# Learning First-Order Rules with Differentiable Logic Program Semantics

arXiv: https://arxiv.org/abs/2204.13570 · HTML: https://mlsanity.com/paper/2204.13570/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 28 Apr 2022.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Inductive Relation Prediction by Subgraph Reasoning](https://mlsanity.com/paper/1911.06962/) (2019)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [RNNLogic: Learning Logic Rules for Reasoning on Knowledge Graphs](https://mlsanity.com/paper/2010.04029/) (2020)
- [Turning 30: New Ideas in Inductive Logic Programming](https://mlsanity.com/paper/2002.11002/) (2020)
- [Ontology Reasoning with Deep Neural Networks](https://mlsanity.com/paper/1808.07980/) (2018)
- [Making sense of sensory input](https://mlsanity.com/paper/1910.02227/) (2019)
- [Exploiting Answer Set Programming with External Sources for Meta-Interpretive Learning](https://mlsanity.com/paper/1805.00068/) (2018)
- [Neural Networks for Relational Data](https://mlsanity.com/paper/1909.04723/) (2019)

## Built on by (most cited first, 5 of 5)

- [Variable Assignment Invariant Neural Networks for Learning Logic Programs](https://mlsanity.com/paper/2408.10709/) (2024)
- [Differentiable Rule Induction from Raw Sequence Inputs](https://mlsanity.com/paper/2602.13583/) (2026)
- [A Noise-tolerant Differentiable Learning Approach for Single Occurrence Regular Expression with Interleaving](https://mlsanity.com/paper/2212.00373/) (2022)
- [Visual Perceptual to Conceptual First-Order Rule Learning Networks](https://mlsanity.com/paper/2604.07897/) (2026)
- [ANDRE: An Attention-based Neuro-symbolic Differentiable Rule Extractor for Inductive Logic Programming](https://mlsanity.com/paper/2605.04193/) (2026)

## Abstract (arXiv)

Learning first-order logic programs (LPs) from relational facts which yields intuitive insights into the data is a challenging topic in neuro-symbolic research. We introduce a novel differentiable inductive logic programming (ILP) model, called differentiable first-order rule learner (DFOL), which finds the correct LPs from relational facts by searching for the interpretable matrix representations of LPs. These interpretable matrices are deemed as trainable tensors in neural networks (NNs). The NNs are devised according to the differentiable semantics of LPs. Specifically, we first adopt a novel propositionalization method that transfers facts to NN-readable vector pairs representing interpretation pairs. We replace the immediate consequence operator with NN constraint functions consisting of algebraic operations and a sigmoid-like activation function. We map the symbolic forward-chained format of LPs into NN constraint functions consisting of operations between subsymbolic vector representations of atoms. By applying gradient descent, the trained well parameters of NNs can be decoded into precise symbolic LPs in forward-chained logic format. We demonstrate that DFOL can perform on several standard ILP datasets, knowledge bases, and probabilistic relation facts and outperform several well-known differentiable ILP models. Experimental results indicate that DFOL is a precise, robust, scalable, and computationally cheap differentiable ILP model.