# Mathematical Models for Behavior

Claude L. Bullard (bullardc@source.asset.com)
Thu, 30 Mar 1995 10:30:51 -0500

Mark:

| The idea is centred around virtual worlds which exist in terms of
| mathematic constructs , algorithms and other mathematical theory.
| The multiple actors and agents who exist in this world are not
| defined as geometric representations of any form , but exist as
| variable mathematical entities which through their changes affect
| and influence the mathematical whole.

Consider the work of Mandelbrot's definitions of view dimensions
in fractals and apply them to an event model (fractal behavior).
A fractal is a mathematical object with theoretically infinite levels
of detail whose measure is subject to the observer. Apply this to a
recursive system definition (in VR, could be nested worlds as
a nested geometric space (i.e., nested cartesian systems
as complex number systems a la Gauss)). Within each
space, a particular topic dominates and other topics are
subdominant. This is similar to the effect of cadence in
music in which tonic, dominant subdominant sequences
occur because of the frequency relationships expressed
as intervals whose aggregate occurence create the
sequential order. Therefore, within each space, a
measure of predictable but variable order occurs. Fractals and
markov sequences express this predictability and it
is found in all evolving systems from music to tidal basin cycles.

Each system (world, whatever) is defined as a member of
a recursive system set.

<!ELEMENT system - - (system, topic1, topic2) >.

For each system a set of topic are generated whose form
is invariant but whose membership in a populated space
can vary as the units of information aggregate by variable
measures of frequency and order of occurence. The view
in the space is chosen by an observer whos selects a level of
system and a level of topicality (topics may also contain
themselves recursively). Movement is defined by complex
number operations. Any operation that creates an address
that escapes the boundary of the local view is chaotic in
that dimension. This model in which a hierarchy of generative
productions is scaled in the event phase space by the
an independent axis (the recursive system axis) is a cascade.
It is useful to consider the instance of the model as an
ascending cascade in which dispersed elements aggregate
to consitute a larger element, the system. The model can
be considered a descending cascade in which the original
element is made up of smaller and smaller self-similar
but variant instances of itself. N-dimensional space
can be defined to express relationships that involve
aligned systems (e.g., like a z-axis relationship). This technique
can be used to create alternate or variant worlds that
share features and are identified with intersecting vector

The processing model resembles a set of
limit-cycle oscillators coupled by common conditions that
align events. Their behavior is modeled in a phase
space whose coorediantes are determined by the alignment of
objects along the phase space axes. For example, two
behaviors that require a common resource are contrained
from phase locks. A scheduler (sequencer) that is sensitive to
the initial conditions desynchronizes these cycles to ensure
objects in a shared space do not conflict. Interference is represented
as chaotic oscillation and can be detected by negative coordinates
in the event schedule. A negative coordinate indicates
"missing information", essentially, a Boltzmann conflict. The effects
of oscillation in the model might be expressed in the real world
as communications saturation, incomplete task closure, overruns,
sudden increases in energy demands. Late binding systems
exist at the edge of instability which is the source of their
dynamic evolution. The degree of resolution (detail in the
view) determines the rate of complexity growth. Sparsely
populated views survive longer than densely populated
views. The boundary is infinite but truncated in practice
by the measures of detail in the view. Hidden systems
whose effects can cross view dimensions are sources of
change, dynamism, and chaos. Positive and negative
feedback are used to dampen behaviors.

Have fun.

Len