Projecting knowledge graph queries into an embedding space using geometric models (points, boxes and spheres) can help to answer queries for large incomplete knowledge graphs. In this work, we propose a symbolic learning-free approach using fuzzy logic to address the shape-closure problem that restricted geometric-based embedding models to only a few shapes (e.g. ConE) for answering complex logical queries. The use of symbolic approach facilitates non-closure geometric models (e.g. point, box) to handle logical operators (including negation). This enabled our newly proposed spherical embeddings (SpherE) in this work to use a polar coordinate system to effectively represent hierarchical relation. Results show that the SpherE model can answer existential positive first-order logic and negation queries. We show that SpherE significantly outperforms the point and box embeddings approaches while generating semantically meaningful hierarchy-aware embeddings.

Recent geometric-based approaches have been shown to efficiently model complex logical queries (including the intersection operation) over Knowledge Graphs based on the natural representation of Venn diagram.Existing geometric-based models (using points, boxes embeddings), however, cannot handle the logical negation operation.Further, those using cones embeddings are limited to representing queries by two-dimensional shapes, which reduced their effectiveness in capturing entities query relations for correct answers.To overcome this challenge, we propose unbounded cylinder embeddings (namely CylE), which is a novel geometric-based model based on threedimensional shapes.Our approach can handle a complete set of basic first-order logic operations (conjunctions, disjunctions and negations).CylE considers queries as Cartesian products of unbounded sector-cylinders and consider a set of nearest boxes corresponds to the set of answer entities.Precisely, the conjunctions can be represented via the intersections of unbounded sector-cylinders.Transforming queries to Disjunctive Normal Form can handle queries with disjunctions.The negations can be represented by considering the closure of complement for an arbitrary unbounded sector-cylinder.Empirical results show that the performance of multihop reasoning task using CylE significantly increases over state-of-the-art geometric-based query embedding models for queries without negation.For queries with negation operations, though the performance is on a par with the best performing geometric-based model, CylE significantly outperforms a recent distributionbased model.