1. Modeling with Alternative Arithmetic
Steve Stevenson

2. Issues in Modeling Symbolic representation of the system
Introduce uncertainty
Simulation of uncertain system

3. 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
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.
A dynamical system has a state determined by a collection of real numbers. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.

4. 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.

5. (F+F)(x+x) = y+y [sim]
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.

6. (F+F+F)(x+x+x) = y+y+y [sim]
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”.

7. 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” ()

8. (F+(F))(x+ x) = y+ y
So we get …
(F+(F))(x+ x) = y+ y
[u-sim]
Now the question is, how do the numbers work?

9. Types of Arithmetic Standards And lots, lots more.
Formal algebraic systems: natural numbers, …
IEEE ?54 Floating point.
Interval arithmetic.
Fuzzy arithmetic.
And lots, lots more.

10. 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.

11. 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.

12. 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.

13. 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.

14. Fuzzy Arithmetic II Can be built on interval arithmetic.