Behavior Modes
Meir Kalech
Partially Based on slides of Brian Williams, Luca Console and Peter
struss
Outline


Last lecture:
1.
Generation of tests/probes
2.
Measurement Selection
3.
Probabilities of Diagnoses
Today’s lecture:
1.
Models of correct + faulty behavior
2.
Sherlock engine
3.
Abductive diagnosis
4.
Qualitative models
Exploiting models of correct/faulty
behavior
Initial proposal: using only models of correct behavior
 They are those that are in strict accordance with
the goals (easy to acquire, e.g., from design)
 But unfortunately they are not always sufficient
 Need of exploiting also fault models of some form
 predictive models
[Struss, Dressler, 89] – GDE+
[de Kleer, Williams 89] - SHERLOCK
 “weak” models of physical impossibility
[Friedrich et al. 90]
 behavioral models
[Console, Torasso, 91]
Diagnosis With Only the Unknown
0
A
X
Inverter(i):
 G(i):
 U(i):
B
Y
C
0
Out(i) = not(In(i))
• Isolates surprises
• Doesn’t explain
Nominal and Unknown Modes
Diagnosis With Only the Known
0
A
X
Inverter(i):
 G(i):
 S1(i):
 S0(i):
B
Y
C
Out(i) = not(In(i))
Out(i) = 1
Out(i) = 0
Exhaustive Fault Modes
0
• No surprises
• Explains
Solution: Diagnosis as
Estimating Behavior Modes
0
A
X
Inverter(i):
 G(i):
 S1(i):
 S0(i):
 U(i):
B
Y
C
Out(i) = not(In(i))
Out(i) = 1
Out(i) = 0
0
• Isolates surprises
• Explains
Nominal, Fault and Unknown Modes
Measurement motivation to use
Behavior modes
Knowledge of failure modes is important to
decide what measurement to make next.
 If all faults were equally likely, measuring X or
Y provides equal information.
 Suppose:



Inverters A and B almost always fail by stuck-at-1.
Inverter C almost always fails by stuck at-0.
It is unlikely that inverter A is failing.
 The likely failures of inverters B and C are
consistent with the symptom

Behavior Modes
• System comprises a (finite) set of components
COMPS = { Ci }
• Each Ci has a (finite) set of behavior modes
modes(Ci) = { mij(Ci)}
• E.g.
- (unique) correct behavior: ok(Ci)
- (any) faulty behavior:  ok(Ci)
Definition
- a specific fault: stuck-closed(valve
)i (Mode Assignment)
• COMPS’
 COMPS
• Behavior mode  operating mode
(of correct
behavior)
• E.g. blocking mode of a diode• MA = {mij(Ci)  Ci  COMPS’ }

• or MA = Ci  COMPS’ mij(Ci)
• MA complete: COMPS’=COMPS
Diagnoses as Assignments of Fault
Modes
Definition (Diagnosis):
• A complete mode assignment MA that is consistent with
the observations:
 SD  MA OBS
^
Definition (Mode Assignment)
• COMPS’  COMPS
• MA = {mij(Ci)  Ci  COMPS’ }

