1. Partial Implications, etc.
Using Partial Implications for Redundancy Identification and Fault Equivalence
Vishwani D. Agrawal
Agere Systems, Murray Hill, New Jersey
2002
Partial Implications, etc.

2. Partial Implications, etc.
Talk Outline
Problem statement
Background
Implication graph
Partial implications
Transitive closure
Redundancy identification
A new method for fault equivalence
Conclusion
2002
Partial Implications, etc.

3. Partial Implications, etc.
Problem Statement
Many problems can be solved by implication graphs and transitive closure.
We will study two problems:
Redundancy identification
Fault equivalence
2002
Partial Implications, etc.

4. Partial Implications, etc.
Background
Implication graphs:
Chakradhar, et al., Book’90
Larrabee, IEEE-TCAD’92
Transitive closure:
ATPG: Chakradhar, et al., IEEE-TCAD’93
Redundancy, Agrawal, et al., ATS’96
Partial implications:
Henftling, et al., EDAC’95
Gaur, et al., DELTA’02, Gaur, MS Thesis’02
Fault equivalence:
Bushnell and Agrawal, Book’00
2002
Partial Implications, etc.

5. Partial Implications, etc.
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 “a=1” implies “b=1”.
2002
Partial Implications, etc.

6. Building an Implication Graph
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.
2002
Partial Implications, etc.

7. Partial Implications, etc.B A B C C
AB + C = 0 A B C
AC + BC + ABC = 0
Reference: Henftling, et al., EDAC, 1995
2002
Partial Implications, etc.

8. 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 OB
(PO)
OCB + OA = 0
OCOA + BOA + OCBOA = 0
2002
Partial Implications, etc.

9. Adding Observability Variables to Implication GraphOC OA
OCOA + BOA + OCBOA = 0 B A B C OC OA A B C OC OA
OB can be added similarly.
2002
Partial Implications, etc.

10. Partial Implications, etc.
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. b a b a c d c d
A graph
Transitive closure
2002
Partial Implications, etc.

11. Partial Implications, etc.
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
2002
Partial Implications, etc.

12. Partial Implications, etc.
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
2002
Partial Implications, etc.

13. 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.
2002
Partial Implications, etc.

14. Redundancy Identification by Transitive Closure
s-a-0 e
s-a-0 b d Oc Od
Circuit with two
redundant faults
Implication graph (some nodes
and edges not shown)
Implication
Partial implication
Transitive closure edge
2002
Partial Implications, etc.

15. Partial Implications, etc.
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.
2002
Partial Implications, etc.

16. Classification of Redundant Faults by TCAND
Circuit
Total red.
+aborted faults*
Tot. red. faults ident.
Unexcit-able red. faults
Unpropag-atable red. faults
Undriv-able red. faults
C1908
6+3 2
C2670
99+18
23+2 3
17+2
C3540
99+40 74 8 66
C5315 59 32 1 30
C6288 33 31 15 16
C7552
67+76 34
*HITEC, Nierman and Patel, EDAC’91
2002
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17. Partial Implications, etc.
Complexity of TCAND
SUN
Sparc 5
2002
Partial Implications, etc.

18. Partial Implications, etc.
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
Two redundant
stem faults not identified
by transitive closure
Three redundant s-a-0
faults identified by
transitive closure
2002
Partial Implications, etc.

19. Partial Implications, etc.
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.
2002
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20. Partial Implications, etc.
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.
2002
Partial Implications, etc.

21. Gate Fault Equivalence
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa1
sa0
sa0
sa0
sa0
sa1
sa1
sa1
sa0
sa1
sa1
sa0
sa1
2002
Partial Implications, etc.

22. Sa0 and Sa1 Variables Variable Asa0 assumes a true state
when A=1 and OA is true
s-a-0 A B A
Asa0 C OA
Asa1 is similarly defined.
2002
Partial Implications, etc.

23. Equivalence by Imp. Graph
s-a-0
s-a-0 A C
Asa0
Csa0 B A B C OC OA A B C OC OA
2002
Partial Implications, etc.

24. Partial Implications, etc.
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
2002
Partial Implications, etc.

25. Partial Implications, etc.
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.
2002
Partial Implications, etc.

26. Partial Implications, etc.
THANK YOU
2002
Partial Implications, etc.