Transcript

MATHEMATICAL FOUNDATIONS OF

QUALITATIVE REASONINGLouise-Travé-Massuyès, Liliana Ironi,

Philippe Dague

Presented by Nuri Taşdemir

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Overview

• Different formalisms for modeling physical systems

• Mathematical aspects of processes, potential and limitations

• Benefits of QR in system identification

• Open research issues

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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 system’s behaviour

• more intuitive interpretation

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QR

• Combine discrete states-continous dynamics

• Finite no. of states – transitions obeying continuity constraints

• Behaviour: sequence of states

• Domain abstraction

• Function abstraction

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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 CQ(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))

CQ|Q(C)C

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Reasoning about Signs

• Direction of change• S={-,0,+,?}• Qualitative equality (≈)

a,b S, (a ≈ b iff (a = b or a = ? or b = ?))

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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

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Absolute Orders of Magnitude

• S1 = { NL,NM,NS,0,PS,PM,PL }

• S = S1 {[X,Y] S1-{0} and X<Y}, where X < Y means x X and y Y, x < 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

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Semi-Lattice Structure

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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 xx Co y → y Co xx Co y, y Vo z → x Co zx Ne y → (x + y) Vo y

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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

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Q-SIM

• Variables in form <x,dx/dt>

• transitions obtained by MVT and IVT

• P-transitions: one time point time interval

I-transitions:time interval one time point

• Temporal branching

• Allen’s algebra does not fit to qualitative simulation

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Allen’s Algebra

The “Allen Calculus” specifies the results of combining intervals. There are precisely 13 possible combinations including symmetries (6 * 2 + 1)

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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(t+1)

uncertainty in x causes more uncertainty in dx/dt so use sign algebra for dx/dt

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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)

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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)QRA:qualitative response abstraction

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Gray-Box Sytems

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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

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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

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THANKS FOR LISTENING!


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