MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

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MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

• Louise-Travé-Massuyès, Liliana Ironi, Philippe
  Dague
• Presented by Nuri Tasdemir

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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 systems 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 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))

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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 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

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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 x
• x Co y ? y Co x
• x Co y, y Vo z ? x Co z
• x 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 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

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Allens 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(t1)
• 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)
• QRAqualitative 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!