## Abstract

The combination of learning and reasoning is an essential and challenging topic in neuro-symbolic research. Differentiable inductive logic programming is a technique for learning a symbolic knowledge representation from either complete, mislabeled, or incomplete observed facts using neural networks. In this paper, we propose a novel differentiable inductive logic programming system called differentiable learning from interpretation transition (D-LFIT) for learning logic programs through the proposed embeddings of logic programs, neural networks, optimization algorithms, and an adapted algebraic method to compute the logic program semantics. The proposed model has several characteristics, including a small number of parameters, the ability to generate logic programs in a curriculum-learning setting, and linear time complexity for the extraction of trained neural networks. The well-known bottom clause positionalization algorithm is incorporated when the proposed system learns from relational datasets. We compare our model with NN-LFIT, which extracts propositional logic rules from retuned connected networks, the highly accurate rule learner RIPPER, the purely symbolic LFIT system LF1T, and CILP++, which integrates neural networks and the propositionalization method to handle first-order logic knowledge. From the experimental results, we conclude that D-LFIT yields comparable accuracy with respect to the baselines when given complete, incomplete, and mislabeled data. Our experimental results indicate that D-LFIT not only learns symbolic logic programs quickly and precisely but also performs robustly when processing mislabeled and incomplete datasets.

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## Funding

This work was supported by the National Key R&D Program of China under Grants 2018YFC1314200 and 2018YFB1003904, and the National Natural Science Foundation of China under Grants 61772035, 61972005, and 61932001; this work was also supported in part by the NII international internship program and JSPS KAKENHI Grant Number JP17H00763.

## Author information

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### Contributions

Conceptualization: Kun Gao, Katsumi Inoue; Methodology: Kun Gao; Formal analysis and investigation: Kun Gao; Writing - original draft preparation: Kun Gao; Writing - review and editing: Katsumi Inoue, Kun Gao, Yongzhi Cao; Funding acquisition: Hanpin Wang, Yongzhi Cao, Katsumi Inoue; Resources: Hanpin Wang, Yongzhi Cao, Katsumi Inoue; Supervision: Hanpin Wang, Yongzhi Cao, Katsumi Inoue.

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The authors declare that they have no conflict of interest.

### Code availability

https://drive.google.com/drive/folders/1SKB05Jka5jaJjfVi8LZmeo2smf0ujMeM?usp=sharing (Need to download the LF1T by Python)

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Editors: Nikos Katzouris, Alexander Artikis, Luc De Raedt, Artur d’Avila Garcez, Sebastijan Dumančić, Ute Schmid, Jay Pujara.

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### Cite this article

Gao, K., Wang, H., Cao, Y. *et al.* Learning from interpretation transition using differentiable logic programming semantics.
*Mach Learn* (2021). https://doi.org/10.1007/s10994-021-06058-8

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### Keywords

- Machine learning
- Differentiable inductive logic programming
- Explainability
- Neuro-symbolic method
- Learning from interpretation transition