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

2. Outline  Last lecture: 1. What is a diagnosis? 2. Expert systems 3. Model-based systems 4. Case Based Reasoning (CBR) 5. Inductive learning 6. Probabilistic reasoning  Today’s lecture: 1. Knowledge-based systems and diagnosis 2. Some definitions for model-based diagnosis 3. Reiter’s MBD algorithm using HS-trees 4. Causal form

3. Knowledge-based Systems are not simply  Systems based on knowledge are not simply  Systems based on knowledge but  Systems grounding their solution on a knowledge base but  Systems grounding their solution on a knowledge base Problem solver Knowledge base  Model-based systems are knowledge-based systems

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

5. For instance: diagnosis 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 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 under- braked under- braked over- braked over- braked harder

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

7.  Notation entails does not entail  inconsistency, “false” is consistent is inconsistent ^ 

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

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

10. Outline  Last lecture: 1. What is a diagnosis? 2. Expert systems 3. Model-based systems 4. Case Based Reasoning (CBR) 5. Inductive learning 6. Probabilistic reasoning  Today’s lecture: 1. Knowledge-based systems and diagnosis 2. Some definitions for model-based diagnosis 3. Reiter’s MBD algorithm using HS-trees

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

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

13. Example: System

14. Etc…

15. 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:

16. 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 ⊢⊥

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

18. Definition: Diagnosis A diagnosis for (SD, COMP, OBS) is a minimal set ∆ ∈ COMP such that: SD  {AB(C i )|C i ∈ Δ}  {¬AB(C i )|C i ∈ COMP- Δ}  OBS ⊢⊥ Example: ∆ 1 ={X 1 }, ∆ 2 ={X 2, O 1 }, ∆ 3 ={X 2, A 2 }

19. Example: ∆ 1 ={X 1 } 1 0 1 1 0 1 1 0 0 1 0 0

20. Example: ∆ 2 ={X 2, O 1 } 1 0 1 1 0 1 1 0 0 1

21. Example: ∆ 3 ={X 2, A 2 } 1 0 1 1 0 1 1 0 0 1 0

22. Definition: Conflict set A conflict set for (SD, COMP, OBS) is a set {c 1 …c k }  COMP such that: SD  OBS  {¬AB(C 1 )…¬AB(C k )} ⊢⊥ A conflict set is minimal iff no proper subset of it is a conflict set.

23. 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: 1. the components that are normal ( ¬AB(C i )) could not be a conflict set 2. a conflict set must contain at least one component of the diagnosis.

24. Definition: Hitting set Suppose C is a collection of sets. A hitting set for C is a set H   S ∈ C S such 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}

25. Δ  COMP is a diagnosis for (SD,COMP, OBS) iff Δ is 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}. Theorem of diagnosis

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

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

28. How to compute 1. Conflict sets 2. Diagnosis

29. Outline  Last lecture: 1. What is a diagnosis? 2. Expert systems 3. Model-based systems 4. Case Based Reasoning (CBR) 5. Inductive learning 6. Probabilistic reasoning  Today’s lecture: 1. Knowledge-based systems and diagnosis 2. Some definitions for model-based diagnosis 3. Reiter’s MBD algorithm using HS-trees

30. 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:

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

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

33. Pruning 1. If node n is labelled by √ and node n’ is such that H(n)  H(n') then close n’. 2. If node n has been generated and node n' is such that H(n')= H(n) then close n'.

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

35. Pruning 1. If node n is labelled by √ and node n’ is such that H(n)  H(n'), close n’. 2. If node n has been generated and node n' is such that H(n')= H(n) then close n'. 3. 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 .

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

37. Finally tree after pruning

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

39. 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.