View
215
Download
0
Category
Tags:
Preview:
Citation preview
1
Consistent-based Diagnosis
Yuhong YANNRC-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
3
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
4
Model
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)...
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)
Mult1
Mult2
Mult3
Add2
23
23
23
Add1 A
B
5
Diagnosis on models of structure and function
actualdevice
observedbehaviour
modelof the device
predicted behaviour
diagnosis
design textbook first principles ....
model of the structureof the device and of the (nominal) behaviourof each type ofcomponent
diagnosis = removing discrepanciesbetween the nominal predicted behaviourand the observed one
From Luca Console
6
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: , :-
7
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
8
Consistent There is an interpretation that makes a set
of formulas true. Example:
Consistent: {AB, AB} Inconsistent: {AB,AB, AB, AB}
Connect formula sets with union operator Example: {AB, AB} {AB}
9
Some explanations Consistency means “AND” of the formulas
in the sets Consistency of SDOBSAB(.)¬AB(.)
10
Definition: diagnosisA diagnosis for (SD, COMPONENTS, OBS) is a
minimal set COMPONENTS such that
SDOBS{AB(c)|c}{¬AB(c)|cCOMPONENTS- }
is consistent
is the smallest set of components• SD is the right modes
11
Diagnosis {} is a diagnosis if all components are in
right modes, i.e.SDOBS{¬AB(c)|cCOMPONENTS} is consistent Before computing diagnosis
Should have enough observables Is a NP-hard problem
12
Proposition 3.3If is a diagnosis for (SD, COMPONENTS, OBS), then
for each ci.
SDOBS{¬AB(c)|cCOMPONENTS- }|=AB(ci)
faulty components are logically determined by the normal components
13
Compute diagnoses Direct compute Compute conflict set (by using ATMS) then
compute diagnoses from conflict set
14
Definition: Conflict setA conflict set for (SD, COMPONENTS, OBS) is
a set {c1, …,ck}COMPONENTS such thatSDOBS{¬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)
15
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)
16
Minimal hitting setA hitting set for a collection of sets C is a set
H SCS such that HS{ } for each SC.
A hitting set is minimal iff no proper subset of it is a hitting set for C.
17
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
18
Mult1
Mult2
Mult3
Add2
23
23
23
Add1 A
B
Prediction: A=12, B=12
12
12Observation A=10, B=12
10
12
A=10 generates two conflicts:{A1, M1, M2}{A1, M1, M3, A2}
Example
19
HS Tree{M1,M2,A1}
{M3,A2,M1,A1}
M1M2
A1
M3A2 M1
A1
20
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
21
Algorithm: outline Generate a HS-tree Return {H(n)|n is a node labeled }
22
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 xC, such that H(n)x={}, label n by x. If such x doesn’t 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)
23
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
24
New Measurements A diagnosis can predict some behavior
Example SD OBS {¬AB(c)|cCOMPONENTS-}|=
Confirming measurements preserve diagnoses (which predict )
Disconfirming measurements reject diagnoses (which predict ¬)
25
Observation vs. Prediction
Mult1
Mult2
Mult3
Add2
2
3
23
23
Add1 A
B(12)
(12)10
12
SD OBS {M1}|={out(Mult1)=4, out(Mult2)=6}
26
Summary of Reiter’s Paper The diagnosis is the minimal hitting set of
conflict sets An algorithm for computing minimal
hitting set
27
Assignment 1Use 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
28
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
29
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}}
30
Oral Test Read paperRandall Davis, “Diagnostic Reasoning Based
on Structure and Behavior”, Artificial Intelligence 24 (1984), 347-410
Half hour presentation, half hour q&a Audit students are invited to present
Recommended