A New Transitive Closure Algorithm with Application to Redundancy Identification

1
A New Transitive Closure Algorithm with
Application to Redundancy Identification

• Vivek Gaur
• Avant! Corp., Fremont, CA 94538, USA
• vgaur_at_avanticorp.com
• Vishwani D. Agrawal
• Agere Systems, Murray Hill, NJ 07974, USA
• va_at_agere.com
• http//cm.bell-labs.com/cm/cs/who/va
• Michael L. Bushnell
• Rutgers University, Dept. of ECE, Piscataway, NJ
  08854, USA
• bushnell_at_caip.rutgers.edu

2
Talk Outline

• Problem Statement
• Background Redundancy Identification
• Implication graph
• Partial implications
• Transitive closure
• Redundancy identification
• Results
• Conclusion

3
Problem Statement

• Many problems can be solved by implication graphs
  and transitive closure.
• We will study the problem of redundancy
  identification.
• Redundancy identification has applications in
  testing and in circuit optimization.

4
Redundancy Identification

• ATPG based method
• Exhaustive test pattern generation to find
  whether or not a target fault has a test.
• All redundant faults can be found, but ATPG cost
  is exponential in circuit size.
• Fault independent method
• Method analyzes circuit topology and function
  locally no specific fault targeted.
• Many (not all) redundant faults can be found at a
  lower cost.
• FIRE, Iyer and Abramovici, VLSI Design94.
• TC, Agrawal, Bushnell and Qing, ATS96.

5
Use of Implication Graphs

• 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, MS Thesis02, Rutgers University

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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.

8
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 Fault Redundancy

• Detection of a fault requires the fault to be
  activated and its effect observed at a PO.
• Example Fault a s-a-1 is detectable, iff
  following conditions can be simultaneously
  satisfied
• a 0
• Oa 1

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Redundancy Identification by Transitive Closure
c
a
s-a-0
b
c
a
d
e
s-a-0
b
d
Circuit with two redundant faults not found by
FIRE or TC
Od
Oc
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
• Path-tracing algorithm (see this paper).
• Matrix method (see, Gaur, MS Thesis, Rutgers U.,
  2002).
• 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.

15
Classification of Redundant Faults by TCAND
HITEC, Nierman and Patel, EDAC91
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FIRE and Transitive Clo.
Iyer and Abramovici, VLSI Design94
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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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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.
• 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.