  Answering complex first-order logic (FOL) queries on knowledge graphs is a
fundamental task for multi-hop reasoning. Traditional symbolic methods traverse
a complete knowledge graph to extract the answers, which provides good
interpretation for each step. Recent neural methods learn geometric embeddings
for complex queries. These methods can generalize to incomplete knowledge
graphs, but their reasoning process is hard to interpret. In this paper, we
propose Graph Neural Network Query Executor (GNN-QE), a neural-symbolic model
that enjoys the advantages of both worlds. GNN-QE decomposes a complex FOL
query into relation projections and logical operations over fuzzy sets, which
provides interpretability for intermediate variables. To reason about the
missing links, GNN-QE adapts a graph neural network from knowledge graph
completion to execute the relation projections, and models the logical
operations with product fuzzy logic. Experiments on 3 datasets show that GNN-QE
significantly improves over previous state-of-the-art models in answering FOL
queries. Meanwhile, GNN-QE can predict the number of answers without explicit
supervision, and provide visualizations for intermediate variables.