• or MA = Ci  COMPS’ mij(Ci)
• MA complete: COMPS’=COMPS
Yet Another Simple Example
 Head lights work
 Starter and rear light don’t
 Obvious diagnosis:
Starter and rear light are
broken
Battery
Starter
RLight
HLight
Fault Localization for the Simple
Example - Conflicts 1 and 2
• ok(Battery)  ok(Wire1)
 ok(Wire2)  ok(Starter)
 active(Starter)
• OBS   active(Starter)
•  Conflict
ok(Battery)  ok(Wire1)
 ok(Wire2)  ok(Starter)
Battery
Wire1
Wire2
Starter
Wire3
Wire4
RLight
Analogously:
• ok(Battery) ok(Wire1)
 ok(Wire2) ok(Wire3)
 ok(Wire4)  ok(RLight)
Wire5
Wire6
HLight
Fault Localization for the Simple
Example - Conflicts 3 and 4
• lit(HLight)  ok(HLight) 
ok(Wire5)
 ok(Wire6)  ok(RLight)
 lit(RLight)
• OBS   lit(RLight)
•  Conflict
ok(HLight)  ok(Wire5)
 ok(Wire6)  ok(RLight)
Battery
Wire1
Wire2
Starter
Wire3
Wire4
RLight
Analogously:
•  ok(HLight)  ok(Wire5)
 ok(Wire6)  ok(Wire3)
 ok(Wire4)  ok(Starter)
Wire5
Wire6
HLight
Fault Localization for the Simple
Example - Hitting Sets
• {Battery, Wire1, Wire2, Starter}
• {Battery, Wire1, Wire2, Wire3,
Wire4, Rlight}
• {HLight, Wire5, Wire6, Rlight}
• {Hlight, Wire5, Wire6, Wire3,
Wire4, Starter}
• {Starter, Rlight}
• {Battery, HLight}
• {Wire1, Wire5}
Battery
Wire1
Wire2
Starter
Wire3
?!
• + 19 NONSENSES more!
Wire4
RLight
Wire5
Wire6
HLight
What Makes Most of the Fault
Localizations Implausible?
• If the battery were broken, the
headlights would not be lit
• Broken headlights cannot be lit
•  Knowledge about faults can
reduce the set of fault
localizations
• {Starter, Rlight}
• {Battery, HLight}
• {Wire1, Wire5}
• + 19 more!
Battery
Wire1
Wire2
Starter
Wire3
?!
Wire4
RLight
Wire5
Wire6
HLight
Fault Models - “Physical Negation”
Fault models are needed
predictive: bulb(B)  AB(B) 
light(B,off)
voltageIn(B,X) 
Outline


Last lecture:
1.
Generation of tests/probes
2.
Measurement Selection
3.
Probabilities of Diagnoses
Today’s lecture:
1.
Models of correct + faulty behavior
2.
Sherlock engine
3.
Abductive diagnosis
4.
Qualitative models
Search Guided by Probabilities:
SHERLOCK ([de Kleer- Williams 89])
• Basic Idea: Search for the most probable explanations of the
observations
• Fault models for each component (type)
• Possible: unknown fault mode
•
•
•
•
Modes have prior probability
 Mode assignments have a probability
SHERLOCK: best first search for consistent mode assignments
termination criteria
Leading Diagnoses


