  In this article we introduce the notion of a \textit{regular partition} of a
Coxeter group. We develop the theory of these partitions, and show that the
class of regular partitions is essentially equivalent to the class of automata
(not necessarily finite state) recognising the language of reduced words in the
Coxeter group. As an application of this theory we prove that each cone type in
a Coxeter group has a unique minimal length representative. This result can be
seen as an analogue of Shi's classical result that each component of the Shi
arrangement of an affine Coxeter group has a unique minimal length element. We
further develop the theory of cone types in Coxeter groups by identifying the
minimal set of roots required to express a cone type as an intersection of
half-spaces. This set of \textit{boundary roots} is closely related to the
elementary inversion sets of Brink and Howlett, and also to the notion of the
base of an inversion set introduced by Dyer.
