Using Partial Implications for Redundancy Identification and Fault Equivalence

1
Using Partial Implications for Redundancy
Identification and Fault Equivalence

• Vishwani D. Agrawal
• Agere Systems, Murray Hill, New Jersey
• va_at_agere.com
• http//cm.bell-labs.com/cm/cs/who/va

2
Talk Outline

• Problem statement
• Background
• Implication graph
• Partial implications
• Transitive closure
• Redundancy identification
• A new method for fault equivalence
• Conclusion

3
Problem Statement

• Many problems can be solved by implication graphs
  and transitive closure.
• We will study two problems
• Redundancy identification
• Fault equivalence

4
Background

• Implication graphs
• Chakradhar, et al., Book90
• Larrabee, IEEE-TCAD92
• Transitive closure
• ATPG Chakradhar, et al., IEEE-TCAD93
• Redundancy, Agrawal, et al., ATS96
• Partial implications
• Henftling, et al., EDAC95
• Gaur, et al., DELTA02, Gaur, MS Thesis02
• Fault equivalence
• Bushnell and Agrawal, Book00

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Implication graph
An implication graph is a representation of
logical implications between pairs of signals of
a digital circuit.

• Nodes
• Two nodes per signal nodes a and a correspond to
  signal a.
• A node has two states (true,false) represents
  the signal state.
• Edges
• A directed edge from node a to b means a1
  implies b1.

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Building an Implication Graph
A B
B
C
A
C
AB C 0
A
B
C
AC BC ABC 0

• If C is 1 then that implies that A and B must
  be 1, but the reverse is not true. Similarly,
  if either A or B is 0 then C will be 0. But
  if we want to represent the implications of A and
  B on C then partial implications are necessary.

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Partial Implications
A B
B
C
A
C
AB C 0
A
B
C
AC BC ABC 0

Reference Henftling, et al., EDAC, 1995
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Observability Variables

• Observability variable of a signal represents
  whether or not that signal is observable at a PO.
  It can be true or false.

OA
OC 1
OC
A B
OA
C
B
(PO)
OB
OCB OA 0
OCOA BOA OCBOA 0
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Adding Observability Variables to Implication
Graph
OC

OA
OCOA BOA OCBOA 0
B
OA
OC
B
C
A
OC
OA
A
B
C
OB can be added similarly.
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Transitive Closure

• Transitive closure of a directed graph contains
  the same set of nodes as the original graph.
• If there is a directed path from node a to b,
  then the transitive closure contains an edge from
  a to b.

a
b
a
b
c
c
d
d
Transitive closure
A graph
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Stuck-at Faults

• This is a type of fault, which causes a line to
  hold a constant logic value, irrespective of
  change of state at previous stages.
• There are two types of stuck-at-faults
• Stuck-at-1
• Stuck-at-0
• Detection of a fault requires the fault to be
  activated and its effect observable at a PO.
• Fault a s-a-1 is detectable, iff following
  conditions can be simultaneously satisfied
• a 0
• Oa 1

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Redundant Faults

• A fault that has no test is called an untestable
  fault.
• Any untestable fault in a combinational circuit
  is a redundant fault because it does not cause
  any change in the input/output logic function of
  the circuit.
• Identification of redundant faults is useful
  because they can be removed
• from testing consideration, or
• from hardware

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Redundancy Identification

• ATPG based method
• It uses exhaustive test pattern generation to
  determine whether or not a target fault has a
  test.
• All redundant faults can be found, but the ATPG
  cost is high (exponential in circuit size).
• Fault independent method
• This method analyzes circuit topology and
  function locally without targeting a specific
  fault.
• Many (not all) redundant faults can be found at a
  lower cost.

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Redundancy Identification by Transitive Closure
b
c
a
d
c
a
s-a-0
e
s-a-0
b
Od
Oc
d
Circuit with two redundant faults
Implication graph (some nodes and edges not shown)
Implication Partial implication Transitive
closure edge
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Method Summarized

• Obtain an implication graph from the circuit
  topology and compute transitive closure.
• Examine all nodes
• S-a-0 is redundant if the signal implies its
  complement.
• S-a-1 is redundant if the complement of the
  signal implies the signal.
• Both faults are redundant if the signal and its
  complement imply each other.
• S-a-0 is redundant if the signal implies its
  false observability variable.
• S-a-1 is redundant if the complement of the
  signal implies its false observability variable.
• S-a-0 is redundant if the observability variable
  implies the complement of the signal.
• S-a-1 is redundant if the observability variable
  implies the signal.
• Both faults are redundant if the observability
  variable and its complement imply each other.

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Classification of Redundant Faults by TCAND
HITEC, Nierman and Patel, EDAC91
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Complexity of TCAND
SUN Sparc 5
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Limitation of Method

• Observability variable of a fanout stem is not
  analyzed.
• Only the redundant faults due to false
  controllability of fanout stem can be identified.

s-a-1
s-a-0
Three redundant s-a-0 faults identified
by transitive closure
Two redundant stem faults not identified by
transitive closure
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Some Conclusions

• Partial implications help find more redundant
  faults than the transitive closure without
  partial implications.
• The results have been compared with FIRE (another
  fault-independent program), which did equal to or
  worse that transitive closure with partial
  implication.
• Transitive closure computation run times were
  linear in the number of nodes for the implication
  graphs of benchmark circuits, although the known
  worst-case complexity is O(N3) for N nodes.

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Fault Equivalence

• Two faults of a Boolean circuit are called
  equivalent if they transform the circuit such
  that the two faulty circuits have identical
  output functions.
• Equivalent faults are indistinguishable and have
  exactly the same set of tests.
• Many fault equivalences can be determined by
  structural analysis based on gate fault
  equivalences and circuit interconnects this is
  done in practice.
• Some fault equivalences require equivalence
  checking of faulty circuits too complex and not
  generally done.

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Gate Fault Equivalence
sa1
sa0
sa0
sa1
sa1
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa0
sa1
sa1
sa0
sa0
sa0
sa0
sa1
sa0
sa1
sa1
sa1
sa1
sa0
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Sa0 and Sa1 Variables
Variable Asa0 assumes a true state when A1 and
OA is true
s-a-0
A
A B
Asa0
C
OA
Asa1 is similarly defined.
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Equivalence by Imp. Graph
s-a-0
s-a-0
A
C
Asa0
Csa0
B
OA
OC
B
C
A
OC
OA
A
B
C
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An Open Question

• Can implication graph/transitive closure
  recognize functional fault equivalence. These
  are not found by structural analysis.
• Example e b, for both faults.

s-a-1
a
e
s-a-1
b
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Conclusion

• Partial implications improve redundancy
  identification.
• Present limitation of the method is the
  identification of redundancy due to the false
  observability of fanout stem open problem.
• New formulation of fault equivalence may find
  some functional equivalences.

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• THANK YOU