Probabilistic logic programming under the distribution semantics has been very useful in machine learning. However, inference is expensive so machine learning algorithms may turn out to be slow. In this paper we consider a restriction of the language called hierarchical PLP in which clauses and predicates are hierarchically organized. In this case the language becomes truth-functional and inference reduces to the evaluation of formulas in the product fuzzy logic. Programs in this language can also be seen as arithmetic circuits or deep neural networks and inference can be reperformed quickly when the parameters change. Learning can then be performed by EM or backpropagation.