Introduction to Model-Based

DiagnosisMeir Kalech

Partially based on the slides of Peter Struss

Outline Last lecture:

1. What is a diagnosis?2. Expert systems3. Model-based systems4. Case Based Reasoning (CBR)5. Inductive learning6. Probabilistic reasoning

Today’s lecture:1. Knowledge-based systems and diagnosis2. Some definitions for model-based diagnosis3. Reiter’s MBD algorithm using HS-trees4. Causal form

Knowledge-based Systems

are not simply Systems based on knowledge

but Systems grounding their

solution on a knowledge base

Problem solver

Knowledge base

Model-based systems are knowledge-based systems

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

Knowledge Base and Problem Solver

Problem solver

Knowledge base

hydraulic unit

front left wheel

rear right wheel

brake pedal

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

under-braked

over-braked

harder

hydraulic unit

front left wheel

rear right wheel

brake pedal

under-braked

over-braked

harderKnowledge 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: „What“ and „How“

Diagnosis

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

Model-based Diagnosis

Task: Determine system models that are consistent with the observations

OBS?

MODEL

Outline Last lecture:

1. What is a diagnosis?2. Expert systems3. Model-based systems4. Case Based Reasoning (CBR)5. Inductive learning6. Probabilistic reasoning

Today’s lecture:1. Knowledge-based systems and diagnosis2. Some definitions for model-based diagnosis

3. Reiter’s MBD algorithm using HS-trees

Model-Based Diagnosis – Formal

Based on:

R. Reiter, A theory of diagnosis from first principles, Artificial Intelligence 32 (1) (1987) 57--95.

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.

Example: System

Example: System

Etc…

Etc…

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:

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)|CiCOMP} OBS ⊢⊥

Example: System is faulty

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Definition: DiagnosisA 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}

Example: ∆1={X1}

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Example: ∆2={X2, O1}

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Example: ∆3={X2, A2}

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Definition: Conflict setA 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.

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(Ci))

could not be a conflict set2. a conflict set must contain at least one

component of the diagnosis.

Definition: Hitting setSuppose C is a collection of sets. A hitting set for

C is a set H S∈CS such that HS{ } 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}

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

Example: conflict set {X1, X2}

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Example: conflict set {X1,A2,O1}

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How to compute

1.Conflict sets2.Diagnosis

Outline Last lecture:

1. What is a diagnosis?2. Expert systems3. Model-based systems4. Case Based Reasoning (CBR)5. Inductive learning6. Probabilistic reasoning

Today’s lecture:1. Knowledge-based systems and diagnosis2. Some definitions for model-based diagnosis3. Reiter’s MBD algorithm using HS-trees

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

Pruning 1. If node n is labelled by √ and node n’ is such

that H(n)H(n'), close n’.

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

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'.

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

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 eachS-S' mark as redundant the edge from node n labelled by .

Pruning 3n10={2,4}, n0={2,4,5}, n10 n0: mark 5 as

redundant since {2,4} is not hit by it.

Finally tree after pruning

Diagnosis: {H(n)|n is labelled by √}

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.