Diagnosis of realistic bridging faults with single stuck-at information.
Abstract: Precise failure analysis requires accurate fault diagnosis. A previously proposed method for diagnosing bridging faults using single stuck-at dictionaries was applied only to small circuits, produced large and imprecise diagnoses, and did not take into account the Byzantine Generals Problem for bridging faults. We analyze the original technique and improve it by introducing the concepts of match restriction, match requirement, and failure recovery. Our new technique, which requires no information other than that used by standard stuck-at methods, produces diagnoses that are an order of magnitude smaller than those produced by the original technique and produces many fewer misleading diagnoses than that of traditional stuck-at diagnosis.
Introduction
Accurate fault diagnosis of realistic defects is an integral
part of failure analysis. The majority of spot
defects in modern CMOS technologies cause changes
in the circuit description that result in electrical
shorts [9], which implies that many failures are bridging
faults [16]. However, most fault diagnosis techniques
use the single stuck-at fault model to diagnose
faulty ICs.
Diagnosing bridging faults with single stuck-at fault
information is an appealing idea, but this approach
can lead to unusably large diagnoses or an unacceptable
percentage of misleading diagnoses. To address
these deficiencies, Aitken and Maxwell built dictionaries
comprised of realistic faults [3]. While there are
obvious advantages to this approach, the number of
realistic faults in a circuit is significantly larger than
the number of single stuck-at faults for a circuit, and
the cost of simulating each individual realistic fault is
frequently much greater. In summary, while diagnosis
using a realistic fault model is an ideal, achieving similar
results using single stuck-at information is a very
desirable goal.
In this paper we investigate and improve a bridging
fault diagnosis technique using the single stuck-at
fault model that was proposed by Millman, Mc-
Cluskey, and Acken (henceforth called the MMA tech-
nique) [17]. The MMA technique has many advan-
tages, the most notable of which are the ubiquitous
single stuck-at fault model, the obviation of the need
for additional circuit information for bridging fault di-
agnosis, and the absence of misleading diagnoses. How-
ever, the MMA technique, like other techniques, has
the disadvantages of intractable diagnosis size and assumption
of identical down-stream logic thresholds [5].
In addition, the MMA technique was originally demonstrated
only on circuits smaller than any of those in the
ISCAS-85 benchmark circuits [4].
By removing vectors that cannot detect a bridging
fault from its composite signature (match restriction),
and requiring vectors that should detect it (match re-
quirement), we have reduced the average diagnosis size
by more than an order of magnitude. Together with
failure recovery, our improvements allow us to provide
a diagnosis no larger than ten faults for more than
80% of the thousands of diagnostic trials we have performed
on the ISCAS-85 circuits. This improvement
has transformed a theoretical technique into one that
also has practical value.
In Sections 2 and 3 we describe the original MMA
technique and some of its shortcomings, in Sections 4
and Section 5 we present an experimental analysis of
the original technique using faulty responses obtained
from very accurate fault simulation. In Section 6 we
describe our improvements to the original technique
and demonstrate the improved diagnoses obtained. In
Section 7 we compare our technique to a traditional
diagnosis method and demonstrate the advantages of
our method.
2 The MMA algorithm
The original MMA technique disregarded bridge re-
sistance, variable downstream logic thresholds, and
the possibility of state-holding bridging fault behav-
ior. The original paper recognized that if the effect of
a bridging fault is detected at circuit outputs, the two
bridged nodes must carry opposite values in the fault-free
circuit. In the faulty circuit, the two nodes will
have the same value, which means that, under their
chosen assumptions, the faulty circuit will appear to
have a stuck-at fault on one of the two bridged nodes
and no fault on the other node. Therefore, for each
test vector producing an error on the circuit outputs,
a bridge between nodes A and B will appear to be one
of four stuck-at faults: A stuck-at 1, A stuck-at 0, B
stuck-at 1, or B stuck-at 0.
A faulty circuit's response to a test set is called its
signature. The key to the MMA technique is to consider
the composite signature of each potential bridging
fault. The composite signature of a bridging fault is
the union of the four associated single stuck-at signatures

