[Paper Review] End-to-End Differentiable Proving

Tim Rocktäschel, Sebastian Riedel|arXiv (Cornell University)|May 31, 2017

Topic Modeling72 citations

TL;DR

This paper introduces Neural Theorem Provers (NTPs), a differentiable neural network architecture that combines symbolic backward chaining with subsymbolic vector representations for end-to-end learning in knowledge bases. By replacing symbolic unification with a differentiable radial basis function kernel on vector embeddings, NTPs learn to infer facts, induce interpretable first-order logic rules, and outperform ComplEx on three out of four benchmark KBs while enabling multi-hop reasoning and gradient-based training.

ABSTRACT

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.

Motivation & Objective

• To bridge symbolic reasoning and subsymbolic learning by enabling end-to-end differentiable proving in knowledge bases.
• To address the limitations of neural link prediction models in handling complex, multi-hop reasoning patterns.
• To combine the interpretability of symbolic theorem provers with the generalization capability of distributed vector representations.
• To learn both vector representations of symbols and function-free first-order logic rules via gradient descent.
• To enable integration of domain-specific rules and rule templates into a differentiable reasoning framework.

Proposed method

• NTPs are recursively constructed using Prolog’s backward chaining algorithm as a blueprint, replacing symbolic unification with a differentiable operation on vector representations.
• A radial basis function kernel computes similarity between vector embeddings of symbols, enabling differentiable unification.
• The proof success score is differentiable with respect to symbol embeddings and rule parameters, allowing end-to-end training via gradient descent.
• Joint training with a neural link prediction model (ComplEx) as an auxiliary loss improves representation quality and performance.
• Batch proving and approximate gradient calculation are used to scale the training process.
• Induced rules are decoded from learned parameters and validated for interpretability and logical consistency.

Experimental results

Research questions

• RQ1 Can a neural network be trained to perform multi-hop reasoning in knowledge bases using only differentiable operations on vector representations?
• RQ2 Can such a model learn to induce interpretable, function-free first-order logic rules from incomplete knowledge bases?
• RQ3 How does end-to-end differentiable proving compare to state-of-the-art neural link prediction models like ComplEx in terms of reasoning performance?
• RQ4 Can auxiliary losses from neural link prediction improve the quality of subsymbolic representations in a differentiable prover?
• RQ5 To what extent can learned rules generalize across semantically similar symbols and support transfer learning in reasoning?

Key findings

• NTPs outperform ComplEx on three out of four benchmark knowledge bases—Countries S3, Nations, and UMLS—while achieving comparable or better performance on Kinship.
• The NTPλ variant, which jointly trains with ComplEx as an auxiliary loss, achieves the highest performance, with a statistically significant improvement in AUC-PR on all Countries tasks (p < 0.0001).
• The model successfully induces interpretable, human-readable first-order logic rules, such as transitivity rules on UMLS and task-specific rules on Countries.
• Induced rules demonstrate generalization: a single rule can be applied to multiple queries involving semantically similar symbols.
• The differentiable unification mechanism enables gradient-based optimization of both symbol embeddings and logical rule parameters.
• The approach enables multi-hop reasoning by composing proofs through recursive application of rules, with proof paths being explicitly traceable and interpretable.