Complexity of diagnoses space:
n-#components, m-#modes
GDE 2n  SHERLOCK: mn
To reduce the high complexity, generate only
leading diagnoses:
1.
2.
3.
4.
Diagnoses are those with the highest probabilities.
No more than k1 (=5) leading diagnoses.
Candidates with probability less than 1/k2(=100)
of the best diagnosis are not considered
The diagnoses need not include more than k3
(=0.75) of the total probability mass of the
candidates.
SHERLOCK - Example: Two
Inverters
0
A
1
B
Model of Inverter X
Mode
Behavior
Prior
XN: Normal
X1: Stuck at 1
Out =In
Out = 1
.99
.006
X0: Stuck at 0
XU: Unknown
Out = 0
.003
.001
SHERLOCK - Example: Two
Inverters
0
A
1
B
In slide 25 see how to
generate conflicts and
diagnoses
• Conflicts: {AN, BN}, {B0}, {A1, BN}
Model of Inverter X
Mode
Behavior
Prior
XN: Normal
X1: Stuck at 1
Out =In
Out = 1
.99
.006
X0: Stuck at 0
XU: Unknown
Out = 0
.003
.001
SHERLOCK - Example: Two
Inverters
0
A
B
1
• Conflicts: {AN, BN}, {B0}, {A1, BN}
• Inspired by GDE + modes
• Diagnosis is an explanation: SDMAOBS ┴ , where
MA=CiCOMPSmij(Ci), (rather than a set of faulty
components)
• I.E., the diagnosis set contains all the combinations
of the components’ modes except of conflicts.
SHERLOCK - Example: Two
Inverters
0
A
B
1
Generated by ATMS
Conflicts: {AN, BN}, {B0}, {A1, BN}
Full diagnostic explanations with probabilities:
[[AN , B1]]
.00594
[[AN , BU]]
.00099
[[A1 , B1]]
.00004
[[A1 , BU]]
.00001
[[A0 , BN]]
.00297
[[A0 , B1]]
.00002
[[A0 , BU]]
.000003
[[AU , BN]]
.00099
[[AU , B1]]
.00001
[[AU , BU]]
.000001
Diagnoses set does not contain:
{AN, BN}, {A1, BN}, and supersets of {B0}
SHERLOCK - Example: Two
Inverters
0
A
B
1
Generated by ATMS
Conflicts: {AN, BN}, {B0}, {A1, BN}
Full diagnostic explanations with probabilities:
[[AN , B1]]
.00594
[[AN , BU]]
.00099
[[A1 , B1]]
.00004
[[A1 , BU]]
.00001
[[A0 , BN]]
.00297
[[A0 , B1]]
.00002
[[A0 , BU]]
.000003
[[AU , BN]]
.00099
[[AU , B1]]
.00001
[[AU , BU]]
.000001
• Exhaustive search impossible
• Perform best first search
SHERLOCK - Example: Search
Strategy
{An,Bn} .98010 3 x
0
A
1
B
{An}
.99
2
{An,B1} .00594 5
{An,B0} .00297 8 x
{An,Bu} .00099
{}
1.0
1
{A1,Bn} .00594 6 x
{A1}
.006 4
{A1,B1} .00004
{A1,B0} .00002
x
{A1,Bu} .00001
{A0,Bn} .00297 9
{A0}
Legend:
MA Probability Step #
x
Consistent ?
.003 7
{A0,B1} .00002
{A0,B0} .000009
{A0,Bu} .000003
inconsistent
{Au} the
.001
Generate
next explanation with the
highest probability
x
SHERLOCK, Process in details:
1. Find Symptoms & Conflicts
0
A
X
B
0
G
1
G
Y
0
1
C
0
G
0
Conflict:
not (G(A) and G(B) and G(C))
Finding conflict through ATMS, but generate
focus environments
More Symptoms & Conflicts
0
A
X
B
0
S1
1
G
Y
0
C
0
G
0
1
not (S1(A) and G(B) and G(C)
More Symptoms & Conflicts
0
0
A
X
B
S0
Y
0
1
not (S0(B) and G(C))
C
0
G
0
More Symptoms & Conflicts
0
A
X
B
0
not S1(C)
Y
C
0
S1
1
0
All Minimal Conflicts

< S1(C) >

< S0(B), G(C) >

< S1(A), G(B), G(C) >

< G(A), G(B), G(C) >
2. Constituent Diagnoses from
Conflicts

Diagnosis is an explanation, so it must contains no
conflict:

< S1(C) >: not S1(C):
=> G(C) or S0(C) or U(C)

<S0(B) and G(C)>:not (S0(B) and G(C)) => not S0(B) or not G(C)
=> G(B) or S1(B) or U(B) or S1(C) or S0(C) or U(C)

< S1(A), G(B), G(C) >
=> G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C) or U(C)

< G(A), G(B), G(C) >
=> S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C) or U(C)
3. Generating Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[U(C)]
3. Generating Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]


[U(C)]
[S0(C)]
3. Generating Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[U(C)]
[S0(C)]

[U(B),G(C)]

3. Generating Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[U(C)]
[S0(C)]

[U(B),G(C)]


[S1(B),G(C)]
3. Generating Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[U(C)]
[S0(C)]

[U(B),G(C]


[S1(B),G(C)]

[U(A),G(B),G(C)]
3. Generate Kernel Diagnoses

[G(C),S0(C),U(C)]

[G(B),S1(B),U(B),S1(C),S0(C),U(C)]

