Mathematical Foundations of Qualitative Reasoning

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Mathematical Foundations of Qualitative Reasoning

• Orhan Sönmez

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Why Qualitative Reasoning?

• Handles uncertain and incomplete data
• Infinity of numeric runs
• Qualitative distinctions in the behavior of the
  system
• More intuitive explanation

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Aspects of Qualitative Reasoning 1

• Finite states
• Transition between states supplies continuity
• Discrete events and continuous models

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Aspects of Qualitative Reasoning 2

• Domain abstraction
• Function abstraction
• Example
• Envisionment

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Domain Abstraction

• Real domain values ? Finite ordered landmarks
• Value of a variable is either a landmark or an
  interval
• Quantity space is the finite ordered set of all
  possible qualitative values

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Sound Complete

• Set of constraints C
• Abstracted into Q(C)
• Q-Sol and Sol
• Q is sound iff Q(Sol(C)) ? Q-Sol(Q(C))
• All solutions are found
• Q is complete iff Q-Sol(Q(C)) ? Q(Sol(C))
• No spurious

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Signs 1

• Defining a relation on S -,0,
• Addition and multiplication of signs

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Signs 2

• Qualitative Equality
• For any a and b in S, a b iff a b or a ? Or
  b ?
• Quasi-transitivity
• If a b and b c and b ? ? then a c
• Compatibility
• a b c is equal to a c b
• Qualitative Resolution
• x y a and x z b and x ? ? then y z a
  b

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Orders of Magnitude

• Absolute orders of magnitude (AOM)
• Generalized sign model
• Relative orders of magnitude (ROM)
• Comparative relations between quantities

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Absolute orders of magnitude

• Quantity space
• Semi-lattice structure

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(No Transcript)
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Relative order of magnitude

• Negligible (Ne)
• A B
• Comparable (Co)
• A B
• Close (Vo)
• A B

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Relative order of Magnitude

• Example
• Momentum MVi mvi constant
• Conservation MVi2 mvi2 constant
• what we know m Ne M V Vo v
• what we estimate Vi decreases a bit
• vi changes sign and increases

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Qualitative Simulation

• Component-based approach
• ENVISION (Kleer and Brown,1984)
• Process-based approach
• QPT (Forbus,1984)
• Individual views Processes
• Constraint-based approach
• QSIM (Kuipers,1986)

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Time Representation 1

• Should time be qualitatively abstracted?
• State-based
• Observations at given sampled time points
• Local static models
• Continuity or Differentiability
• Allen Calculus
• 13 qualitative combinations of time intervals

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Time Representation 2

• Time-point
• Instant that qualitative state changes
• Not mapped into physical time
• Time intervals
• Transactions
• P (point to interval) and I (interval to point)
• Temporal branching
• Which event first? Or simultaneously?

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Functional Abstraction

• Constraint satisfaction techniques (QSIM)
• Arithmetic ADD, MULT, MINUS
• Differential DERIV
• Functional M, M-
• Corresponding values
• Landmark tuples appearing in the constraint
• Limited domain validity
• Quantity space restriction
• Region transactions (modeling discontinuity)

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Combining Quantitative and Qualitative Knowledge

• Semi-quantitative simulation approach
• Q2 and Q3 (Kuipers)
• Qualitative phase-space approach
• Systems theory
• Qualitative reasoning with engineering
• Numeric simulation
• System identification

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System Identification 1

• Deriving a quantitative model of a dynamic system
  from input-output observations
• Experimental data and model space
• Structural identification
• Selection within the model space
• Parameter estimation
• Evaluation of unknown parameters

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System Identification 2

• Gray-box
• Underlying physics of the system
• Initial guess of model
• Black-box
• Knowledge of internal structure in incomplete
• Construction of a relation between input-output
  variables
• Choice of suitable function complexity

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Gray Box 1

• Example RHEOLO
• Specific domain behaviour of viscoelastic
  materials
• Instantaneous and delayed elasticity has the same
  ODE
• But different qualitative behaviours

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Gray Box 2

• QR has significant contributions
• Model space characterization
• Partition into model classes according to
  qualitative behaviours

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Black Box

• Requires sufficient data set
• Resulting model does not capture any structural
  knowledge
• Generally qualitative physical knowledge is
  available
• Example Fuzzy systems with QR
• Meaningful fuzzy rules

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Open Issues

• Hybrid approaches
• Not pure qualitative neither pure quantitative
• Determining landmarks from numeric input-output
  observations
• Compositional modellings

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Conclusion

• Mimicing human qualitative reasoning
• Significant modeling methodology
• Some limitations due to the weakness of
  qualitative knowledge

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Reference

• Mathematical Foundations of
• Qualitative Reasoning
• Louise Trave-Massuyes, Liliana Ironi, Philippe
  Dague
• AI Magazine Winter 2004
• Vol. 24, Issue 4 p. 91

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Questions

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