A composite signature containing errors that are a
superset of the errors contained in the observed faulty
behavior is said to be a match. The MMA diagnosis
of a bridging fault is a list of candidate bridging faults
having composite signatures that match the observed
faulty behavior. This technique does not require explicit
simulation of bridging faults.
All matches correspond to bridging faults made up of
two nodes. There are three types of matches: A correct
match correctly identifies both of the nodes involved in
the bridge, a partial match correctly identifies only one
of the nodes involved in the bridge, and a misleading
match identifies neither of the nodes involved in the
bridge.
Having defined the types of matches, a diagnosis is
labeled in one of three ways based on the matches used
to construct it: An exact diagnosis is derived from only
the correct match, a partial diagnosis is derived from
the correct match in addition to other matches, and an
incorrect diagnosis is derived only from matches other
than the correct match. Incorrect diagnoses can be further
divided into three categories: an incomplete diagnosis
is derived from partial matches but not the correct
match, a misleading diagnosis is derived only from
misleading matches, and a failed diagnosis is empty.
Although all types of incorrect diagnoses are undesir-
able, it is much better to have a failed diagnosis than
a misleading diagnosis; a failed diagnosis is clearly incorrect
and cannot mislead an engineer.
If a bridging fault can create a feedback loop in the
circuit, some test vectors may cause the circuit to os-
cillate. Such a vector is said to only possibly detect the
bridging fault. The inclusion of possibly detecting test
vectors can lead to misleading and failed diagnoses,
but if the possibly detecting vectors are ignored, misleading
and failed diagnoses will not occur, and the
correct fault will always be part of the diagnosis. This
is stated as a theorem in the MMA paper:
When possibly detecting patterns are ig-
nored, the fault signature of a bridging fault 1
must be contained in its composite signature.
The MMA theorem guarantees that incorrect diagnoses
will not occur, but it places no bound on the
size of the diagnosis. A diagnosis with many misleading
or partial matches is undesirable; it can lead an
engineer to search portions of the chip not involved
in the fault. This is potentially frustrating because
physical investigation of the failed part sometimes requires
destruction of the layers above the site of the
suspected defect. Once these layers are gone, nearby
suspected sites cannot be investigated. In Section 5 we
show that the size of the average diagnosis using the
MMA technique on the ISCAS-85 benchmark circuits
is at least 36 matches (for the C880) and can reach
over 200 matches (for the C7552).
The MMA theorem guarantees that the correct
match will appear in the diagnosis-as long as the observed
behavior of the fault is not affected by variable
logic thresholds, which commonly affect the behavior
of faulty CMOS circuits [8, 15].
3 Byzantine Generals Problem
In order to be detected with a logic test, a bridging
fault must create an error that is propagated to one
or more circuit outputs. At the fault site, this error is
1 Throughout this paper we will use the term observed faulty
behavior rather than fault signature when referring to the output
of the faulty circuit being diagnosed.
a voltage that is subject to interpretation as variable
logic values by downstream logic gates. Because gate
input logic thresholds are not identical, different down-stream
gates can interpret the voltage as different logic
values: This phenomenon is known as the Byzantine
Generals Problem for bridging faults [1, 2]. Figure 1
shows a simple example of voltage interpretation in the
presence of variable logic thresholds.
threshold: 2.4 V
2.5 V
logic 1
logic 1
logic
A
threshold: 2.6 V

