Sajjad Siddiqi and Jinbo Huang National ICT Australia and Australian National University

1. Hierarchical Diagnosis of Multiple Faults Sajjad Siddiqi and Jinbo Huang National ICT Australia and Australian National University

2. Consistency-based Diagnosis C Abnormal observation : A  B  D A X D Y B Which gate(s) are broken?

3. Consistency-based Diagnosis C A X D Y B System model : okX  (A  C) okY  (B  C)  D Health variables: okX, okY Observables: A, B, D Nonobservable: C

4. Consistency-based Diagnosis C Abnormal observation : A  B  D A X D Y B System model : okX  (A  C) okY  (B  C)  D Find values of (okX, okY) consistent with   : (0, 0), (0, 1), (1, 0)

5. Consistency-based Diagnosis System model  over health variables (okX, okY, …) observables nonobservables Given observation , diagnosis is assignment to health variables consistent with    Consider minimum-cardinality diagnoses

6. 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

7. Compilation-based Approach System Model Tractable Form Compile Query Evaluator

8. Compilation-based Approach 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

9. Decomposable Negation Normal Form (DNNF) or DAG of nested and/or Conjuncts share no variable (decomposable) and or X3 and X1 X2

10. Diagnosis Using DNNF C A X D Y B 1 1 Observation: A  B  D System model : okX  (A  C) okY  (B  C)  D 0 1 1 0 1 0

11. Diagnosis Using DNNF C A X D Y B or Observation: A  B  D System model : okX  (A  C) okY  (B  C)  D okX okY

12. Compilation-based Approach System Model Tractable Form Compile Bottleneck Query Evaluator

13. 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

14. Hierarchical Diagnosis

15. Hierarchical Diagnosis

16. Hierarchical Diagnosis

17. 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

18. Abstraction of Circuit C = {T, U, V, A, B, C}

19. Top-level Diagnosis Diagnosis: {A, B, C}

20. 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

21. 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}

22. 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

23. 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)

24. Experiments Use ISCAS85 circuits Observations (inputs/outputs) randomly generated Multiple instances per circuit Use tool from (Huang and Darwiche, 2005) as baseline diagnoser

25. Results

26. 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