Consistent-based Diagnosis

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Consistent-based Diagnosis

• Yuhong YAN
• NRC-IIT

2
Main concepts in this paper

• (Minimal) Diagnosis
• Conflict Set
• Proposition 3.3
• Corollary 4.5
• How to calculate minimal hitting set (algorithm
  revised!)
• Not required default logic (section 6) and
  section 7

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Scope of this paper

• Only uses nominal/right/OK model
• OK/right modes / right behaviour
• Faulty modes / faulty behaviour
• Only consider minimal diagnosis (the minimal set
  of abnormal components)
• Is logic foundation for using first order logic
  for diagnosis
• Its revise
• Its extension

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Model
Structural model Multiplier(Mult1),
Multiplier(Mult2), Multiplier(Mult3),
Adder(Add1), Adder(Add2) out(Mult1)
inp1(Add1) out(Mult2) inp2(Add1)
inp1(Add2) out(Mult3) inp2(Add2)

• Behavioral model of each type of component
• Adder(X) ? not AB(X) ? out(X) inp1(X)
  inp2(X)
• Multiplier(X) ?? not AB(X) ? out(X) inp1(X)
  inp2(X)
• ...

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Diagnosis on models of structure and function
....
design
textbook
first principles
actual
model
device
of the device
observed
predicted
model of the structure
behaviour
behaviour
of the device and of
the (nominal) behaviour
of each type of component
diagnosis
From Luca Console
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Symbols used for system

• SD System description, a set of first-order
  sentences
• COMPONENTS a finite set of constants
• System a pair (SD, COMPONENTS)
• AB(.) unary predicate, abnormal
• ? imply, other forms ? or ?, other direction
  ?, -

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Symbol used for observation

• OBS observations, a finite set of first-order
  sentence, value assignments to some variables.
• Observables the variables can be
  observed/measured

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Consistent

• There is an interpretation that makes a set of
  formulas true.
• Example
• Consistent A?B, A?B
• Inconsistent A?B,A??B, ?A?B, ?A??B
• Connect formula sets with union operator
• Example A?B, A?B ? ?A?B

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Some explanations

• Consistency means AND of the formulas in the
  sets
• Consistency of SD?OBS?AB(.)?AB(.)

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Definition diagnosis

• A diagnosis for (SD, COMPONENTS, OBS) is a
  minimal set ? ? COMPONENTS such that
• SD?OBS?AB(c)c???AB(c)c?COMPONENTS- ?
• is consistent
• ? is the smallest set of components
• SD is the right modes

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Diagnosis

• is a diagnosis if all components are in right
  modes, i.e.
• SD?OBS?AB(c)c?COMPONENTS is consistent
• Before computing diagnosis
• Should have enough observables
• Is a NP-hard problem

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Proposition 3.3

• If ? is a diagnosis for (SD, COMPONENTS, OBS),
  then for each ci??.
• SD?OBS?AB(c)c?COMPONENTS- ?AB(ci)
• faulty components are logically determined by the
  normal components

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Compute diagnoses

• Direct compute
• Compute conflict set (by using ATMS) then compute
  diagnoses from conflict set

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Definition Conflict set

• A conflict set for (SD, COMPONENTS, OBS) is a set
  c1, ,ck?COMPONENTS such that
• SD?OBS?AB(c1), , AB(ck)
• is inconsistent
• A conflict set for (SD, COMPONENTS, OBS) is
  minimal iff no proper subset of it a conflict set
  for (SD, COMPONENTS, OBS)

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How to compute diagnosis

• Theorem4.4 ??COMPONENTS is a diagnosis for (SD,
  COMPONENTS, OBS) iff ? is a minimal hitting set
  for the collection of conflict sets for (SD,
  COMPONENTS, OBS)
• (A diagnosis is the minimal hitting set of
  conflict sets)
• Corollary4.5 ??COMPONENTS is a diagnosis for
  (SD, COMPONENTS, OBS) iff ? is a minimal hitting
  set for the collection of minimal conflict sets
  for (SD, COMPONENTS, OBS)

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Minimal hitting set

• A hitting set for a collection of sets C is a set
  H ? ?S?CS such that H?S? for each S?C.
• A hitting set is minimal iff no proper subset of
  it is a hitting set for C.

