# Logic Embeddings for Complex Query Answering

arXiv: https://arxiv.org/abs/2103.00418 · HTML: https://mlsanity.com/paper/2103.00418/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 28 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [DeepPath: A Reinforcement Learning Method for Knowledge Graph Reasoning](https://mlsanity.com/paper/1707.06690/) (2017)
- [Query2box: Reasoning over Knowledge Graphs in Vector Space using Box Embeddings](https://mlsanity.com/paper/2002.05969/) (2020)
- [Go for a Walk and Arrive at the Answer: Reasoning Over Paths in Knowledge Bases using Reinforcement Learning](https://mlsanity.com/paper/1711.05851/) (2017)
- [Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs](https://mlsanity.com/paper/2010.11465/) (2020)
- [Embedding Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/1806.01445/) (2018)
- [Logical Neural Networks](https://mlsanity.com/paper/2006.13155/) (2020)
- [Complex Query Answering with Neural Link Predictors](https://mlsanity.com/paper/2011.03459/) (2020)
- [Traversing Knowledge Graphs in Vector Space](https://mlsanity.com/paper/1506.01094/) (2015)
- [Neural Subgraph Matching](https://mlsanity.com/paper/2007.03092/) (2020)
- [Scalable Neural Methods for Reasoning With a Symbolic Knowledge Base](https://mlsanity.com/paper/2002.06115/) (2020)
- [Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors](https://mlsanity.com/paper/1304.5823/) (2013)
- [Information-Geometric Set Embeddings (IGSE): From Sets to Probability Distributions](https://mlsanity.com/paper/1911.12463/) (2019)

## Built on by (most cited first, 17 of 17)

- [Neural Graph Reasoning: Complex Logical Query Answering Meets Graph Databases](https://mlsanity.com/paper/2303.14617/) (2023)
- [Benchmarking the Combinatorial Generalizability of Complex Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2109.08925/) (2021)
- [Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport](https://mlsanity.com/paper/2305.04034/) (2023)
- [Type-aware Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2205.00782/) (2022)
- [TFLEX: Temporal Feature-Logic Embedding Framework for Complex Reasoning over Temporal Knowledge Graph](https://mlsanity.com/paper/2205.14307/) (2022)
- [Rethinking Complex Queries on Knowledge Graphs with Neural Link Predictors](https://mlsanity.com/paper/2304.07063/) (2023)
- [EFOk-CQA: Towards Knowledge Graph Complex Query Answering beyond Set Operation](https://mlsanity.com/paper/2307.13701/) (2023)
- [StaRPO: Stability-Augmented Reinforcement Policy Optimization](https://mlsanity.com/paper/2604.08905/) (2026)
- [FLEX: Feature-Logic Embedding Framework for CompleX Knowledge Graph Reasoning](https://mlsanity.com/paper/2205.11039/) (2022)
- [TAR: Neural Logical Reasoning across TBox and ABox](https://mlsanity.com/paper/2205.14591/) (2022)
- [FALCON: Scalable Reasoning over Inconsistent ALC Ontologies](https://mlsanity.com/paper/2208.07628/) (2022)
- [LogicRec: Recommendation with Users' Logical Requirements](https://mlsanity.com/paper/2304.11722/) (2023)
- [Extending Complex Logical Queries on Uncertain Knowledge Graphs](https://mlsanity.com/paper/2403.01508/) (2024)
- [Meta Operator for Complex Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2403.10110/) (2024)
- [Pathformer: Recursive Path Query Encoding for Complex Logical Query Answering](https://mlsanity.com/paper/2406.14880/) (2024)
- [KGCompiler: Deep Learning Compilation Optimization for Knowledge Graph Complex Logical Query Answering](https://mlsanity.com/paper/2503.02172/) (2025)
- [Neural Scalable Symbolic Search Framework for Complex Logical Queries with Multiple Free Variables](https://mlsanity.com/paper/2605.25985/) (2026)

## Abstract (arXiv)

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.