introduction to model-based diagnosis
DESCRIPTION
Introduction to Model-Based Diagnosis. Meir Kalech Partially based on the slides of Peter Struss. Outline. Last lecture: What is a diagnosis? Expert systems Model-based systems Case Based Reasoning (CBR) Inductive learning Probabilistic reasoning Today’s lecture: - PowerPoint PPT PresentationTRANSCRIPT
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.