Figure

1: An instance of the Byzantine Generals Problem
for bridging faults. Each gate interprets the voltage
as a different logic value.
This behavior has important implications for diagno-
sis: a bridging fault need not mimic a single stuck-at
fault for any one test vector. The faulty voltage on
node B in Figure 1 will not cause the circuit to behave
as if B were stuck-at 1 or stuck-at 0-each circuit output
reports evidence that there is a different value on
node B.
The Byzantine Generals problem can affect diagnosis
in several ways. It might cause the error introduced
by the bridging fault to be propagated to fewer circuit
outputs than would be affected by a single stuck-at
fault. Alternatively, because of reconvergent fanout,
the error introduced by the bridging fault may be propagated
to more circuit outputs than would be affected
by a single stuck-at fault-or even to an output that
would never hold an error for any one of the four single
stuck-at faults. The Byzantine Generals Problem
may also cause errors to occur downstream from both
of the bridged nodes at the same time. Each of these
is an example of how variable logic thresholds nullify
the previously stated MMA theorem and cause incorrect
diagnoses to result for real circuits. In Section 5
we show that up to 10% of our diagnostic trials resulted
in incorrect diagnoses because of the Byzantine
Generals Problem.
4 Experimental considerations
We implemented the MMA technique and applied it to
the MCNC layouts of the ISCAS-85 circuits [4]. Before
reporting on the quality of our diagnoses, we describe
how we created composite signatures, selected
the faulty circuits to be considered in diagnosis exper-
iments, determined the behavior of the faulty circuits,
created the initial stuck-at signatures, and interpreted
the MMA theorem.
4.1 Realistic bridging faults
The first issue to be addressed when implementing the
MMA technique is the construction of the composite
signatures. As described, a single entry is created from
stuck-at signatures by concatenating the four fault signatures
of two distinct nodes. There are, however, \Gamma
possible bridging faults, making both the construction
and use of all possible composite signatures impractical
for most circuits.
The obvious solution is to limit the construction of
composite signatures to only realistic bridging faults.
A realistic bridging fault is one that is considered likely
to occur in the fabricated circuit due to the physical
locations of the wires on the IC. For this research, realistic
bridging fault lists were created for the MCNC
layouts of the ISCAS-85 circuits using the program
Carafe [12, 13]. Restricting the composite signatures to
realistic bridging faults cuts the fault lists to a manageable
length (for the ISCAS-85 circuits, there are three
to eight times as many realistic bridging faults as single
stuck-at faults [10]). For diagnostic trials, we created
faulty output responses for the 10% of realistic bridging
faults that are most likely to occur based on layout
and defect density information, although the process
of diagnosis considers all realistic faults as potential
candidates.
4.2 Bridging fault simulation
The Nemesis bridging fault simulator [7, 14] produced
the observed faulty behaviors used to evaluate the
MMA technique. Note that bridging fault simulation
is used only to determine faulty output responses to
be diagnosed and not as part of the diagnostic pro-
cedure. The behavioral model used by Nemesis for
bridging faults is two-component simulation, in which
the gates driving the bridged nodes are SPICE simulated
to determine the bridge voltage, which is then
compared against the SPICE-computed logic thresholds
of downstream gates in order to model the effect
of the Byzantine Generals Problem.
The simulator also extensively models feedback
bridging faults. If a feedback bridging fault evidences
the potential to oscillate or hold state, as verified by the
Test Guarantee Theorem [7], the simulator biases the
bridge voltage in favor of the fault free value on the rear
bridged node, thereby disallowing oscillation and state-holding
behavior. This approximation is very accurate
when the feedback path is short. When the feedback
path is long, disallowing oscillation and state-holding
behavior is an optimistic assumption that makes the
fault easier to diagnose. (Since the MMA theorem requires
that potentially detecting vectors be ignored,
disallowing potential detections gives us more information
to use for diagnosis.)
Before performing the experiment presented in the
next section, we replicated the original MMA results by
setting all gate logic thresholds to the same value. As
expected, no incorrect diagnoses occurred under these
conditions.
4.3 Stuck-at signatures
Since stuck-at signatures are used to create the MMA
composite signatures, the diagnostic ability of the technique
is limited by the diagnostic ability of the stuck-at
signatures. For this research, we have used diagnostic
test pattern generation to assure the best possible
stuck-at diagnosis information [6, 11].
4.4 MMA theorem interpretation
The Byzantine Generals Problem may cause errors to
appear on more or fewer circuit outputs than would
otherwise be the case; whether more or fewer, the evidence
presented by the application of one vector to the
faulty circuit will not be identical to the expected output
for any of the four single stuck-at faults associated
with the two bridged nodes. Whether or not the faulty
signature is contained in the composite signature is a
question of the definition of "contained."
The strict definition is that, for each vector, a faulty
response is contained in the composite signature only
if it is indistinguishable from the response of one of the
four stuck-at faults. The relaxed definition is that the
faulty response is contained if its affected outputs are
a subset of the affected outputs of the four associated
stuck-at faults. If the Byzantine Generals Problem
were not a factor, the strict definition would always be
the superior choice: it minimizes the chance of a misleading
match-thus reducing the number of matches
in a diagnosis. However, the strict definition coupled
with the Byzantine Generals Problem will cause more
incorrect diagnoses.
Throughout this paper, we report relaxed results.
We performed experiments using both definitions. For
both the original MMA technique and our improved
technique, the strict results have, on average, half
again as many incorrect diagnoses as the relaxed re-
sults, and the strict average match sizes are, on aver-
age, 60% of the relaxed average match sizes.
5 Baseline results
Circuit Total Average Percent Percent
Trials Matches Exact
C1355 443 134.2 19.6% 43.8%
C7522 5379 248.6 17.0% 47.7%

