Formulating and answering logical queries is a standard communication interface for knowledge graphs (KGs). Alleviating the notorious incompleteness of real-world KGs, neural methods achieved impressive results in link prediction and complex query answering tasks by learning representations of entities, relations, and queries. Still, most existing query answering methods rely on transductive entity embeddings and cannot generalize to KGs containing new entities without retraining the entity embeddings. In this work, we study the inductive query answering task where inference is performed on a graph containing new entities with queries over both seen and unseen entities. To this end, we devise two mechanisms leveraging inductive node and relational structure representations powered by graph neural networks (GNNs). Experimentally, we show that inductive models are able to perform logical reasoning at inference time over unseen nodes generalizing to graphs up to 500% larger than training ones. Exploring the efficiency--effectiveness trade-off, we find the inductive relational structure representation method generally achieves higher performance, while the inductive node representation method is able to answer complex queries in the inference-only regime without any training on queries and scales to graphs of millions of nodes. Code is available at https://github.com/DeepGraphLearning/InductiveQE.
Answering complex queries over knowledge graphs (KG) is an important yet challenging task because of the KG incompleteness issue and cascading errors during reasoning. Recent query embedding (QE) approaches to embed the entities and relations in a KG and the first-order logic (FOL) queries into a low dimensional space, answering queries by dense similarity search. However, previous works mainly concentrate on the target answers, ignoring intermediate entities' usefulness, which is essential for relieving the cascading error problem in logical query answering. In addition, these methods are usually designed with their own geometric or distributional embeddings to handle logical operators like union, intersection, and negation, with the sacrifice of the accuracy of the basic operator - projection, and they could not absorb other embedding methods to their models. In this work, we propose a Neural and Symbolic Entangled framework (ENeSy) for complex query answering, which enables the neural and symbolic reasoning to enhance each other to alleviate the cascading error and KG incompleteness. The projection operator in ENeSy could be any embedding method with the capability of link prediction, and the other FOL operators are handled without parameters. With both neural and symbolic reasoning results contained, ENeSy answers queries in ensembles. ENeSy achieves the SOTA performance on several benchmarks, especially in the setting of the training model only with the link prediction task.
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.
Answering complex logical queries on incomplete knowledge graphs (KGs) with missing edges is a fundamental and important task for knowledge graph reasoning. The query embedding method is proposed to answer these queries by jointly encoding queries and entities to the same embedding space. Then the answer entities are selected according to the similarities between the entity embeddings and the query embedding. As the answers to a complex query are obtained from a combination of logical operations over sub-queries, the embeddings of the answer entities may not always follow a uni-modal distribution in the embedding space. Thus, it is challenging to simultaneously retrieve a set of diverse answers from the embedding space using a single and concentrated query representation such as a vector or a hyper-rectangle. To better cope with queries with diversified answers, we propose Query2Particles (Q2P), a complex KG query answering method. Q2P encodes each query into multiple vectors, named particle embeddings. By doing so, the candidate answers can be retrieved from different areas over the embedding space using the maximal similarities between the entity embeddings and any of the particle embeddings. Meanwhile, the corresponding neural logic operations are defined to support its reasoning over arbitrary first-order logic queries. The experiments show that Query2Particles achieves state-of-the-art performance on the complex query answering tasks on FB15k, FB15K-237, and NELL knowledge graphs.
Complex Query Answering (CQA) is an important reasoning task on knowledge graphs. Current CQA learning models have been shown to be able to generalize from atomic operators to more complex formulas, which can be regarded as the combinatorial generalizability. In this paper, we present EFO-1-QA, a new dataset to benchmark the combinatorial generalizability of CQA models by including 301 different queries types, which is 20 times larger than existing datasets. Besides, our work, for the first time, provides a benchmark to evaluate and analyze the impact of different operators and normal forms by using (a) 7 choices of the operator systems and (b) 9 forms of complex queries. Specifically, we provide the detailed study of the combinatorial generalizability of two commonly used operators, i.e., projection and intersection, and justify the impact of the forms of queries given the canonical choice of operators. Our code and data can provide an effective pipeline to benchmark CQA models.
Answering logical queries over incomplete knowledge bases is challenging because: 1) it calls for implicit link prediction, and 2) brute force answering of existential first-order logic queries is exponential in the number of existential variables. Recent work of query embeddings provides fast querying, but most approaches model set logic with closed regions, so lack negation. Query embeddings that do support negation use densities that suffer drawbacks: 1) only improvise logic, 2) use expensive distributions, and 3) poorly model answer uncertainty. In this paper, we propose Logic Embeddings, a new approach to embedding complex queries that uses Skolemisation to eliminate existential variables for efficient querying. It supports negation, but improves on density approaches: 1) integrates well-studied t-norm logic and directly evaluates satisfiability, 2) simplifies modeling with truth values, and 3) models uncertainty with truth bounds. Logic Embeddings are competitively fast and accurate in query answering over large, incomplete knowledge graphs, outperform on negation queries, and in particular, provide improved modeling of answer uncertainty as evidenced by a superior correlation between answer set size and embedding entropy.
