## Formal Approaches to a Definition of Agents

##### Abstract

This thesis is a contribution to the formalisation of the notion of an agent
within the class of finite multivariate Markov chains. In accordance with
the literature agents are are seen as entities that act, perceive, and are goaldirected.
We present a new measure that can be used to identify entities
(called i-entities). The intuition behind this is that entities are spatiotemporal
patterns for which every part makes every other part more probable. The
measure, complete local integration (CLI), is formally investigated within the
more general setting of Bayesian networks. It is based on the specific local
integration (SLI) which is measured with respect to a partition. CLI is the
minimum value of SLI over all partitions. Upper bounds are constructively
proven and a possible lower bound is proposed. We also prove a theorem
that shows that completely locally integrated spatiotemporal patterns occur
as blocks in specific partitions of the global trajectory. Conversely we can
identify partitions of global trajectories for which every block is completely
locally integrated. These global partitions are the finest partitions that achieve
a SLI less or equal to their own SLI. We also establish the transformation behaviour
of SLI under permutations of the nodes in the Bayesian network.
We then go on to present three conditions on general definitions of entities.
These are most prominently not fulfilled by sets of random variables i.e. the
perception-action loop, which is often used to model agents, is too restrictive a
setting. We instead propose that any general entity definition should in effect
specify a subset of the set of all spatiotemporal patterns of a given multivariate
Markov chain. Any such definition will then define what we call an entity set.
The set of all completely locally integrated spatiotemporal patterns is one
example of such a set. Importantly the perception-action loop also naturally
induces such an entity set. We then propose formal definitions of actions and
perceptions for arbitrary entity sets. We show that these are generalisations of
notions defined for the perception-action loop by plugging the entity-set of the
perception-action loop into our definitions. We also clearly state the properties
that general entity-sets have but the perception-action loop entity set does not.
This elucidates in what way we are generalising the perception-action loop.
Finally we look at some very simple examples of bivariate Markov chains.
We present the disintegration hierarchy, explain it via symmetries, and calculate
the i-entities. Then we apply our definitions of perception and action to
these i-entities.