[G(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(A),S0(A),U(A),S1(B),S0(B),U(B),S1(C),S0(C),U(C)]

[S1(B),G(C)]

[U(C)]
[S0(C)]

[U(A),G(B),G(C)]

[U(B),G(C]

[S0(A),G(B),G(C)]




These are the kernel diagnoses
But for [U(C)] (for instance), what are the modes of A and B?
The best first search finds the most likely modes of A and B.
Candidate Initial (prior)
Probabilities
p(c)   p(m)
m c
A
B
C
p(G)
.99
.99
.99
p(S1)
.008
.008
.001
p(S0)
.001
.001
.008
p(U)
.001
.001
.001
No observations

With no observations Sherlock finds the single
leading diagnosis

The unfocused Sherlock finds 43 diagnoses
Input I=0

Sherlock computes the following environments:

The focused Sherlock finds no label for X=0 as it
does not hold in the single leading diagnosis.
S
0(
C
)
(C
S
1(
)
A
),
S
0(
C
)
)
Y
U
(B
B
U
)
S
1(
B
)
(A
X
U
A
S
0(
A
)
O
K
1.2
1
0.8
0.6
0.4
0.2
0
0
C
0

1.
2.
3.
4.

A
X
B
Y
C
0
Suppose O=0  Minimal conflicts are:
< S1(C) >
< S0(B), G(C) >
Next highest probability:
< S1(A), G(B), G(C) >
< G(A), G(B), G(C) >
Leading candidates:
For instance:
These are the most likely
modes of A and B, beyond
the kernel diagnosis U(C)
0
A
X
B
Y
C
0

Now, the posterior behavior modes probabilities are:

ATMS labels are:
0
A
B
X
C
Y
0
0.5
0.4
0.3
0.2
0.1
)
A
)S
0(
C
)
)
)
U
(B
U
(C
S
1(
Top 6 of 64 = 98.6% of P
U
(A
C
)
S
0(
B
)
S
1(
A
)
S
0(
O
K
0
Outline


Last lecture:
1.
Generation of tests/probes
2.
Measurement Selection
3.
Probabilities of Diagnoses
Today’s lecture:
1.
Models of correct + faulty behavior
2.
Sherlock engine
3.
Abductive diagnosis
4.
Qualitative models
Abductive diagnosis

The definition above is based on consistency:




explanation  consistent with the observations
Weak notion of explanation
A diagnosis D explains an observation m if it does
not contradict m (D does not support not m).
Abductive diagnosis:




A stronger notion of explanation
Explanation  implies the observations
A diagnosis D explains an observation m if it
supports m (D  m).
Abductive diagnosis [Poole 87][Console, Torasso, 89]
Abductive diagnosis – Poole et al. 87
A different concept:

A diagnosis is not a logical consequence of our
observations.
Exactly the opposite:

The observation should be shown to be logical
consequences of our knowledge and diagnosis.
Abductive diagnosis - definition

Given




Cxt: inputs.
The data that let the diagnoser to
make prediction about the behavior
of the system
SD
Modes
Observations, with the distinction between
contextual data (Cxt) and observations (Obs)
Determine

An assignment of behavior modes to components
D = {mi(ci) | mi Modes(ci) }
such that:
1.
SD  Cxt  D |= Obs
(SD  Cxt  D consistent)
2.
A continuum of definitions
Console and Torasso 91
Given OBS, partition it into Obs1  Obs2
SD  Cxt  D  Obs2 |= Obs1
SD  Cxt  D  Obs1  Obs2 consistent
Since Abduction diagnosis is more restrict than
consistency:
Abduction provides a subset of the solutions
provided by consistency-based diagnosis
Varying Obs1 we have a continuum of definitions:
Obs1=OBS,Obs2= Ø  abduction diagnosis of Poole86
Obs1=Ø, Obs2=OBS  consistency based diagnosis
Abductive or Consistency?
Criteria to select the most appropriate definition



