# FOLD-R++: A Scalable Toolset for Automated Inductive Learning of Default Theories from Mixed Data

arXiv: https://arxiv.org/abs/2110.07843 · HTML: https://mlsanity.com/paper/2110.07843/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 37 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Constraint Answer Set Programming without Grounding](https://mlsanity.com/paper/1804.11162/) (2018)
- [A New Algorithm to Automate Inductive Learning of Default Theories](https://mlsanity.com/paper/1707.02693/) (2017)
- [Justifications for Goal-Directed Constraint Answer Set Programming](https://mlsanity.com/paper/2009.10238/) (2020)
- [Heuristic Based Induction of Answer Set Programs: From Default theories to combinatorial problems](https://mlsanity.com/paper/1802.06462/) (2018)
- [Generating Explainable Rule Sets from Tree-Ensemble Learning Methods by Answer Set Programming](https://mlsanity.com/paper/2109.08290/) (2021)

## Built on by (most cited first, 10 of 10)

- [FOLD-SE: An Efficient Rule-based Machine Learning Algorithm with Scalable Explainability](https://mlsanity.com/paper/2208.07912/) (2022)
- [CON-FOLD -- Explainable Machine Learning with Confidence](https://mlsanity.com/paper/2408.07854/) (2024)
- [Interpretable Rule-Based System for Radar-Based Gesture Sensing: Enhancing Transparency and Personalization in AI](https://mlsanity.com/paper/2410.12806/) (2024)
- [MC3G: Model Agnostic Causally Constrained Counterfactual Generation](https://mlsanity.com/paper/2508.17221/) (2025)
- [A Foundation Model for Zero-Shot Logical Rule Induction](https://mlsanity.com/paper/2605.04916/) (2026)
- [Counterfactual Explanation Generation with s(CASP)](https://mlsanity.com/paper/2310.14497/) (2023)
- [Counterfactual Generation with Answer Set Programming](https://mlsanity.com/paper/2402.04382/) (2024)
- [CFGs: Causality Constrained Counterfactual Explanations using goal-directed ASP](https://mlsanity.com/paper/2405.15956/) (2024)
- [Interpretable Hybrid Machine Learning Models Using FOLD-R++ and Answer Set Programming](https://mlsanity.com/paper/2506.19573/) (2025)
- [Comparative Analysis of FOLD-SE vs. FOLD-R++ in Binary Classification and XGBoost in Multi-Category Classification](https://mlsanity.com/paper/2509.18139/) (2025)

## Abstract (arXiv)

FOLD-R is an automated inductive learning algorithm for learning default rules for mixed (numerical and categorical) data. It generates an (explainable) answer set programming (ASP) rule set for classification tasks. We present an improved FOLD-R algorithm, called FOLD-R++, that significantly increases the efficiency and scalability of FOLD-R by orders of magnitude. FOLD-R++ improves upon FOLD-R without compromising or losing information in the input training data during the encoding or feature selection phase. The FOLD-R++ algorithm is competitive in performance with the widely-used XGBoost algorithm, however, unlike XGBoost, the FOLD-R++ algorithm produces an explainable model. FOLD-R++ is also competitive in performance with the RIPPER system, however, on large datasets FOLD-R++ outperforms RIPPER. We also create a powerful tool-set by combining FOLD-R++ with s(CASP)-a goal-directed ASP execution engine-to make predictions on new data samples using the answer set program generated by FOLD-R++. The s(CASP) system also produces a justification for the prediction. Experiments presented in this paper show that our improved FOLD-R++ algorithm is a significant improvement over the original design and that the s(CASP) system can make predictions in an efficient manner as well.