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Sajjad Siddiqi and Jinbo HuangNational ICT Australia and
Australian National University
Hierarchical Diagnosis of Multiple Faults
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Consistency-based Diagnosis
CDA
YX
B
Abnormal observation :
A B D
Which gate(s) are broken?
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Consistency-based Diagnosis
CDA
YX
B
System model :
okX (A C)
okY (B C) D
Health variables: okX, okY
Observables: A, B, D
Nonobservable: C
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Consistency-based Diagnosis
CDA
YX
B
Abnormal observation :
A B D
Find values of (okX, okY) consistent with :
(0, 0), (0, 1), (1, 0)
System model :
okX (A C)
okY (B C) D
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Consistency-based Diagnosis
System model overhealth variables (okX, okY, …)observablesnonobservables
Given observation , diagnosis is assignment to health variables consistent with
Consider minimum-cardinality diagnoses
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Search-based Approach
Search for diagnoses consistent with
Reduced to finding solutions to SAT instance
Cardinality enforced by extra constraints
Often restricted to single/double faults
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SystemModel
CompileTractable
Form
Query Evaluator
Compilation-based Approach
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Most work is done off-line
On-line diagnosis is efficient
Off-line work is amortized over multiple queries
Can handle arbitrary cardinality
Off-line compilation can be bottleneck
Compilation-based Approach
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DAG of nested and/or
Conjuncts share no variable (decomposable)
or
and
or andX3
X1 X2
Decomposable Negation Normal Form (DNNF)
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CDA
YX
B
Observation: A B D
10
0 011
11
System model :
okX (A C)
okY (B C) D
Diagnosis Using DNNF
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CDA
YX
B
Observation: A B Dor
okX okY
System model :
okX (A C)
okY (B C) D
Diagnosis Using DNNF
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SystemModel
CompileTractable
Form
Query Evaluator
Bottleneck
Compilation-based Approach
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Compilation-based Approach
Requires a health variable for each component
c1908 has 880 gates; basic encoding fails to compile
New technique to reduce number of health variables
Preserves soundness and completeness w.r.t. min-cardinality diagnoses
Requires only 160 health variables for c1908
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Hierarchical Diagnosis
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Hierarchical Diagnosis
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Hierarchical Diagnosis
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Identifying Cones
Gate G dominates gate X if any path from X to output of circuit contains G
All gates dominated by G form a cone
Dominators found by breath-first traversal of circuit
Treat maximal cones as blackboxes
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Abstraction of Circuit
C = {T, U, V, A, B, C}
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Top-level Diagnosis
Diagnosis: {A, B, C}
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Diagnosis of Cone
Need to set inputs/output of cone according to top-level diagnosis
Rest is similar, but not a simple recursive call (to avoid redundancy)
Once cone diagnoses found, global diagnoses obtained by substitution
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Diagnosis of Cone
Top-level diagnosis:{A, B, C}
3 diagnoses for cone A:{A}, {D}, {E}
3 global diagnoses by substitution:
{A, B, C}{D, B, C}{E, B, C}
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Soundness
Top-level diagnoses have same cardinality. Substitutions do not alter cardinality (cones do not overlap).
Remains to show that cardinality of these diagnoses, d, is smallest. Proof by contradiction:
Suppose there is diagnosis |P| < d. Replace every gate in P with its highest dominator to obtain P’.
P’ is a valid top-level diagnosis, contradicting soundness of baseline diagnoser
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Completeness
Need to show every min-cardinality diagnosis is found
Given diagnosis P of min cardinality d, replace every gate in P with its highest dominator to obtain P’
P’ has cardinality d, and only mentions gates in top-level abstraction, and hence will be found by top-level diagnosis (by completeness of baseline diagnoser)
P itself will be found by substitution (by completeness of cone diagnosis)
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Experiments
Use ISCAS85 circuits
Observations (inputs/outputs) randomly generated
Multiple instances per circuit
Use tool from (Huang and Darwiche, 2005) as baseline diagnoser
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Results
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Conclusion
New technique for compilation-based diagnosis to scale up
Preserves soundness and completeness w.r.t. min-cardinality diagnoses
For further scalability, hybrid of search and compilation is possible