# Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs

arXiv: https://arxiv.org/abs/2010.11465 · HTML: https://mlsanity.com/paper/2010.11465/

## Key facts

- Cited by 129 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2020.

## Built on by (most cited first, 30 of 129)

- [QA-GNN: Reasoning with Language Models and Knowledge Graphs for Question Answering](https://mlsanity.com/paper/2104.06378/) (2021)
- [GreaseLM: Graph REASoning Enhanced Language Models for Question Answering](https://mlsanity.com/paper/2201.08860/) (2022)
- [ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13715/) (2021)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [Temporal Knowledge Graph Forecasting with Neural ODE](https://mlsanity.com/paper/2101.05151/) (2021)
- [Neural-Symbolic Models for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2205.10128/) (2022)
- [Mask and Reason: Pre-Training Knowledge Graph Transformers for Complex Logical Queries](https://mlsanity.com/paper/2208.07638/) (2022)
- [Neural Graph Reasoning: Complex Logical Query Answering Meets Graph Databases](https://mlsanity.com/paper/2303.14617/) (2023)
- [Answering Complex Logical Queries on Knowledge Graphs via Query Computation Tree Optimization](https://mlsanity.com/paper/2212.09567/) (2022)
- [Probabilistic Entity Representation Model for Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13522/) (2021)
- [Logical Message Passing Networks with One-hop Inference on Atomic Formulas](https://mlsanity.com/paper/2301.08859/) (2023)
- [Complex Logical Reasoning over Knowledge Graphs using Large Language Models](https://mlsanity.com/paper/2305.01157/) (2023)
- ["I'm Not Mad": Commonsense Implications of Negation and Contradiction](https://mlsanity.com/paper/2104.06511/) (2021)
- [Inductive Logical Query Answering in Knowledge Graphs](https://mlsanity.com/paper/2210.08008/) (2022)
- [Faithiful Embeddings for EL++ Knowledge Bases](https://mlsanity.com/paper/2201.09919/) (2022)
- [Query2Particles: Knowledge Graph Reasoning with Particle Embeddings](https://mlsanity.com/paper/2204.12847/) (2022)
- [Learning to Plan for Retrieval-Augmented Large Language Models from Knowledge Graphs](https://mlsanity.com/paper/2406.14282/) (2024)
- [Hallucination Detection in Large Language Models with Metamorphic Relations](https://mlsanity.com/paper/2502.15844/) (2025)
- [Query Embedding on Hyper-relational Knowledge Graphs](https://mlsanity.com/paper/2106.08166/) (2021)
- [Benchmarking the Combinatorial Generalizability of Complex Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2109.08925/) (2021)
- [GammaE: Gamma Embeddings for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2210.15578/) (2022)
- [Complex Query Answering on Eventuality Knowledge Graph with Implicit Logical Constraints](https://mlsanity.com/paper/2305.19068/) (2023)
- [Knowledge Graph Reasoning with Logics and Embeddings: Survey and Perspective](https://mlsanity.com/paper/2202.07412/) (2022)
- [Neural-Symbolic Entangled Framework for Complex Query Answering](https://mlsanity.com/paper/2209.08779/) (2022)
- [Logic Embeddings for Complex Query Answering](https://mlsanity.com/paper/2103.00418/) (2021)
- [SMORE: Knowledge Graph Completion and Multi-hop Reasoning in Massive Knowledge Graphs](https://mlsanity.com/paper/2110.14890/) (2021)
- [Rethinking Knowledge Graph Evaluation Under the Open-World Assumption](https://mlsanity.com/paper/2209.08858/) (2022)
- [Neural Methods for Logical Reasoning Over Knowledge Graphs](https://mlsanity.com/paper/2209.14464/) (2022)
- [Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport](https://mlsanity.com/paper/2305.04034/) (2023)
- [A Unified Framework for Rank-based Evaluation Metrics for Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/2203.07544/) (2022)

## Abstract (arXiv)

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.