Figure

2: Total number of experimental trials, the average
number of matches per diagnosis, percentage of
diagnoses that are exact, and percentage of diagnoses
of size 10 or smaller.
Circuit Incorrect Incomp Mislead Failed
C432 1.9% 1.9% 0% 0%
C1355 2.9% 0.7% 0% 2.3%
C1908 7.2% 0.4% 0.2% 6.5%
C2670 9.2% 2.4% 0.1% 6.6%
C6288 4.5% 1.0% 0.3% 3.2%
C7522 10.5% 3.3% 0.6% 6.5%

Figure

3: Percentage of diagnoses that are incorrect:
comprised of failed, incomplete, and misleading diagnoses

The diagnoses returned by the MMA technique are
frequently unusable because of their size. Figure 2
shows that the diagnoses returned by the MMA technique
range from an average of 36 faults for the C880
to over 200 faults for the C7552. The average number
of faults is less than those reported by Millman, Mc-
Cluskey, and Acken-even though their circuits were
smaller-because we limit the list of candidate faults
to realistic faults. The number of exact diagnoses is
less than 30% for each circuit. Generally, around 50%
of the diagnoses are of size ten or less and contain the
correct fault. In order for the technique to be useful
as a practical diagnostic tool, the percentage of the
diagnoses that are small and contain the correct fault
must be much larger than that afforded by the basic
technique.
Incorrect diagnoses can occur when the bridging
faults to be diagnosed are impacted by the Byzantine
Generals Problem. Figure 3 shows that an incorrect
diagnosis occurs less than 10% of the time. An incorrect
diagnosis can be a failed diagnosis, an incomplete
diagnosis, or a misleading diagnosis. Incorrect diagnoses
are dominated by failed diagnoses. Incomplete
diagnoses are rare, and misleading diagnoses are almost
non-existent. The domination of failed diagnoses
in the incorrect diagnoses is an important feature of the
MMA technique: when the technique does not provide
the right answer, it rarely misleads the user.
6 Improvements
We present three methods that significantly improve
the diagnostic potential and precision of the original
technique.
6.1 Match restrictions
The weakness of the MMA technique is that a faulty
signature is likely to be contained in a large number
of composite signatures. The larger a composite signature
is, the broader the range of potential matches,
and the less likely it is to match only a faulty signature
representing a bridge between those two nodes. If unreasonable
portions of the composite signature could
be identified and removed, the result would be fewer
matches per diagnosis and a commensurate increase in
diagnostic precision.
The restriction employed here is to eliminate from
a composite signature any entries that cannot be used
to detect the described bridging fault. In order for a
bridging fault to be detected, a test vector must stimulate
opposite logic values on the two bridged nodes.
Removing vectors that place identical values on the
bridged nodes results in a composite signature that
more precisely contains the possible behavior of the
bridging fault.
Any vector in a composite signature that detects the
same-valued stuck-at fault on both bridged nodes must
stimulate the same value on both nodes. Such a vector
cannot detect the bridging fault and can be dropped
from the composite signature. For example, a vector
that detects both A stuck-at 0 and B stuck-at 0
cannot detect A bridged to B, and this vector can be
removed from the composite signature for A bridged
to B. This method is not exhaustive-there are probably
other vectors that place identical values on the
bridged nodes-but this improvement requires no more
information than that contained in the stuck-at signatures

