21
• Path-Delay Fault Simulation

2
Outline

• General problem definition
• The non-enumerative problem
• Outline of existing methods
• Non-enumerative PDF coverage estimation (Pomeranz
  Reddy, IEEE TCAD 1994)
• Exact PDF Coverage (Padmanaban et al, IEEE TCAD
  2003 to appear)

3
General Problem Definition

• Given a circuit C and a set of pairs of patterns
  T (tests), find the number of PDFs detected
• Straight forward solution
• Generate path-delay fault list
• Simulate tests and mark the PDFs covered
• Use fault simulation methods from Chapter 5

4
General Problem Definition (Cont.)

• Given a circuit C and a set of pairs of patterns
  T (tests), find the number of PDFs detected
• Straight forward solution
• Generate path-delay fault list
• Simulate tests and mark the PDFs covered
• Use fault simulation methods from Chapter 5

PDF Classification - Robustly testable -
Non-Robust testable - Functionally
Sensitizable - Functionally Unsensitizable
5
General Problem Definition (Cont.)

• Given a circuit C and a set of pairs of patterns
  T (tests), find the number of PDFs detected
• Straight forward solution
• Generate path-delay fault list
• Simulate tests and mark the PDFs covered
• Use fault simulation methods from Chapter 5
• Is this really practical? NOPDF
  enumeration is impossible for path-intensive
  circuits

6
Non-Enumerative PDF simulation

• Input C (combinational or fully-scanned
  circuit) and T (test set)
• Goal Find number of PDFs detected without
  enumerating the faults
• Truly non-enumerative methods must avoid
  enumeration of both the fault list and the list
  of detected faults

7
Previous Work

• Estimation of PDF coverage

• - Pomeranz Reddy, IEEE TCAD94
• - Heragu et al, IEEE TCAD97
• - Kagaris Tragoudas, IEEE TCAD97
• - Tragoudas, ITC99

8
Previous Work

• Estimation of PDF coverage
• Exact PDF coverage

• - Kapoor, EDTC95
• - Gharaybeh et al, ITC96
• Deodhar Tragoudas, ISQED01
• Padmanaban et al, IEEE TCAD03

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Pomeranz Reddy, IEEE TCAD94

• First non-enumerative approach
• Calculates an estimate of the number of PDFs
  without fault enumeration
• Estimate is pessimistic, i.e., no smaller than
  the actual
• Estimation accuracy increases with increased time
  complexity

10
Basic properties

• Path counting algorithm to calculate of total
  and detected faults - linear
• Simulation of single test - linear
• For every test, ONLY newly detected PDFs are
  counted, by considering new lines

11
Basic path counting algorithm

• Same as the one discussed in class (see lectures
  56) with the difference that it starts from the
  primary outputs (POs) and proceeds towards the
  primary inputs (PIs)

12
Basic path counting algorithmExample (Figure 1)

1 1
8 1
4 1
16 1
3 4
2 2
13 1
12 2
10 2
5 3
14 1
9 3
17 1
11 1
6 3
15 1
7 1
line ID/name X paths from to POs
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Single test simulation

• Start from PIs, simulate test using appropriate
  algebra (i.e., 3-value for non-robust, 5-value
  for robust, etc)
• While simulating, keep IDs of immediate
  sensitized predecessors (will result in marking
  PDFs covered by test)
• Count only marked paths ? exact number of PDFs
  covered

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Single test simulationExample (Figure 2)
1 1?0 (1)

8 0?1 (1)
4 1?1
16 1?0 (8)
13 1?1
3 1?1
2 0?0
10 0?1(9)
12 1?1
5 1?1
17 0?1 (15)
14 1?1
9 0?1(6)
6 1?0 (6)
11 0?1(9)
15 1?0 1(7,11)
line ID/name X test pattern ( ) list of
sensitized immediate predecessors
7 0?1(7)
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Single test simulationExample (Figure 2)
1 1?0 (1)

