Exploring Complex Non-Linear
Systems Using Interval Arithmetic Constraints
Complex systems often tend to exhibit
non-linear effects and that non- linearity makes such
systems difficult to study using standard techniques.
In this talk I presented a new approach to studying
non-linear systems that relies on interval arithmetic
constraint solving to infer properties of these systems.
This work grows out of seminal research by Moore in
the 1960s on application of interval arithmetic as well
as work in the 1980s and 1990s on high-level computer
languages based on logic and constraints.
The key idea behind this work is to build
a system that will use numerical techniques to automatically
make "provably correct" inferences about the parameters
that appear in mathematical systems. For example, one
could specify an ordinary differential equation with
some parameters (known only to lie in some specific
intervals) together with values of the solution function
at certain points (again known to lie in specific intervals).
The constraint solver would then try to shrink these
intervals without removing any possible solutions to
the system. In this way the parameters could be constrained
to fairly small intervals by giving a large set of measured
values (with explicit error intervals).
The techniques we use to implement such
a solver decompose the original complex constraint into
a large number of primitive constraints by introducing
additional variables representing intermediate quantities.
These new variables are initially assigned to the intervals
[-infinity, infinity] and the constraint solvers for
each of the primitive constraints is repetitively called
to narrow its variables intervals. This process continues
until there is no longer any change. The beauty of this
technique is its simplicity. It allows one to work with
extremely general constraints. The downside is that
the constraint solver may not be able to make any progress
on some constraints. There are several approaches to
making this technique more robust. One method we have
investigated is building meta- level solvers on top
of the underlying solver.
There is much work to be done in this
area. We are currently using these techniques to study
hybrid systems. These are systems in which a digital
controller interacts with a physical environment. The
environment is generally modeled as a non-linear ODE
or PDE and the constraint solvers we consider are well
suited to this type of problem. We are also looking
for examples of complex systems that arise when studying
neural assemblies as they may provide interesting non-linear
case studies for the solver.
