Answering complex logical queries is a challenging task for knowledge graph (KG) reasoning. Recently, query embedding (QE) has been proposed to encode queries and entities into the same vector space, and obtain answers based on numerical computation. However, such models obtain the node representations of a query only based on its predecessor nodes, which ignore the information contained in successor nodes. In this paper, we proposed a Bi-directional Directed Acyclic Graph neural network (BiDAG) that splits the reasoning process into prediction and calibration. The joint probability of all nodes is considered by applying a graph neural network (GNN) to the query graph in the calibration process. By the prediction in the first layer and the calibration in deep layers of GNN, BiDAG can outperform previous QE based methods on FB15k, FB15k-237, and NELL995.

Complex Query Answering (CQA) is a challenge task of Knowledge Graph (KG). Due to the incompleteness of KGs, query embedding (QE) methods have been proposed to encode queries and entities into the same embedding space, and treat logical operators as neural set operators to obtain answers. However, these methods train KG embeddings and neural set operators concurrently on both simple (one-hop) and complex (multi-hop and logical) queries, which causes performance degradation on simple queries and low training efficiency. In this paper, we propose Query to Triple (Q2T), a novel approach that decouples the training for simple and complex queries. Q2T divides the training into two stages: (1) Pre-training the neural link predictor on simple queries to predict tail entities based on the head entity and relation. (2) Training the query encoder on complex queries to encode diverse complex queries into a unified triple form that can be efficiently solved by the pretrained link predictor. Our proposed Q2T is not only efficient to train, but also modular, thus easily adaptable to various neural link predictors that have been studied well. Extensive experiments demonstrate that, even without explicit modeling for neural set operators, Q2T still achieves state-of-the-art performance on diverse complex queries over three public benchmarks.