  Reasoning on knowledge graphs is a challenging task because it utilizes
observed information to predict the missing one. Particularly, answering
complex queries based on first-order logic is one of the crucial tasks to
verify learning to reason abilities for generalization and composition.
Recently, the prevailing method is query embedding which learns the embedding
of a set of entities and treats logic operations as set operations and has
shown great empirical success. Though there has been much research following
the same formulation, many of its claims lack a formal and systematic
inspection. In this paper, we rethink this formulation and justify many of the
previous claims by characterizing the scope of queries investigated previously
and precisely identifying the gap between its formulation and its goal, as well
as providing complexity analysis for the currently investigated queries.
Moreover, we develop a new dataset containing ten new types of queries with
features that have never been considered and therefore can provide a thorough
investigation of complex queries. Finally, we propose a new neural-symbolic
method, Fuzzy Inference with Truth value (FIT), where we equip the neural link
predictors with fuzzy logic theory to support end-to-end learning using complex
queries with provable reasoning capability. Empirical results show that our
method outperforms previous methods significantly in the new dataset and also
surpasses previous methods in the existing dataset at the same time.