8 0?1 (1)
4 1?1
16 1?0 (8)
13 1?1
3 1?1
2 0?0
10 0?1(9)
12 1?1
5 1?1
17 0?1 (15)
14 1?1
9 0?1(6)
6 1?0 (6)
11 0?1(9)
15 1?0 1(7,11)
line ID/name X test pattern ( ) list of
sensitized immediate predecessors
7 0?1(7)
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Single test simulationExample (Figure 3)
1 1

8 1
4 0
16 1
13 0
3 0
2 0
12 0
10 0
5 0
14 0
9 1
17 1
6 1
11 1
15 1
7 1
line ID/name X PDFs detected
17
Test set simulation

• For every test, count ONLY the number of newly
  detected PDFs
• Newly detected PDF a PDF that contains at least
  one new line
• New line a circuit line with a or
  transition (logical line), not included in ANY
  previously detected PDF

18
Test set simulation (Cont.)Example (Figure 2)
1 1?0 (1)
8 0?1 (1)
4 1?1
16 1?0 (8)
13 1?1
3 1?1
2 0?0
10 0?1(9)
12 1?1
5 1?1
17 0?1 (15)
14 1?1
9 0?1(6)
6 1?0 (6)
11 0?1(9)
PDFs detected 1-8-16 6-9-11-15-17 7-15-17
15 1?0 1(7,11)
7 0?1(7)
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Test set simulation (Cont.)Example (Figure 4)
1 0?0
8 1?1
4 1?0 (3)
16 0?0
13 1?1
3 1?0 (3)
2 0?0
10 0?1(9)
12 1?1
9 0?1(5)
14 1?1
5 1?0 (3)
6 1?1
17 0?1 (15)
11 0?1(9)
Old PDFs detected 1-8-16 6-9-11-15-17
7-15-17
15 1?0 1(7,11)
7 0?1(7)
PDFs detected 3-5-9-11-15-17 7-15-17
New lines 3, 5
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Test set simulation (Cont.)

• Modify single test simulation method to only
  count of new PDFs
• Introduce 1 more label per line
• Keep (Nn(i), No(i)) per line i
• Nn(i) -- new PDFs at line I
• No(i) -- old PDFs at line I
• ? Zero-Approximation method

21
Test set simulation (Cont.)Example (Figure 4)
1 0?0
8 1?1
4 1?0 (3)
16 0?0
13 1?1
3 1?0 (3) L2
2 0?0
10 0?1(9)
12 1?1
9 0?1(5) L1
14 1?1
5 1?0 (3) L2
6 1?1
17 0?1 (15) L1
11 0?1(9) L1
Old PDFs detected 1-8-16 6-9-11-15-17
7-15-17
15 1?0 1(7,11) L1
7 0?1(7) L1
New PDFs detected 3-5-9-11-15-17
L1(0,1) and L2(0,1)
22
Zero-Approximation Method

• Major disadvantage tends to miss a lot of new
  PDFs
• All lines are saturated with t1 and t2! No new
  PDFs will ever be counted

1
3
Let t1 cover 1-3, 2-4 and t2 cover 1-4, 2-3
2
4
of paths 4 of PDFs 8
23
Order One Approximation Method

• Partition circuit using cuts
• Cut set of lines s.t. every I/O path passes
  exactly through one line in the cut
• How to form partitions
• Get subcircuit induced by every line in the cut ?
  paths in different subcircuits are disjoint,
  i.e., sum of all paths in subcircuits is the
  total number of circuit paths
• Apply zero-approximation per subcircuit
• Larger cuts give better estimates !

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Zero-Approximation Method

• Zero-approximation can be modified to increase
  accuracy on newly detected PDFs, in the expense
  of computational complexity
• ? Order One (Two, Three ) Approximation

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Padmanaban, Michael, Tragoudas, IEEE TCAD03 to
appear

• First practical exact non-enumerative approach
• Reduces the problem to a set combinatorial
  problem
• PDFs are stored implicitly using Zero-Suppressed
  BDDs (ZBDDs)
• Requires a polynomial number of ZBDD operations

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Reduction to a Combinatorial Problem

• Express a path as a combination of variables

adf
a
d
b
f
c
e
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Reduction to a Combinatorial Problem