Exhaustive information, however, may be available,
usually from a logic simulator or from the fault simulator
used to generate the test set. In the next section we
present the improvement in diagnostic precision that
can be achieved using such information.
One strength of this improvement is that it is not
affected by the Byzantine Generals Problem-it can
never increase the number of incorrect diagnoses.
6.2 Match requirement
While the match restriction of the previous section relied
on identifying test vectors which cannot detect a
particular bridging fault, the improvement presented
in this section is based on vectors that should detect
a bridging fault-namely, those vectors that place opposite
logical values on the bridged nodes and detect
single stuck-at faults on both of the bridged nodes. The
second improvement to the MMA technique is based
on identifying such vectors in the composite signatures,
and then enforcing a match requirement on those vectors

If during the construction of a composite signature
a single vector detects both A stuck-at 0 and B stuck-at
stuck-at 1), it is marked
as a required vector. In order for the composite signature
to match an observed faulty signature, the faulty
signature must contain errors for all required vectors.
Unlike the previous improvement, this improvement
can eliminate the correct match from a diagnosis. A
vector may detect opposite stuck-at values and still
fail to detect the bridging fault because the Byzantine
Generals Problem could prevent fault propagation. In
addition, if the bridge has a comparatively large resis-
tance, certain vectors may not cause a propagatable
error, which may also result in the elimination of the
correct match from the diagnosis.
6.3 Failure recovery
As discussed in Section 3 and demonstrated in Section
5, the Byzantine Generals Problem causes incorrect
diagnoses. Fortunately, most of the incorrect diagnoses
are failures-empty diagnoses. Because a failed
diagnosis is a clear indication that the algorithm has
not identified the correct match, a failed diagnosis is
a good opportunity to attempt to recover the correct
match. A straightforward approach is as follows:
Choose a "best-guess" diagnosis size, S, based on diagnosis
requirements. Rank candidate faults according
to the number of bit-positions in which the corresponding
composite signature fails to predict the errors that
were observed. The S signatures with the least number
of bit mis-matches constitute the diagnosis.
6.4 Improved results
Circuit Total Average Percent Percent
Trials Matches Exact
C1355 443 1.6 75.6% 95.5%
C7522 5379 10.5 55.2% 72.1%

Figure

4: After improvement, using only single stuck-at
information: total number of experimental trials,
the average number of matches per diagnosis, percentage
of diagnoses that are exact, and percentage of diagnoses
that have ten or fewer matches and contain
the correct match.
The diagnoses returned by our improved technique
are a substantial improvement over the original technique

Figure

4 shows that for each circuit, the size
of the average diagnosis is less than one twelfth of its
previous value; in some cases the average diagnosis is
ninety times smaller than before. For five of the benchmark
circuits, the correct match is part of a small diagnosis
(size ten or less) more than ninety percent of
the time. For these experiments, the "best-guess" size,
S, was set to ten faults.
Circuit Incorrect Incomp Mislead Failed Recover
C432 2.5% 0.6% 0% 1.9% 0.6%
C1355 3.4% 0% 0% 3.4% 3.4%
C1908 8.9% 1.5% 0% 7.4% 6.5%
C6288 9.8% 0% 0% 9.8% 9.6%
C7522 17.1% 0.9% 0.7% 15.4% 13.6%

Figure

5: After improvement, using only single stuck-at
information: percentage of diagnoses that are incor-
rect: comprised of incomplete, misleading, and failed
diagnoses, followed by percentage of total diagnoses
that succeed via failure recovery.

