Interpretable machine learning is essential in high-stakes domains like healthcare. Rule lists are a popular choice due to their transparency and accuracy, but learning them effectively remains a challenge. Existing methods require feature pre-discretization, constrain rule complexity or ordering, or struggle to scale. We present NEURULES, a novel end-to-end framework that overcomes these limitations. At its core, NEURULES transforms the inherently combinatorial task of rule list learning into a differentiable optimization problem, enabling gradient-based learning. It simultaneously discovers feature conditions, assembles them into conjunctive rules, and determines their order--without pre-processing or manual constraints. A key contribution here is a gradient shaping technique that steers learning toward sparse rules with strong predictive performance. To produce ordered lists, we introduce a differentiable relaxation that, through simulated annealing, converges to a strict rule list. Extensive experiments show that NEURULES consistently outperforms combinatorial and neural baselines on binary as well as multi-class classification tasks across a wide range of datasets.

Neurules are a type of neuro-symbolic rules integrating neurocomputing and production rules. Each neurule is represented as an adaline unit. Neurules exhibit characteristics such as modularity, naturalness and ability to perform interactive and integrated inferences. One way of producing a neurule base is through conversion of an existing symbolic rule base yielding an equivalent but more compact rule base. The conversion process merges symbolic rules having the same conclusion into one or more neurules. Due to the inability of the adaline unit to handle inseparability, more than one neurule for each conclusion may be produced. In this paper, we define criteria concerning the ability or inability to convert a rule set into a single neurule. Definition of criteria determining whether a set of symbolic rules can (or cannot) be converted into a single, equivalent but more compact rule is of general representational interest. With application of such criteria, the conversion process of symbolic rules into neurules becomes more time- and space-efficient by omitting useless trainings. Experimental results are promising.