• Express a path as a combination of variables
• A set of paths is a combination set

adf, cef
a
d
b
f
c
e
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Reduction to a Combinatorial Problem

• Express a path as a combination of variables
• A set of paths is a combination set
• For PDFs, use 2 variables per primary input

ardf, cfef
a
d
b
f
c
e
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Reduction to a Combinatorial Problem

• Combination Sets Boolean Functions

• Use Zero-Suppressed BDDs for the functions

- Canonical forms
- Compact
- More efficient than BDDs for sparse
combinations
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Exact PDF Coverage

• For every test in the test set

- simulate based on PDF type
- generate set of PDFs covered
- update set of total PDFs covered
Method requires a polynomial number of ZBDD
operations per test
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Exact PDF Coverage - Example
Simulate first test (T1)
a
d
b
f
c
e
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T1
a
b
c
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T1
af
a
b

c
cf
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T1
af
a
afd
d
b

c
e
cfe
cf
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T1
a
d
afd
b
f
c
e
cfe
PDFs under T1 afdf, cfef
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Exact PDF Coverage Example (Cont.)
Simulate second test (T2)
a
d
b
f
c
e
37
Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T2

a
d
b
bf
c
e
cf
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T2

a
d
bfd
b
bf
c
e
cfe
cf
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Exact PDF Coverage Example (Cont.)
Generate set of PDFs covered by T2
a
d
bfd
b
f
c
e
cfe
PDFs under T2 bfdf, cfef
40
Exact PDF Coverage Example (Cont.)
Calculate total number of PDFs covered
a
d
b
f
c
e
PDFs under T1 afdf, cfef PDFs under T2
bfdf, cfef
of PDFs afdf, cfef ? bfdf, cfef 3
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Exact PDF Coverage
What if the ZBDD cannot be stored ?
42
PDF Coverage using circuit partitioning

• Partitioning can reduce path count drastically

• Basic Steps

- find independent cut
- derive virtual partitions
- apply exact PDF grading on partitions

• calculate of PDFs covered accurately

• More partitions ? higher accuracy

43
PDF Coverage using circuit partitioning - Example
Generate circuit partitions
a
d
b
f
c
e
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PDF Coverage using circuit partitioning Example
(Cont.)
Generate circuit partitions
a
d
L1
b
f
L3
c
e
L2
Independent Cut Set L1, L2
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PDF Coverage using circuit partitioning Example
(Cont.)
Simulate test
a
d
L1
b
f
L3
c
e
L2
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PDF Coverage using circuit partitioning Example
(Cont.)
Generate sets of PDFs covered
af
afd
a
d
L1
b

c
e
L2
cfe
cf
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PDF Coverage using circuit partitioning Example
(Cont.)
Store set of PDFs at Cut lines
afd
a
d
L1
b
c
e
L2
cfe
48
PDF Coverage using circuit partitioning Example
(Cont.)
Simulate and Store set of PDFs at Outputs
afd
a
d
L1
f
b
f
L3
c
e
L2
cfe
49
PDF Coverage using circuit partitioning Example
(Cont.)
Number of new PDFs through line pair (L1,L3)
bfd
afd
f
a
d
L1
f
b
f
L3
c
e
L2

• Use history

. f ? f
afd
. f \ f .
afd \ bfd
1
1
0
1
50
PDF Coverage using circuit partitioning Example
(Cont.)
Number of new PDFs through line pair (L1,L3)
bfd, afd
afd
f
a
d
L1
f
b
f
L3
c
e
L2

• Use history

. f ? f
afd
. f \ f
afd \ bfd, afd
1
1
0
0
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PDF Coverage using circuit partitioning

• Estimation is correct (no double counting)
• The probability of failing to detect a new
  PDF is very low, and decreases with more
  partitions

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Summary

• PDF simulation is a much harder problem than
  stuck-at fault simulation
• Only non-enumerative techniques are practical for
  PDF simulation
• Existing methods are divided into exact and
  estimation methods
• PomReddy estimation accuracy increases with
  time complexity
• Padmanaban et al finds exact coverage with
  polynomial number of ZBDDs operations