Figure

5 shows that the number of diagnoses that
do not contain the correct match has increased by a
few percent because of the interaction between the
match requirement technique and the Byzantine Generals
Problem, but the vast majority of the increase
comes from failed diagnoses and not misleading diag-
noses. The increased number of failures is more than
offset by the increase in usable diagnoses. Failed diagnoses
allow for the use of failure recovery: most of the
failed diagnoses are successfully recovered.
Circuit Total Average Percent Percent
Trials Matches Exact
C1355 443 1.3 86.5% 95.5%
C7522 5379 3.0 67.1% 79.7%

Figure

Using logic simulation for match restrictions:
total number of experimental trials, the average number
of matches per diagnosis, percentage of diagnoses
that are exact, and percentage of diagnoses that have
ten or fewer matches and contain the correct match.
As shown in Figures 6 and 7, our results improve
even further if we include information about internal
node values from logic simulation when establishing
match restrictions. Using match restrictions, match
requirements, and failure recovery, more than 90% of
the diagnostic trials result in a list of of size ten or less
Circuit Incorrect Incomp Mislead Failed Recover
C432 2.5% 0.6% 0% 1.9% 1.9%
C1355 3.4% 0% 0% 3.4% 3.4%
C1908 8.9% 0.4% 0.4% 8.0% 7.4%
C6288 9.8% 0% 0% 9.8% 9.6%
C7522 17.1% 0.3% 0.3% 16.3% 15.3%

Figure

7: Using logic simulation for match restrictions:
percentage of diagnoses that are incorrect: comprised
of incomplete, misleading, and failed diagnoses, followed
by percentage of total diagnoses that succeed
via failure recovery.
containing the correct fault. Figure 8 shows that when
failure recovery is successful, the correct fault appears,
on average, in the first three faults on the list.
Recovered Ave Pos Recovered Ave Pos
2.3
C7522 732 1.4 823 1.5

Figure

8: Number of failed diagnoses recovered and the
average position of the correct fault in the recovered
diagnoses
7 Another approach
In order to compare our results to standard diagnosis
methods, we performed an experiment modeled after
the Teradyne fault diagnosis system [18]. The fault ordering
method penalizes candidate stuck-at faults for
each predicted failure that did not actually occur and
for each failure that occurs without being predicted by
the candidate. This procedure produces a ranked list
of stuck-at faults. Given this ranking, if any of the
four stuck-at faults associated with the two bridged
nodes appears among the ten highest-ranked faults,
we consider the diagnosis a success (otherwise it is
counted as a misleading diagnosis). We used the same
observed faulty behaviors as we did for our previous
experiments.
Avg N1 Avg N2 Misleading
Circuit position position Diagnoses
C432 2.1 48.2 0.6%
C7522 2.2 107.8 3.0%

Figure

9: Fault ordering: average position of the first
node, average position of the second node, and the
percentage of misleading diagnoses

Figure

9 shows the results of fault ordering. The average
position of the first node is in the first ten faults,
but the average position of the second node is far behind
the first node. The number of diagnoses where
neither node appears in the top ten nodes is substantially
larger than the number of misleading diagnoses
for either the original or improved MMA techniques.
The improved MMA technique is better than the
fault ordering technique in two respects. First, the improved
MMA technique provides the exact two nodes
of interest in a set of ten or fewer most of the time. Providing
the exact pair is superior to providing individual
candidate stuck-at faults: every candidate stuck-at
fault represents one node that could be involved with
many potential realistic defects. Second, while the improved
MMA technique may produce an incorrect di-
agnoses, an unrecovered incorrect diagnosis occurs less
than 4% of the time. All fault ordering diagnoses appear
to be the same; there is no way to distinguish a
misleading diagnosis from good diagnosis.
Conclusions
We have shown that the original MMA technique is
unusable for even small CMOS circuits because of the
large size of the average diagnosis. Also, we have shown
that the Byzantine Generals Problem causes incorrect
diagnoses, although the number of misleading diagnoses
is very low.
We have improved the MMA technique in several
ways. First, we consider as candidate faults only those
determined to be realistic through inductive fault anal-
ysis. The number of realistic faults is much smaller
than the number of all theoretically possible bridges.
Second, we impose match restrictions and match requirements
in order to minimize diagnosis size. Finally,
we perform failure recovery to increase the number of
correct diagnoses. Using all our improvements, at least
90% of the time the correct match is found in a diagnosis
of size ten or less.



--R

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