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MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

- Louise-Travé-Massuyès, Liliana Ironi, Philippe

Dague - Presented by Nuri Tasdemir

Overview

- Different formalisms for modeling physical

systems - Mathematical aspects of processes, potential and

limitations - Benefits of QR in system identification
- Open research issues

QR as a good alternative for modeling

- cope with uncertain and incomplete knowledge
- qualitative output corresponds to infinitely many

quantitative output - qualitative predictions provide qualitative

distinction in systems behaviour - more intuitive interpretation

QR

- Combine discrete states-continous dynamics
- Finite no. of states transitions obeying

continuity constraints - Behaviour sequence of states
- Domain abstraction
- Function abstraction

Domain Abstraction and Computation of Qualitative

States

- Real numbers ? finite no. of ordered symbols
- quantity space totally ordered set of all

possible qualitative values - Qualititativization of quantitave operators
- a Q-op b Q(x op y) Q(x) a and Q(y) b
- C set of real valued constraints
- Sol(C) real solutions to C
- Q(C) set of qualitative constraints obtained

from C - Soundness ? C, Q(Sol(C)) ? Q-Sol(Q(C))
- Completeness ?Q-C, Q-Sol(Q-C) ?

Q(Sol(C))

Reasoning about Signs

- Direction of change
- S-,0,,?
- Qualitative equality ()
- ?a,b S, (a b iff (a b or a ? or b

?))

Reasoning about Signs

- Quasi-transitivity
- If a b and b c and b ? ? then a c
- Compatibility of addition
- a b c iff a c - b
- Qualitative resolution rule
- If x y a and x z b and x ? ?
- then y z a b

Absolute Orders of Magnitude

- S1 NL,NM,NS,0,PS,PM,PL
- S S1 ? X,Y ? S1-0 and XltY, where X lt Y

means ? x ? X and ? y ? Y, x lt y - S is semilattice under ordering ?
- define q-sum and q-product in lattice
- commutative, associative, is distributive

over - (S, , , ) is defined as Q-Algebra

Semi-Lattice Structure

Relative Order of Magnitude

- Invariant by translation
- Invariant by homothety (proportional transf.)
- A Vo B A is close to B
- A Co B A is comparable to B
- A Ne B A is negligible with respect to B
- x Vo y ? y Vo x
- x Co y ? y Co x
- x Co y, y Vo z ? x Co z
- x Ne y ? (x y) Vo y

Qualitative Simulation

- Three approaches
- 1-the component-centered approach of ENVISION by

de Kleer and Brown - 2-the process-centered approach of QPT by Forbus
- 3-the constraint-centered approach of QSIM by

Kuipers

Q-SIM

- Variables in form ltx,dx/dtgt
- transitions obtained by MVT and IVT
- P-transitions one time point ? time interval
- I-transitionstime interval ? one time point
- Temporal branching
- Allens algebra does not fit to qualitative

simulation

(No Transcript)

Allens Algebra

The Allen Calculus specifies the results of

combining intervals. There are precisely 13

possible combinations including symmetries (6 2

1)

Time Representation

- Should time be abstracted qualitatively?
- State-based approach(Struss) sensors give

information at sampled time points - Use continuity and differentiability to constrain

variables - Use linear interpolation to combine x(t), dx/dt,

x(t1) - uncertainty in x causes more uncertainty in dx/dt

so use sign algebra for dx/dt

System Identification

- Aim deriving quantitative model looking at input

and output - involves experimental data and a model space
- underlying physics of system (gray box)
- incomplete knowledge about internal system

structure ( black box) - Two steps
- (1) structural identification(selection within

the model space of the equation form) - (2) parameter estimation(evaluation of the

numeric values of the equation unknown parameters

from the observations)

Gray-Box Sytems

- RHEOLO ? specific domain behaviour of

viscoelastic materials - instantaneous and delayed elasticity is modeled

with same ODE - Either
- (1)the experimental assesment of material (high

costs and poor informative content) or - (2) a blind search over a possibly incomplete

model space (might fail to capture material

complexity andmaterial features - QR ? brings generality to model space M (model

classes) - S structure of material
- Compare QB(S) with Q(S)
- QRAqualitative response abstraction

Gray-Box Sytems

Black-Box Sytems

- given input and output find f
- difficult when inadequate input
- Alternative to NNs, multi-variate splines, fuzzy

systems - used successfully in construction of fuzzy rule

base

Conclusion and Open Issues

- QR as a significant modeling methodology
- limitations due to weakness of qualitative

information - Open issues
- - Automation of modeling process
- - determining landmarks
- - Compositional Modeling

THANKS FOR LISTENING!