  We consider the problem of exact probabilistic inference for Union of
Conjunctive Queries (UCQs) on tuple-independent databases. For this problem,
two approaches currently coexist. In the extensional method, query evaluation
is performed by exploiting the structure of the query, and relies heavily on
the use of the inclusion-exclusion principle. In the intensional method, one
first builds a representation of the lineage of the query in a tractable
formalism of knowledge compilation. The chosen formalism should then ensure
that the probability can be efficiently computed using simple disjointness and
independence assumptions, without the need of performing inclusion-exclusion.
The extensional approach has long been thought to be strictly more powerful
than the intensional approach, the reason being that for some queries, the use
of inclusion-exclusion seemed unavoidable. In this paper we introduce a new
technique to construct lineage representations as deterministic decomposable
circuits in polynomial time. We prove that this technique applies to a class of
UCQs that had been conjectured to separate the complexity of the two
approaches. In essence, we show that relying on the inclusion-exclusion formula
can be avoided by using negation. This result brings back hope to prove that
the intensional approach can handle all tractable UCQs.
