Logical query answering over Knowledge Graphs (KGs) is a fundamental yet complex task. A promising approach to achieve this is to embed queries and entities jointly into the same embedding space. Research along this line suggests that using multi-modal distribution to represent answer entities is more suitable than uni-modal distribution, as a single query may contain multiple disjoint answer subsets due to the compositional nature of multi-hop queries and the varying latent semantics of relations. However, existing methods based on multi-modal distribution roughly represent each subset without capturing its accurate cardinality, or even degenerate into uni-modal distribution learning during the reasoning process due to the lack of an effective similarity measure. To better model queries with diversified answers, we propose Query2GMM for answering logical queries over knowledge graphs. In Query2GMM, we present the GMM embedding to represent each query using a univariate Gaussian Mixture Model (GMM). Each subset of a query is encoded by its cardinality, semantic center and dispersion degree, allowing for precise representation of multiple subsets. Then we design specific neural networks for each operator to handle the inherent complexity that comes with multi-modal distribution while alleviating the cascading errors. Last, we design a new similarity measure to assess the relationships between an entity and a query's multi-answer subsets, enabling effective multi-modal distribution learning for reasoning. Comprehensive experimental results show that Query2GMM outperforms the best competitor by an absolute average of 6.35%.

Logical queries on Knowledge Graphs (KGs) is a fundamental sub-task of knowledge graph reasoning. A promising paradigm for answering logical queries, recently, has been proposed based on versatile deep learning techniques. In this line, the query is first broken down into a series of first-order logical predicates, and then both the query and knowledge graph entities are jointly encoded in the same embedding space. Some approaches are able to support the full range of traditional First-Order Logic (FOL) operations for complex queries in real-world scenarios, while others have attempted to create a new combination of FOL operations by replacing the negation operation with the difference operation due to the poor performance of the negation operation. Our empirical observations show that the difference operator is more effective for multi-hop reasoning, while the negation operator is better suited for use as the final operation in the query, particularly in single-hop settings. In addition, other fundamental limitations such as linear transformation assumption for negation operator and the fixed-lossy problem for difference operator further degrade the performance of these methods. In light of these, we propose the HaLk, a holistic approach for answering logical queries that, to our knowledge, is the first to support a full set of logical operators in a unified end-to-end framework. In this approach, we propose specific neural models for each operator by considering their own intrinsic properties, based on which HaLk effectively mitigates the cascading error of projection and negation operators as well as delicately provides closed-formed solutions for difference operator. Extensive experimental results on three datasets demonstrate that HaLk outperforms all competitors and achieves up to 32% improvement in accuracy.