  Lifting attempts to speed up probabilistic inference by exploiting symmetries
in the model. Exact lifted inference methods, like their propositional
counterparts, work by recursively decomposing the model and the problem. In the
propositional case, there exist formal structures, such as decomposition trees
(dtrees), that represent such a decomposition and allow us to determine the
complexity of inference a priori. However, there is currently no equivalent
structure nor analogous complexity results for lifted inference. In this paper,
we introduce FO-dtrees, which upgrade propositional dtrees to the first-order
level. We show how these trees can characterize a lifted inference solution for
a probabilistic logical model (in terms of a sequence of lifted operations),
and make a theoretical analysis of the complexity of lifted inference in terms
of the novel notion of lifted width for the tree.
