Introduction to Model-Based Diagnosis

1. Introduction to Model-Based Diagnosis Meir Kalech Partially based on the slides of Peter Struss

2. Outline • Last lecture: • What is a diagnosis? • Expert systems • Model-based systems • Case Based Reasoning (CBR) • Inductive learning • Probabilistic reasoning • Today’s lecture: • Knowledge-based systems and diagnosis • Some definitions for model-based diagnosis • Reiter’s MBD algorithm using HS-trees • Causal form

3. Problem solver Knowledge base Knowledge-based Systems • Model-based systems are knowledge-based systems • are notsimply • Systems based on knowledge • but • Systems grounding their solution on a knowledge base

4. Problem solver Knowledge base Knowledge Base and Problem Solver • Provlem solver: • A usually task-specific, • possibly domain-independent • algorithm which can process the represented knowledge • Knowledge base: • an explicit • declarative • formal representation • of knoweldge about a certain domain and/or class of tasks

5. For instance: diagnosis harder • Observations: • “When braking with ABS, car is yawing to the right, and brake pedal feels harder than normally” • “Yawing”: • under-braking at left side • over-braking at right side over- braked under- braked

6. Diagnosis: „What“ and „How“ harder • Knowledge about the subject • „How is it structured?” • “How does it work?” • Knowledge about • Structure • Componenten behavior • Diagnosis Algorithm • From knowledge about the subject • and observations of the system behavior • infer diagnosis hypotheses over- braked under- braked

7. OBS Diagnosis ? • Task: • Determine, based on a set of observations: • What`s going on in the system?

8. ? MODEL OBS Model-based Diagnosis • Task: • Determine system models • that are consistent with the observations

9. Outline • Last lecture: • What is a diagnosis? • Expert systems • Model-based systems • Case Based Reasoning (CBR) • Inductive learning • Probabilistic reasoning • Today’s lecture: • Knowledge-based systems and diagnosis • Some definitions for model-based diagnosis • Reiter’s MBD algorithm using HS-trees

10. Model-Based Diagnosis – Formal Based on: R. Reiter, A theory of diagnosis from first principles, Artificial Intelligence 32 (1) (1987) 57--95.

11. Definition: System A system is a pair (SD, COMP) where: • (1) SD (system description), is a set of first-order sentences. • (2) COMP={C1,…,Cn}, the system components, is a finite set of constants.

12. Example: System

13. Example: System Etc… Etc…

14. Definition: Observation An observation of a system is a finite set of first-order sentences. We shall write (SD, COMP, OBS) for system (SD, COMP) with observation OBS. Example:

15. Definition: Diagnosis Problem Given SD, COMP and OBS, the observation conflicts with the system description assuming all its components behaving correctly. Formally: SD  {¬AB(Ci)|CiCOMP}  OBS ⊢⊥

16. Example: System is faulty 1 1 1 0 0 1 1 0 0 1

17. Definition: Diagnosis A diagnosis for (SD, COMP, OBS) is a minimal set ∆∈ COMP such that: SD  {AB(Ci)|Ci∈ Δ} {¬AB(Ci)|Ci∈COMP- Δ}  OBS ⊢⊥ Example: ∆1={X1}, ∆2={X2, O1}, ∆3={X2, A2}

18. Example: ∆1={X1} 1 1 1 0 0 0 1 1 0 0 0 1

19. Example: ∆2={X2, O1} 1 1 1 0 0 1 1 0 0 1

20. Example: ∆3={X2, A2} 1 1 1 0 0 1 1 0 0 0 1

21. Definition: Conflict set A conflict set for (SD, COMP, OBS) is a set {c1…ck}  COMP such that: SD  OBS {¬AB(C1)…¬AB(Ck)} ⊢⊥ A conflict set is minimal iff no proper subset of it is a conflict set.

22. The relation between conflict set and diagnosis Δ COMP is a diagnosis for (SD, COMP, OBS) iff Δ is a minimal set such that COMP-Δ is not a conflict set for (SD,COMP, OBS). In other words: • the components that are normal (¬AB(Ci)) could not be a conflict set • a conflict set must contain at least one component of the diagnosis.

23. Definition: Hitting set Suppose C is a collection of sets. A hitting set for C is a set H S∈CSsuch that HS{ } for each S∈C. A hitting set for C is minimal iff no proper subset of it is a hitting set for C. Example: S1={1,2,3} S2={2,4,5} S3={4,6} Minimal: H1={1,5,6}, H2={2,4}, H3={2,6} Not minimal: H4={2,4,6}

24. Theorem of diagnosis Δ COMP is a diagnosis for (SD,COMP, OBS) iff Δis a minimal hitting set for the collection of minimal conflict sets for (SD, COMP, OBS). Example: The full adder has two minimal conflict sets: {X1, X2} and {X1, A2, O1} There are three diagnoses, given by these minimal hitting sets: {X1}, {X2, A2}, {X2, O1}.

25. Example: conflict set {X1, X2} 1 1 0 1

26. Example: conflict set {X1,A2,O1} 1 1 0 1 0

27. How to compute • Conflict sets • Diagnosis

28. Outline • Last lecture: • What is a diagnosis? • Expert systems • Model-based systems • Case Based Reasoning (CBR) • Inductive learning • Probabilistic reasoning • Today’s lecture: • Knowledge-based systems and diagnosis • Some definitions for model-based diagnosis • Reiter’s MBD algorithm using HS-trees

29. Computing diagnosis Assume conflict sets: {2,4,5},{1,2,3},{1,3,5}, {2,4,6},{2,4},{2,3,5},{1,6}. HS-tree:

30. Pruning 1. If node n is labelled by √ and node n’ is such that H(n)H(n'), close n’.

31. Pruning 1 n3={1,2}, n9={1,3,2}, H(n3)H(n9): close n9.

32. Pruning • If node n is labelled by √ and node n’ is such that H(n)H(n') thenclose n’. • If node n has been generated and node n' is such that H(n')= H(n) then close n'.

33. Pruning 2 n6={5,4}, n8={4,5}, H(n6)=H(n8): close n8.

34. Pruning • If node n is labelled by √ and node n’ is such that H(n)H(n'), close n’. • If node n has been generated and node n' is such that H(n')= H(n) then close n'. • If nodes n and n' have been respectively labelled by sets S and S' of F, and if S'S, then for eachS-S' mark as redundant the edge from node n labelled by .

35. Pruning 3 n10={2,4}, n0={2,4,5}, n10  n0: mark 5 as redundant since {2,4} is not hit by it.

36. Finally tree after pruning

37. Diagnosis: {H(n)|n is labelled by √}

38. Computing conflict sets Using “resolution theorem prover”. See the next slides. Homework: 1. Analyse the complexity of the diagnosis process. 2. Read Reiter’s paper (bib 1), describe the algorithm he proposes for calculating the diagnosis and the conflict sets together.