MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

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MATHEMATICAL FOUNDATIONS OF QUALITATIVE REASONING

• Louise-Trave Massuyes, Liliana Ironi, Philippe
  Dague
• Presented by Özgür Yilmaz

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Overview

• Different formalisms of physical systems ?
  qualitative predictions
• Mathematical aspects of processes, potential and
  limitations
• Benefits of QR in system identification
• Open research issues

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QR as good alternative for modeling

• cope with uncertain and incomplete knowledge
• infinity of runs in compact form
• qualitative prediction? qualitative distinction
  in systems behaviour
• more intuitive interpretation

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QR main goals

• Combine discrete events-continous dynamics
• Finite no. of states transitions obeying
  continuity constraints
• envisionment all possible states
• Behaviour sequence of states
• Domain abstraction
• Function abstraction

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Domain Abstraction and Computation of Qualitative
States

• Real domain value of variable ? finite no. of
  ordered symbols (landmarks)
• quantity space (q) totally ordered set of all
  possible qualitative values
• Q R ? quantity space, expected to preserve
  arithmetic operators
• Q(a opR b) Q(a) q-op Q(b)
• C set of real valued constraints
• Sol(C) real solutions to C
• Q(C) set of qual. 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

• define relation on S-,0,
• define function on S-,0,,?
• x sign(x)
• . is not a homeomorphism over addition ( ab
  is not a b )
• no cancellation law

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Reasoning about Signs

• Quasi-transitivity If ab and bc and bis not
  ? then ac
• lack of additive inverses
• define unary negation -s-s
• define subtraction s-t s (-t)
• Qualitative resolution rule If xya and xzb
  and x is not ? then
• yzb

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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 XltY means
  ? x ? X and ? y ? Y ? xltY
• S is semilattice under ordering ?
• define q-sum and q-product in lattice
• commutative, associative,is distributive over
  ? Q-algebra

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Semi-Lattice Structure
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Relative Order of Magnitude

• A ltlt B A is negligible compared to B
• A B A is very close to B
• AB A is the same magnitude as B
• (obtained by adding infinitely small and large
  numbers)
• Assign labels to ratios AltB A/B lt 1 (A
  is slightly smaller than 1)
• k-neglible, k-proximable, k-distant

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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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Component-based approach

• system composed of components
• Behaviour specified by internal laws
• Set of confluences which are linear (since xi x
  or x) ? Ax B (QLS)
• Qualitative linear system (QLS) is sound and
  complete
• Advantages
• - fixed topology ? efficient computation
• - standard representation in traditional
• engineering
• Disadvantages
• - how to construct network models
• - device model can be unnatural

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QPT(Qualitative Process Theory)

• Individual views, representing objects from a
  particular perspective
• Processes, which represent active changes taking
  place
• Advantages
• -objects can come to existence or vanish
• - process provide notion of causality
• -explicit representation of modeling assumptions
• Disadvantages
• -analysis type determines dependent
  independent
• variables
• -more inference to set up model

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QPT
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Time Representation

• Should time be abstracted qualitatively?
• State-based approach(Struss) sensors give
  information at sampled time points
• determine qual. State at each time slice, check
  for consistency (no use of derivatives)
• Better alternative use continuity and
  differentiability to constrain variables
• Use linear interpolation to combine x(t), dx/dt,
  x(t1), but uncertainty in dx/dt ? ,so use sign
  algebra for it

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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
• Goal avoid temporal branching
• Allens axioms no fit to qual. simulation

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Allens Axioms
The Allen Calculus specifies the results of
combining intervals. There are precisely 13
possible combinationsincluding symmetries (6 2
1)
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System Identification

• Aim derive 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
• used successfully in construction of fuzzy rule
  base (Q-SIM generates states distinctions and
  complexity of f)
• (1) to study the dynamics of the blood glucose
  level in diabetic patients in response to insulin
  therapy and meal ingestion (Bellazzi et al. 1998)
• (2) to identify the dynamics of intracellular
  thiamine in the intestine tissue

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Conclusion and Open Issues

• A comprehensive overview of QR
• 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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