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.