Large, heterogeneous datasets are characterized by missing or even erroneous information. This is more evident when they are the product of community effort or automatic fact extraction methods from external sources, such as text. A special case of the aforementioned phenomenon can be seen in knowledge graphs, where this mostly appears in the form of missing or incorrect edges and nodes. Structured querying on such incomplete graphs will result in incomplete sets of answers, even if the correct entities exist in the graph, since one or more edges needed to match the pattern are missing. To overcome this problem, several algorithms for approximate structured query answering have been proposed. Inspired by modern Information Retrieval metrics, these algorithms produce a ranking of all entities in the graph, and their performance is further evaluated based on how high in this ranking the correct answers appear. In this work we take a critical look at this way of evaluation. We argue that performing a ranking-based evaluation is not sufficient to assess methods for complex query answering. To solve this, we introduce Message Passing Query Boxes (MPQB), which takes binary classification metrics back into use and shows the effect this has on the recently proposed query embedding method MPQE.
Neural link predictors are immensely useful for identifying missing edges in large scale Knowledge Graphs. However, it is still not clear how to use these models for answering more complex queries that arise in a number of domains, such as queries using logical conjunctions ($\land$), disjunctions ($\lor$) and existential quantifiers ($\exists$), while accounting for missing edges. In this work, we propose a framework for efficiently answering complex queries on incomplete Knowledge Graphs. We translate each query into an end-to-end differentiable objective, where the truth value of each atom is computed by a pre-trained neural link predictor. We then analyse two solutions to the optimisation problem, including gradient-based and combinatorial search. In our experiments, the proposed approach produces more accurate results than state-of-the-art methods -- black-box neural models trained on millions of generated queries -- without the need of training on a large and diverse set of complex queries. Using orders of magnitude less training data, we obtain relative improvements ranging from 8% up to 40% in Hits@3 across different knowledge graphs containing factual information. Finally, we demonstrate that it is possible to explain the outcome of our model in terms of the intermediate solutions identified for each of the complex query atoms. All our source code and datasets are available online, at https://github.com/uclnlp/cqd.
One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty. Here, we present BetaE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BetaE is the first method that can handle a complete set of first-order logical operations: conjunction ($\wedge$), disjunction ($\vee$), and negation ($\neg$). A key insight of BetaE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BetaE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BetaE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation.
Answering complex logical queries on large-scale incomplete knowledge graphs (KGs) is a fundamental yet challenging task. Recently, a promising approach to this problem has been to embed KG entities as well as the query into a vector space such that entities that answer the query are embedded close to the query. However, prior work models queries as single points in the vector space, which is problematic because a complex query represents a potentially large set of its answer entities, but it is unclear how such a set can be represented as a single point. Furthermore, prior work can only handle queries that use conjunctions ($\wedge$) and existential quantifiers ($\exists$). Handling queries with logical disjunctions ($\vee$) remains an open problem. Here we propose query2box, an embedding-based framework for reasoning over arbitrary queries with $\wedge$, $\vee$, and $\exists$ operators in massive and incomplete KGs. Our main insight is that queries can be embedded as boxes (i.e., hyper-rectangles), where a set of points inside the box corresponds to a set of answer entities of the query. We show that conjunctions can be naturally represented as intersections of boxes and also prove a negative result that handling disjunctions would require embedding with dimension proportional to the number of KG entities. However, we show that by transforming queries into a Disjunctive Normal Form, query2box is capable of handling arbitrary logical queries with $\wedge$, $\vee$, $\exists$ in a scalable manner. We demonstrate the effectiveness of query2box on three large KGs and show that query2box achieves up to 25% relative improvement over the state of the art.
Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple unobserved edges, entities, and variables. For instance, given an incomplete biological knowledge graph, we might want to predict "em what drugs are likely to target proteins involved with both diseases X and Y?" -- a query that requires reasoning about all possible proteins that {\em might} interact with diseases X and Y. Here we introduce a framework to efficiently make predictions about conjunctive logical queries -- a flexible but tractable subset of first-order logic -- on incomplete knowledge graphs. In our approach, we embed graph nodes in a low-dimensional space and represent logical operators as learned geometric operations (e.g., translation, rotation) in this embedding space. By performing logical operations within a low-dimensional embedding space, our approach achieves a time complexity that is linear in the number of query variables, compared to the exponential complexity required by a naive enumeration-based approach. We demonstrate the utility of this framework in two application studies on real-world datasets with millions of relations: predicting logical relationships in a network of drug-gene-disease interactions and in a graph-based representation of social interactions derived from a popular web forum.