# End-to-End Differentiable Proving

arXiv: https://arxiv.org/abs/1705.11040 · HTML: https://mlsanity.com/paper/1705.11040/

## Key facts

- Cited by 161 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2017.

## Built on by (most cited first, 30 of 161)

- [Language Models Are Greedy Reasoners: A Systematic Formal Analysis of Chain-of-Thought](https://mlsanity.com/paper/2210.01240/) (2022)
- [Proof Artifact Co-training for Theorem Proving with Language Models](https://mlsanity.com/paper/2102.06203/) (2021)
- [A Survey of Knowledge Graph Reasoning on Graph Types: Static, Dynamic, and Multimodal](https://mlsanity.com/paper/2212.05767/) (2022)
- [NeurASP: Embracing Neural Networks into Answer Set Programming](https://mlsanity.com/paper/2307.07700/) (2023)
- [Large Language Models can Learn Rules](https://mlsanity.com/paper/2310.07064/) (2023)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [Inductive logic programming at 30](https://mlsanity.com/paper/2102.10556/) (2021)
- [Modular Design Patterns for Hybrid Learning and Reasoning Systems: a taxonomy, patterns and use cases](https://mlsanity.com/paper/2102.11965/) (2021)
- [A Survey on Neural-symbolic Learning Systems](https://mlsanity.com/paper/2111.08164/) (2021)
- [Systematic Generalization with Edge Transformers](https://mlsanity.com/paper/2112.00578/) (2021)
- [TacticZero: Learning to Prove Theorems from Scratch with Deep Reinforcement Learning](https://mlsanity.com/paper/2102.09756/) (2021)
- [Neuro-Symbolic Inductive Logic Programming with Logical Neural Networks](https://mlsanity.com/paper/2112.03324/) (2021)
- [Learning Iterative Reasoning through Energy Diffusion](https://mlsanity.com/paper/2406.11179/) (2024)
- [Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing](https://mlsanity.com/paper/2210.15889/) (2022)
- [NeuPSL: Neural Probabilistic Soft Logic](https://mlsanity.com/paper/2205.14268/) (2022)
- [FaiRR: Faithful and Robust Deductive Reasoning over Natural Language](https://mlsanity.com/paper/2203.10261/) (2022)
- [Logic-induced Diagnostic Reasoning for Semi-supervised Semantic Segmentation](https://mlsanity.com/paper/2308.12595/) (2023)
- [HAKE: A Knowledge Engine Foundation for Human Activity Understanding](https://mlsanity.com/paper/2202.06851/) (2022)
- [Learning Algebraic Representation for Systematic Generalization in Abstract Reasoning](https://mlsanity.com/paper/2111.12990/) (2021)
- [Differentiable Inductive Logic Programming for Structured Examples](https://mlsanity.com/paper/2103.01719/) (2021)
- [Neural Multi-Hop Reasoning With Logical Rules on Biomedical Knowledge Graphs](https://mlsanity.com/paper/2103.10367/) (2021)
- [Neural Probabilistic Logic Programming in Discrete-Continuous Domains](https://mlsanity.com/paper/2303.04660/) (2023)
- [Neural-Logic Human-Object Interaction Detection](https://mlsanity.com/paper/2311.09817/) (2023)
- [Neuro-Symbolic Hierarchical Rule Induction](https://mlsanity.com/paper/2112.13418/) (2021)
- [RobustLR: Evaluating Robustness to Logical Perturbation in Deductive Reasoning](https://mlsanity.com/paper/2205.12598/) (2022)
- [TILP: Differentiable Learning of Temporal Logical Rules on Knowledge Graphs](https://mlsanity.com/paper/2402.12309/) (2024)
- [Deep Symbolic Learning: Discovering Symbols and Rules from Perceptions](https://mlsanity.com/paper/2208.11561/) (2022)
- [Dual Algorithmic Reasoning](https://mlsanity.com/paper/2302.04496/) (2023)
- [Complex Query Answering on Eventuality Knowledge Graph with Implicit Logical Constraints](https://mlsanity.com/paper/2305.19068/) (2023)
- [Neural Compositional Rule Learning for Knowledge Graph Reasoning](https://mlsanity.com/paper/2303.03581/) (2023)

## Abstract (arXiv)

We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. By using gradient descent, the resulting neural network can be trained to infer facts from a given incomplete knowledge base. It learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove queries, (iii) induce logical rules, and (iv) use provided and induced logical rules for multi-hop reasoning. We demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, on three out of four benchmark knowledge bases while at the same time inducing interpretable function-free first-order logic rules.