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Compute Minimal Hitting Set

• Hitting set tree (HS-tree) a smallest
  edge-labeled and node-labeled tree for C a
  collection of sets
• The root is labeled by ? if C is empty, otherwise
  the root is labeled by an arbitrary set of C
• For each node n of T, let H(n) be the set of edge
  labels on the path in T from the root node to n.
  The label for n is any set ? ?C such that ? ?H(n)
  , if such a set ? exists. Otherwise, the
  label for n is ?, If n is labeled by the set ?,
  then for each ?? ?, n has a successor, n?, joined
  to n by an edge labeled by ?

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Example
Prediction A12, B12
Observation A10, B12
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A10 generates two conflicts A1, M1, M2 A1,
M1, M3, A2
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12
12
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HS Tree
M1,M2,A1
M1
A1
M2
M3,A2,M1,A1
?
?
M3
A1
A2
M1
?
?
?
?
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Constructing HS-Tree

• Keep the HS-tree as small as possible
• Calculate only minimal hitting set
• Minimize the number of calls to the underlying
  theorem prover

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Algorithm outline

• Generate a HS-tree
• Return H(n)n is a node labeled ?

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Algorithm more detail

• Select a set from C as root node, label the root
  node with this set
• For each ?, generate an arc labeled by ?
• For a new node n
• Select the first member x?C, such that H(n)?x,
  label n by x. If such x doesnt exist, label n by
  ? .
• If n is labeled by x, x? ?, for each ? ?x,
  generate an arc labeled by ?. If ?m,
  H(m)H(n)??, the arc points to m (a graph)

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Optimization Strategies

• Reusing Nodes
• If ?m, H(m)H(n)??, the arc points to m (a
  graph)
• Closing
• If ?n, n labeled by ? and H(n)?H(n), then
  close n.
• Pruning (Subset problem)
• A old node n, labeled S A new node n, labeled
  ?
• If ??S, relabel n with ?. Remove arcs ??S- ?
• Interchange S and ? in collection

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New Measurements

• A diagnosis can predict some behavior ?
• Example
• SD ?OBS ?AB(c)c?COMPONENTS-? ?
• Confirming measurements preserve diagnoses (which
  predict ?)
• Disconfirming measurements reject diagnoses
  (which predict ?)

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Observation vs. Prediction
(12)
10
(12)
12
SD ?OBS ?M1out(Mult1)4, out(Mult2)6
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Summary of Reiters Paper

• The diagnosis is the minimal hitting set of
  conflict sets
• An algorithm for computing minimal hitting set

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Assignment 1

• Use the revised algorithm to compute minimal
  hitting set
• Language Java
• Inputs F is conflict sets (vector of vectors) in
  a file
• Outputs minimal hitting sets (vector of vectors)
• Option functions
• Display the tree on the screen (good for debug)
• The number of times visiting conflict sets
  (efficiency)
• Single fault diagnoses (only first level)
• Write result to a file
• NO more than required

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Please notice

• The input vectors in different sequences
• Reduce the access times to conflict sets F
• Option functions can add points
• Functions beyond basic functions and option
  functions can NOT add points

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My test cases

• 3-multi-2-adder
• Add1, Mult1, Mult2,Add1, Mult1, Mult3, Add2
• Full adder
• X1,X2,X1,A2,O1
• Case A
• a,b,b,c,a,c,b,d,b
• Case B
• 2,4,5,1,2,3,1,3,5,2,4,6,2,4,1,6

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Oral Test

• Read paper
• Randall Davis, Diagnostic Reasoning Based on
  Structure and Behavior, Artificial Intelligence
  24 (1984), 347-410
• Half hour presentation, half hour qa
• Audit students are invited to present