MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING Louise-Travé-Massuyès, Liliana Ironi, Philippe...
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Transcript of MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING Louise-Travé-Massuyès, Liliana Ironi, Philippe...
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!