1. Task-Dependent Qualitative Model Abstraction
Martin Sachenbacher
September 2003

2. On-board, Real-time Environments

3. Qualitative Model Abstraction
Map real-valued constraints to finite domains
Problem: Find adequate distinctions
avoid unnecessary details

4. Interchangeability [Freuder 91]
Domain values val1, val2 are interchangeable, iff exchanging val1, val2 preserves solutions
Task-independent form of abstraction v2 R
val1
val'
val2 v1

5. Task-Dependent Abstraction
Idea: “Relaxed” version of interchangeability that keeps only distinctions essential for a given task
Granularity of observations and solutions Ç = =
Robs R
R Ç Robs

6. Task-Dependent Abstraction
Observable Distinctions
Partition defining granularity of observations
tobs
tobs (Robs) Ç = =
Robs R
R Ç Robs

7. Task-Dependent Abstraction
Target Distinctions
Partition defining granularity of solutions
tobs
ttarg
ttarg(R Ç tobs(Robs))
tobs (Robs) Ç = =
Rext(v) R
R(v) Ç Rext(v)

8. Task-Dependent Abstraction
Induced Abstraction
(Maximal) partition preserving solutions
tind(R)
tind
Induced
tobs
ttarg ?
ttarg(R Ç tobs(Robs)) Ç = =
Rext(v)
tobs (Robs) R
R(v) Ç Rext(v)

9. Qualitative Abstraction Problem
Find domain abstraction tind for R such that " Robs
Adequacy: ttarg(R Ç tobs(Robs)) = ttarg(tind(R) Ç tind(tobs(Robs)))
Simplicity: any abstraction of tind is not adequate
Definition of Properties
“Obs-complete”, iff each possible Robs can occur
“Sol-complete”, iff each possible solution can occur

10. Determining Induced Abstractions
tobs

11. Determining Induced Abstractions
, ttarg
tobs

12. Determining Induced Abstractions1 1 2 1 2 3 1
1,3 2 4 1
1,3 2

13. Determining Induced Abstractions
Partition S 1
1,3 2

14. Theorem
Induced abstraction tind formed by intersection of interchangeable values for each element of S
Partition S
val2
val1 1
1,3 2

15. Results [Sachenbacher et al. 01]
On-board Vehicle Diagnosis Application
Observable distinctions: sensor values
Target distinction: threshold for air/fuel ratio 

16. Discussion Conclusions Current Work
Generalizes notion of interchangeability
Increases efficiency and/or fidelity of models
Facilitates re-usability of model for several tasks
Current Work
Extend to probabilistic models
Extend to quantitative constraints
Extend to dynamic models (transitions)
Frame as (pareto-) optimization problem

17. Material
Pedal Position Sensor Example

18. Example: Pedal Position Sensor
Potentiometer
Switch
Pedal
vleft
vright
vleft
vright
On-board Diagnosis requires distinction corresponding to switching point
Battery
vpot
vswitch
Control Unit