abduction and consistency are the two extremes of
a spectrum of alternatives
abduction is the most restrictive definition
 it requires “complete” models
 it provides a strong (physical) notion of
explanation
consistency-based is less restrictive
 less constraints on the models
 weaker notion of explanation
Computing abductive diagnoses: an example
oil_cup
normal holed
oil_loss
radiator
normal holed
oil_level
water_level
oil_below_car
normal low
normal low
oil_gauge
normal red
water_temp
normal high
Model:
oil_cup(normal)  oil_level(normal)
oil_cup(holed)  oil_loss(present)
oil_loss(present)  oil_below_car(present)
oil_loss(present)  oil_level(low)
oil_level(normal)  oil_gauge(normal)
oil_level(low)  oil_gauge(red)
oil_level(normal)  water_level(normal) 
engine(on) 
engine_temp(normal)
...
engine_on
normal high
engine_temp
Obs1 = {engine_temp(high)}
Two minimal candidate
explanations
 E1 = { oil_cup(holed) }
 E2 = {radiator(holed)}
Outline


Last lecture:
1.
Generation of tests/probes
2.
Measurement Selection
3.
Probabilities of Diagnoses
Today’s lecture:
1.
Models of correct + faulty behavior
2.
Sherlock engine
3.
Abductive diagnosis
4.
Qualitative models
Qualitative models in MBD
Instead of educate quantity model give qualitative values

In many cases qualitative models are sufficient for
diagnostic purposes.

They are close to the way expert diagnosticians
operate.

In many cases imprecise knowledge makes numerical
models useless.

Usually imprecision in the observations.

Easier to use (in some cases).
Qualitative models in MBD (2)
New problems:

choosing the “right” qualitative models
 choosing
the right form of qualitative reasoning
 choosing the “right” qualitative abstractions
mapping quantities into qualitative values
(e.g., for observations)
 dealing with the ambiguities that arise as a
result of the abstraction

A simple example
pump
1 pipe 2
tank
Pipe (flow direction from 1 to 2):
 interface variables:
P1, P2, F1, F2 (0, +)
constraints

Structure:
Fout= F1, Pout= P1 , P2= Ptank ,
F2= Fin
Pump:






Pipe Ok F2= F1 P1=P2
Pipe occluded F2=0, P1=+
Pipe leaking F2=0, P1=0
interface variables:
Tank:
 Pout, Fout (0, +)
 interface variables: Fin, Ltank, Ptank
(0, +)
constraints (pump ON)
 Pump OK
 internal variable dLtank (0, +, -)
Pout= +  Fout=0,
 constraints
Pout=0  Fout=+
 Tank Ok Ptank=Ltank, dLtank= Fin
 Pump blocked Fout=0
 Pipe leaking Ptank=0, dLtank= -
A simple example (2)
Diagnosis (using pump and pipe only, we will return
later to considering the tank)


Obs {F2=0, P2=0 }
the set of assumptions {ok(pump), ok(pipe)} is a conflict




from P2=0 and ok(pipe) we predict P1=0 and from the
structure Pout=0 (for the pump)
then from ok(pump) we predict Fout=+ i.e., F1=0 (for the pipe)
and finally using again ok(pipe) we predict F2=+
(CONFLICT!!!!)
Three set of assumptions explain the observations (are
candidate diagnoses)

D1  { blocked(pump) }
D2  { occluded(pipe)}

D3  { leaking(pipe)}

A second example
pump1
pipe1
join
Diagnosis

pump2

pipe2
if: Obs = {Fout=0, Pout=0 }
Then several minimal
explanations

Join



interface variables:
 Pout, Fout, Pin1, Fin1
Pin2, Fin2 (0, +)
constraints
 Pump OK




D1  { blocked(pump1) }
D2  { occluded(pipe1)}
D3
D1
D2
D3




Pin1= Pin2= Pout
Fout= Fin1  Fin2
( qualitative add)

0
+
??
0
0
+
+
??
+
+
{
{
{
{
leaking(pipe1)}
blocked(pump2) }
occluded(pipe2)}
leaking(pipe2)}