Modeling with Alternative Arithmetic

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Modeling with Alternative Arithmetic

• Steve Stevenson

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Issues in Modeling

• Symbolic representation of the system
• Introduce uncertainty
• Simulation of uncertain system

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Dynamical Systems

• The general assumption of dynamical systems
• a fixed rule describing time dependence in space.
• Real (well, ok, complex) numbers.
• Often expressed as differential equations
• Real systems very non-linear leading to need for
  simulation

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Types of Uncertainty

• Aleatoric uncertainty is the natural uncertainty
  due to randomness.
• Epistemic uncertainty is uncertainty in
  knowledge.
• While not discussed in the literature, I guess we
  need to computational uncertainty due to
  inexactness of floating point arithmetic.

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Assumptions

• Normal practice is to assume all uncertainty is
  in the numbers before the simulation sees it.
• This means that the uncertainty in the simulation
  is due to the arithmetic.
• (F?F)(x?x) y?y sim
• With the ?s being approximation errors.

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Uncertainty

• Eqn sim puts the simulation outside the
  modeling process because it does not address the
  problem facing the modeler uncertainty.
• So how do we change sim? Simplest
• (F?F?F)(x?x?x) y?y?y sim
• With ? being uncertainty in.

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Too Simplistic

• Numerical issues are not necessarily independent
  of the uncertainty
• Better maybe to invent something like an
  uncertainty operator ?(?) (Upsilon would make
  more sense but is the letter Y) and then have
  uncertain numbers ?(?)

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So we get

• (?F?(?F))(?x? ?x) ?y? ?y
• u-sim
• Now the question is, how do the numbers work?

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Types of Arithmetic

• Standards
• Formal algebraic systems natural numbers,
• IEEE ?54 Floating point.
• Interval arithmetic.
• Fuzzy arithmetic.
• And lots, lots more.

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Interval Arithmetic

• Instead of one number, keep the lower and upper
  bounds
• Inf(x), Sup(x)
• Now define operators to work that way.
• Sun Fortran 95 and C, C compilers all have
  interval as a primitive type.
• http//www.cs.utep.edu/interval-comp/

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Operations

• a,b c,d a c, b d
• a,b - c, d a - d, b -c
• a,b ? c,d min (ac, ad, bc, bd), max (ac,
  ad, bc, bd)
• a,b / c,d min (a/c, a/d, b/c, b/d), max
  (a/c, a/d, b/c, b/d)
• Division by an interval containing zero is not
  defined under the basic interval arithmetic.

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Interpretations

• Floating point interpretation Instead of doing
  error analysis with (x ? error) that is usually
  too pessimistic, keep inf(x),sup(x).
• Epistemic interpretation any value in
  inf(x),sup(x) is equally likely.

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Fuzzy Arithmetic

• Fuzzy arithmetic makes the assumption that the
  uncertainty can be quantified as a distribution.
• Motivation in expert systems, the expert has a
  good estimate of lowest value, highest value, and
  most likely value.

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Fuzzy Arithmetic II

• Can be built on interval arithmetic.
• http//documents.wolfram.com/applications/fuzzylog
  ic/Manual/10.html